# A new space of Generalised vector-valued functions of Bounded Variation

Davide Donati  SISSA, Via Bonomea 265, 34136 Trieste, Italy<EMAIL_ADDRESS>
###### Abstract.

In [15], the authors introduced the space $GBV_{\star}(\Omega)$ to minimise a
class of functionals whose study is motivated by fracture mechanics. In this
paper, we extend the definition of $GBV_{\star}(\Omega)$ to the vectorial
case, introducing the space $GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$. We study
the main properties of $GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ and prove a
lower semicontinuity result useful for minimisation purposes. With the Direct
Method in mind, we adapt the arguments of [15] to show that minimising
sequences in $GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ can be modified to
obtain a minimising sequence converging ${\mathcal{L}}^{d}$-a.e in $\Omega$.

Keywords:  Generalised functions of bounded variation, Minimising sequences,
Fracture Mechanics

2020 MSC:  26A45, 49J45

##  1\. Introduction

In his seminal paper [20], Griffith introduced an influential model for
fracture mechanics, where the stability or growth of the crack path in a
material subject to external forces is determined by the competition of two
energies acting at two different scales: the bulk energy spent to elastically
deform the material and the surface energy spent to widen the pre-existing
crack. In [22], Francfort and Marigo revisited Griffith’s theory and proposed
a model of crack growth based on an energy minimisation principle, renewing
the interest in the study of energy functionals of the type

$\int_{\Omega}f(x,\nabla
u)\,dx+\int_{\mathcal{J}_{u}}g(x,[u],\nu_{u})\,d{\mathcal{H}}^{d-1},$ (1.1)

where the displacement $u\colon\Omega\to{{\mathbb{R}}}^{k}$ can vary among
different spaces of functions with bounded variation, $[u]=u^{+}-u^{-}$
represents the crack opening along the crack surface $\mathcal{J}_{u}$,
$\nu_{u}$ is the unit normal at $\mathcal{J}_{u}$, and $\nabla u$ is the
approximate gradient of $u$.

In [11, 22], the authors mostly consider surface integrands $g$ which are
bounded from below and are independent of the crack opening $[u]$, a choice
which forces choosing $SBV(\Omega,{{\mathbb{R}}}^{k})$ as the ambient space
and choosing, for lower semicontinuity reasons (see [4]), $f$ superlinear.
Many relevant functionals fall in this class. For instance, setting
$f(x,\xi)=f(\xi)=|\xi|^{p}$, $p>1$ and $g\equiv c$, $c>0$, we recover the
widely studied Mumford-Shah functional.

However, the hypothesis that $g$ is independent of the crack opening $[u]$ is
not fully satisfactory in describing the fracture of a considerably large
variety of materials. This observation suggests the introduction of cohesive
models, in which different kinds of surface energies are considered, the most
simple being Dugdale’s energy, where $g(x,\zeta,\nu)=|\zeta|\land 1$. If one
considers functions $g$ satisfying growth conditions of type

$c_{1}(|\zeta|\land 1)\leq g(x,\zeta,\nu)\leq c_{5}(|\zeta|\land 1),$ (1.2)

lower semicontinuity reasons (see [9, Theorem 3.1]) force $f$ to have linear
growth

$c_{1}|\xi|-c_{2}\leq f(x,\xi)\leq c_{3}|\xi|+c_{4}$ (1.3)

and to add another term to functionals of type (1.1). The object of study then
becomes

$\int_{\Omega}f(x,\nabla
u)\,dx+\int_{\Omega}f^{\infty}(x,\frac{D^{c}u}{|D^{c}u|})\,d|D^{c}u|+\int_{\mathcal{J}_{u}}g(x,[u],\nu_{u})\,d{\mathcal{H}}^{d-1},$
(1.4)

where $f^{\infty}$ is the recession function of $f$, $D^{c}u$ is the Cantor
part of the displacement $u$, which one may interpret as a term carrying
information about the formation of microfractures at a diffuse scale, and
$D^{c}u/|D^{c}u|$ is the Radon-Nykodim derivative of $D^{c}u$ with respect to
$|D^{c}u|$.

Functionals of type (1.4), with some particular choice of functions $g$ and
$f$ satisfying (1.2) and (1.3) have been extensively studied (see for instance
[2, 3, 7, 8, 12, 13, 14, 15, 16, 17]).

The choice of the functional space where to study functionals of type (1.1)
under growth conditions of (1.2) and (1.3) is a delicate matter. Since (1.2)
does not control the amplitude of the crack openings $|[u]|$, one is tempted
to consider as ambient space Ambrosio’s $GBV(\Omega)^{k}$ (see [5, Section 1]
and [6, Chapter 4]), where all the quantities appearing in (1.1) are well-
defined, including $D^{c}u$ (see for instance [3, Lemma 2.10]). However, under
the growth conditions (1.2) and (1.3), the sublevels of functionals of type
(1.1) are not precompact in $GBV(\Omega)^{k}$ and compactness is satisfied
only by adding to (1.4) lower order terms.

To address this issue in the antiplane case, Dal Maso and Toader introduced in
[15] the space $GBV_{\star}(\Omega)$. Adapting the arguments of [19], the
authors are able to prove in [15, 16] that for a wide range of minimum
problems, minimising sequences $(u_{n})_{n}\subset GBV_{\star}(\Omega)$ admit
a modification $(y_{n})_{n}\subset GBV_{\star}(\Omega)$ which is still
minimising and converges ${\mathcal{L}}^{d}$-a.e. to $y\in
GBV_{\star}(\Omega)$. They also show that this modification may be performed
while preserving some prescribed boundary conditions.

In this paper, we extend the definition of $GBV_{\star}(\Omega)$ to the
vectorial case and lay down the tools needed to study the problem of
homogenisation for functionals of cohesive type in the vectorial case. This
problem is not treated here and will be the subject of future work.

In particular, we introduce the space $GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$
and we study some of its main properties. We prove in Proposition 3.2 that,
contrary to $GBV(\Omega)^{k}$, $GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ is a
vector space. We prove in Proposition 3.7 that
$GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ coincides with the subspace of
$GBV(\Omega)^{k}$ of vector-valued functions satisfying

$\displaystyle\displaystyle\nabla u\in L^{1}(\Omega,{\mathbb{R}}^{k\times
d}),$ $\displaystyle\displaystyle|D^{c}u|(\Omega)<+\infty,$
$\displaystyle\displaystyle\int_{\mathcal{J}_{u}}|[u]|\land
1\,d{\mathcal{H}}^{d-1}<+\infty.$

We then study the lower semicontinuity of a class of functionals with respect
to the convergence in $L^{0}(\Omega,{{\mathbb{R}}}^{k})$. In Proposition 3.19,
we adapt the non-trivial arguments of [9] to prove the lower semicontinuity in
$L^{0}(\Omega,{{\mathbb{R}}}^{k})$ of certain functionals depending on the
operatorial norm of $\nabla u$ and whose domain is the space
$GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$.

We then prove the following compactness result (Theorem 3.21): if
$(u_{n})_{n}\subset GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ is a minimising
sequence for (1.1) with $g$ and $f$ satisfying (1.2) and(1.3), and if

$\sup_{n\in\mathbb{N}}\int_{\Omega}\psi(|u_{n}|)\,dx<+\infty,$ (1.5)

for some continuous increasing function $\psi$ with $\psi(t)\to+\infty$ for
$t\to+\infty$, then there exists a subsequence of $(u_{n})_{n}$ converging
${\mathcal{L}}^{d}$-a.e. to some vector-valued function $u\in
GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$.

We conclude by showing (Theorem 3.22) that using the argument of [15, Theorem
5.5], every minimising sequence $(u_{n})_{n}\subset
GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ for functionals of the form (1.1),
with $g$ and $f$ satisfying (1.3) and (1.2), admits a modification
$(y_{n})_{n}\subset GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ which is still
minimising and satisfies (1.5).

##  2\. Notation and preliminaries

Throughout the paper $\Omega\subset{{\mathbb{R}}}^{d}$ is a bounded open set.
If $u\colon{{\mathbb{R}}}^{d}\to{{\mathbb{R}}}^{k}$ is a vector-valued
function, we denote by $u^{i}$ the $i$-th component of $u$.

Given a matrix $C\in{\mathbb{R}}^{m\times n}$, we denote by $C^{T}$ its
transpose matrix and by $C_{j}$ the $j$-th row of $C$. Given
$C=(c_{ij})\in{\mathbb{R}}^{m\times n}$, we denote by
$|C|:=(\sum_{i=1}^{m}\sum_{j=1}^{n}c_{ij}^{2})^{1/2}$ the Frobenius norm of
$C$ and by $|C|_{\rm op}:=\sup_{\nu\in\mathbb{S}^{n-1}}|C\cdot\nu|$ the
operatorial norm of $C$.

We denote by $\mathcal{M}_{b}(\Omega,{\mathbb{R}}^{m})$ the space of
${\mathbb{R}}^{m}$ valued bounded Radon measures on $\Omega$. The
$\sigma$-algebra of Borel subsets of $\Omega$ is denoted by
$\mathcal{B}(\Omega)$. The class of open subset of $\Omega$ is denoted by
$\mathcal{A}(\Omega)$. Given a Borel set $B\in\mathcal{B}(\Omega)$ and a
measure $\mu\in\mathcal{M}_{b}(\Omega,{\mathbb{R}}^{m}),$ we denote by
$\mu\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}B$ the Borel measure defined by
$\mu\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}B(E):=\mu(E\cap B)$ for any
$E\in\mathcal{B}(\Omega)$. Given
$\mu\in\mathcal{M}_{b}(\Omega,{\mathbb{R}}^{m\times n})$, we denote by $|\mu|$
the total variation measure of $\mu$ and by $|\mu|_{\rm op}$ the measure
defined for every $B\in\mathcal{B}(\Omega)$ as

$\displaystyle|\mu|_{\rm op}(B):=\sup\Big{\\{}\sum_{i=1}^{n}|\mu(B_{i})|_{\rm
op}:n\in\mathbb{N},$ $\displaystyle B_{i}\subset\subset\Omega\text{ Borel}$
$\displaystyle\text{for }i=1,...,n,\,B_{i}\cap B_{j}=\mathchar 31\relax\text{
for }i\neq j\Big{\\}}.$

If $\mu\in\mathcal{M}_{b}(\Omega,{\mathbb{R}}^{m\times n})$ and
$\lambda\in\mathcal{M}_{b}(\Omega)$ is a positive measure, we denote by
$\mu/\lambda$ the Radon-Nykodim derivative of $\mu$ with respect to $\lambda$.

The Lebesgue measure is denoted by ${\mathcal{L}}^{d}$ and the
$(d-1)$-dimensional Hausdorff measure is denoted by ${\mathcal{H}}^{d-1}$.

For every ${\mathcal{L}}^{d}$-measurable set $E\subset\Omega$,
$\chi_{E}\colon\Omega\to{\mathbb{R}}$ is the characteristic function of the
set $E$, i.e. $\chi_{E}(x)=1$ if $x\in E$, $\chi_{E}(x)=0$ otherwise.

If $E\subset{{\mathbb{R}}}^{d}$ is a ${\mathcal{L}}^{d}$-measurable set with
finite perimeter (see [21] for the general properties of such sets), we denote
by $\partial^{*}E$ its reduced boundary.

We denote by $L^{0}(\Omega,{{\mathbb{R}}}^{k})$ the space of all
${\mathcal{L}}^{d}$-measurable vector-valued functions
$u\colon\Omega\to{{\mathbb{R}}}^{k}$ with the topology induced by the
convergence in measure.

Given a ${\mathcal{L}}^{d}$-measurable set $E\subset\Omega$ and $u\in
L^{0}(E,{{\mathbb{R}}}^{k})$, we say that $a\in{{\mathbb{R}}}^{k}$ is the
approximate limit of $u$ at a point $x\in\Omega$, in symbols

$\underset{\underset{y\in E}{y\to x}}{\text{ap}\lim\,}u(y)=a,$

if for every $\varepsilon>0$ we have

$\lim_{\rho\to 0^{+}}\frac{{\mathcal{L}}^{d}(E\cap
B_{\rho}(x)\cap\\{|u-a|>\varepsilon\\})}{\rho^{d}}=0,$

where $B_{\rho}(x)$ is the ball in ${{\mathbb{R}}}^{d}$ of center $x$ and
radius $\rho$.

Given $u\in L^{0}(\Omega,{{\mathbb{R}}}^{k})$ the jump set $\mathcal{J}_{u}$
is the set of all points $x\in\Omega$ such that there exists a triple
$(u^{+}(x),u^{-}(x),\nu(x))\in{{\mathbb{R}}}^{k}\times{{\mathbb{R}}}^{k}\times\mathbb{S}^{d-1}$,
with $u^{+}(x)\neq u^{-}(x)$ for which, setting

$\Omega^{+}:=\\{y\in\Omega\colon\,(y-x)\cdot\nu(x)>0\\},\,\,\,\Omega^{-}:=\\{y\in\Omega\colon\,(y-x)\cdot\nu(x)<0\\},$

we have

$\underset{\underset{y\in\Omega^{+}}{y\to
x}}{\text{ap}\lim\,}u(y)=u^{+}(x),\,\,\underset{\underset{y\in\Omega^{-}}{y\to
x}}{\text{ap}\lim\,}u(y)=u^{-}(x).$

It is not difficult to see that the triple $(u^{+}(x),u^{-}(x),\nu(x))$ is
uniquely defined up to exchanging the roles of $u^{+}(x)$ and $u^{-}(x)$ and
changing the sign of $\nu(x)$.

We denote by $BV(\Omega,{{\mathbb{R}}}^{k})$ the space of functions of bounded
variation, i.e. the space of all vector-valued functions
$u\colon\Omega\rightarrow{{\mathbb{R}}}^{k}$ such that $u\in
L^{1}(\Omega,{{\mathbb{R}}}^{k})$ and the distributional directional
derivatives $D_{i}u\in\mathcal{M}_{b}(\Omega,{{\mathbb{R}}}^{k})$ for every
$i=1,...,d$. When $k=1$, we simply write $BV(\Omega)$. For an exhaustive
description of such spaces, we refer the reader to [18, Chapter 5] and to [6,
Chapter 3].

We recall that if $u\in BV(\Omega,{{\mathbb{R}}}^{k})$ then for
${\mathcal{H}}^{d-1}$-a.e. $x\in\Omega\\!\setminus\\!\mathcal{J}_{u}$ there
exists

$\tilde{u}(x):=\underset{\underset{y\in\Omega}{y\to x}}{\text{ap}\lim\,}u(y)$

(see [6, Theorem 3.78]). We recall also that if $u\in
BV(\Omega,{{\mathbb{R}}}^{k})$, then $\mathcal{J}_{u}$ is a $(d-1)$-countably
rectifiable set, and it admits a well-defined measure theoretical normal
$\nu_{u}$ (see [6, Theorem 3.77]). Having fixed an orientation for $\nu_{u}$,
we set for $x\in\mathcal{J}_{u}$

$[u](x):=u^{+}(x)-u^{-}(x).$

By the definition of $BV(\Omega,{{\mathbb{R}}}^{k})$, the distributional
jacobian matrix $Du$ can be decomposed as

$Du=D^{a}u+D^{c}u+D^{j}u=\nabla
u{\mathcal{L}}^{d}+D^{c}u+[u]\otimes\nu_{u}{\mathcal{H}}^{d-1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\mathcal{J}_{u},$

where $D^{c}u$ denotes the Cantor part of $Du$ and $\nabla u\in
L^{1}(\Omega,{\mathbb{R}}^{k\times d})$ is the approximate gradient of $u$,
that is, the only matrix-valued function (up to a
${\mathcal{L}}^{d}$-negligible set) such that for ${\mathcal{L}}^{d}$-a.e
$x\in\Omega$ we have

$\underset{\underset{y\in\Omega}{y\to
x}}{\text{ap}\lim\,}\frac{u(y)-\tilde{u}(x)-\nabla u\cdot(y-x)}{|y-x|}=0.$
(2.2)

We briefly recall the properties of slicing of functions in
$BV(\Omega,{{\mathbb{R}}}^{k})$ (see [6, Section 3.12]). For
$\nu\in\mathbb{S}^{d-1}$, we denote by $\Pi_{\nu}:=\\{x\cdot\nu=0\\}$, the
hyperplane through the origin orthogonal to $\nu$ and for every
$y\in\Pi_{\nu}$, $B\in\mathcal{B}(\Omega)$, we set
$B^{\nu}_{y}:=\\{t\in{\mathbb{R}}:\,y+t\nu\in B\\}$. For every $u\in
BV(\Omega,{{\mathbb{R}}}^{k})$, $\nu\in\mathbb{S}^{d-1}$, and $y\in\Pi_{\nu}$,
$u^{\nu}_{y}:\Omega_{\nu}^{y}\to{{\mathbb{R}}}^{k}$ is the function defined
for every $t\in\Omega_{\nu}^{y}$ as $u^{\nu}_{y}(t):=u(y+t\nu)$.

We recall the following one-dimensional slicing theorem (see [6, Theorem
3.108]).

######  Proposition 2.1.

Let $u\in BV(\Omega,{{\mathbb{R}}}^{k})$, $\nu\in\mathbb{S}^{d-1}$. Then

  * (a)

for $\mathcal{L}^{d-1}$-a.e $y\in\Pi_{\nu}$, $u^{\nu}_{y}\in
BV(\Omega_{\nu}^{y},{{\mathbb{R}}}^{k})$;

  * (b)

for every $B\in\mathcal{B}(\Omega)$ we have

$\displaystyle\displaystyle\nabla(u^{\nu}_{y})=(\nabla
u)^{\nu}_{y}\quad\text{for ${\mathcal{H}}^{d-1}$-a.e. }y\in\Pi_{\nu},$
$\displaystyle\displaystyle
D^{j}u\cdot\nu(B)=\int_{\Pi_{\nu}}D^{j}u^{\nu}_{y}(B^{\nu}_{y})\,d{\mathcal{H}}^{d-1}(y),$
$\displaystyle\displaystyle
D^{c}u\cdot\nu(B)=\int_{\Pi_{\nu}}D^{c}u^{\nu}_{y}(B^{\nu}_{y})\,d{\mathcal{H}}^{d-1}(y);$

  * (c)

if $u_{n}\rightharpoonup u$ weakly∗ in $BV(\Omega,{{\mathbb{R}}}^{k})$, then
for ${\mathcal{H}}^{d-1}$-a.e. $y\in\Pi_{\nu}$ we have
$(u_{n})_{y}^{\nu}\rightharpoonup u^{\nu}_{y}$ weakly∗ in
$BV(\Omega_{y}^{\nu},{{\mathbb{R}}}^{k})$.

We now recall the chain rule for the space $BV(\Omega,{{\mathbb{R}}}^{k})$
(see [6, Theorem 3.99]).

######  Theorem 2.2 (Chain Rule in $BV(\Omega,{{\mathbb{R}}}^{k})$).

Let $\varphi\in C^{1}({{\mathbb{R}}}^{k},{{\mathbb{R}}}^{k})$ and $u\in
BV(\Omega,{{\mathbb{R}}}^{k})$. Then, setting $v:=\varphi\circ u$, we have
$v\in BV(\Omega,{{\mathbb{R}}}^{k})$ and

$Dv=\nabla\varphi(u)\nabla
u{\mathcal{L}}^{d}+\nabla\varphi(\tilde{u})D^{c}u+[\varphi(u)]\otimes\nu_{u}{\mathcal{H}}^{d-1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\mathcal{J}_{u}$ (2.3)

We briefly recall the definition and the main properties of the space
$GBV_{\star}(\Omega)$, introduced by Dal Maso and Toader in [15]. For the
proofs of the following results, we refer the reader to [15, Section 2-3]. For
every $t\in{\mathbb{R}}$ and $m>0$, we denote by $t^{(m)}:=(t\land
m)\lor(-m)$, where $a\land b$ and $a\lor b$ denote the minimun between $a$ and
$b$ and the maximum between $a$ and $b$, respectively.

######  Definition 2.3.

Let $u\in L^{0}(\Omega)$. Then

  * (a)

$u\in GBV(\Omega)$ if and only if $u^{(m)}\in BV_{\rm loc}(\Omega)$ for every
$m>0$;

  * (b)

$u\in GBV_{\star}(\Omega)$ if and only if $u^{(m)}\in BV(\Omega)$ for every
$m>0$ and there exists $M>0$ such that

$\sup_{m>0}\int_{\Omega}|\nabla
u^{(m)}|\,dx+|D^{c}u^{(m)}|(\Omega)+\int_{\mathcal{J}_{u^{(m)}}}|[u^{(m)}]|\land
1\,d\mathcal{H}^{d-1}\leq M<+\infty.$ (2.4)

######  Proposition 2.4.

Let $u\in GBV(\Omega)$. Then:

  * (a)

for ${\mathcal{H}}^{d-1}$-a.e $x\in\Omega\\!\setminus\\!\mathcal{J}_{u}$ there
exists finite

$\tilde{u}(x):=\underset{\underset{y\in\Omega}{y\to
x}}{\textup{ap}\lim\,}u(y);$

moreover $u^{+}$ and $u^{-}$ are finite for ${\mathcal{H}}^{d-1}$-a.e.
$x\in\mathcal{J}_{u};$

  * (b)

there exists a Borel function $\nabla u:\Omega\to{{\mathbb{R}}}^{d}$ such that
for ${\mathcal{L}}^{d}$-a.e. $x\in\Omega$ formula (2.2) holds true; moreover,
for all $m>0$ we have

$\nabla u(x)=\nabla u^{(m)}(x)\quad\text{for ${\mathcal{L}}^{d}$-a.e.
}x\in\\{|u|\leq m\\};$

  * (c)

there exists a unique positive measure $|D^{c}u|\in\mathcal{M}_{b}(\Omega)$
such that

$|D^{c}u|(B)=\sup_{m>0}|D^{c}u|(B)\quad\text{for every
}B\in\mathcal{B}(\Omega);$

  * (d)

for every $m>0$ we have $\mathcal{J}_{u^{(m)}}\subset\mathcal{J}_{u}$ up to a
${\mathcal{H}}^{d-1}$-negligible set and $|[u^{(m)}]|\leq|[u]|$
${\mathcal{H}}^{d-1}$-a.e. in $\mathcal{J}_{u^{(m)}}\cap\mathcal{J}_{u}$.
Moreover, ${\mathcal{H}}^{d-1}$-a.e. in $\mathcal{J}_{u}$ there exists
$m_{x}\in\mathbb{N}$ such that $x\in\mathcal{J}_{u^{(m)}}$ for every
$m\in\mathbb{N}$, $m\geq m_{x}$ and $[u^{(m)}](x)\to[u](x)$ as
$m\rightarrow\infty$.

If in addition $u\in GBV_{\star}(\Omega)$, then

  * (e)

there exists a unique Radon measure
$D^{c}u\in\mathcal{M}_{b}(\Omega,{{\mathbb{R}}}^{d})$ such that for every
$m>0$ we have $D^{c}u(B)=D^{c}u^{(m)}(B)$ for every
$B\subset\\{x\in\Omega\setminus\mathcal{J}_{u}\colon\,\tilde{u}(x)\text{
exists and }|\tilde{u}|\leq m\\}$ and $D^{c}u(B)=0$ for every
$B\in\mathcal{B}(\Omega)$ with
${\mathcal{H}}^{d-1}(B\\!\setminus\\!\mathcal{J}_{u})=0;$ moreover, for every
$B\in\mathcal{B}(\Omega)$ we have

$\displaystyle\displaystyle D^{c}(B)=\lim_{m\to\infty}D^{c}u^{(m)}(B),$ (2.5)
$\displaystyle\displaystyle|D^{c}u|(B)=\sup_{m>0}|D^{c}u|(B).$ (2.6)

We conclude this section recalling the vector space properties of
$GBV_{\star}(\Omega)$.

Given $u\in GBV_{\star}(\Omega)$ and $\Gamma\in\mathcal{B}(\Omega)$ such that

$\int_{\mathcal{J}_{u}\setminus\Gamma}|[u]|\,d{\mathcal{H}}^{d-1}<+\infty,$
(2.7)

we denote by
$D^{\Gamma}u\in\mathcal{M}_{b}(\Omega\setminus\Gamma,{{\mathbb{R}}}^{d})$ the
vector measure defined as

$D^{\Gamma}u:=\nabla
u{\mathcal{L}}^{d}+D^{c}u+[u]\nu_{u}{\mathcal{H}}^{d-1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(\mathcal{J}_{u}\setminus\Gamma).$

######  Proposition 2.5 ([15, Theorem 3.9]).

$GBV_{\star}(\Omega)$ is a vector space. Moreover, for every $u,v\in
GBV_{\star}(\Omega)$, $\lambda\in{\mathbb{R}}$, and
$\Gamma\in\mathcal{B}(\Omega)$ satisfying (2.7) for both $u$ and $v$, then

$\displaystyle\displaystyle\nabla(u+v)=\nabla u+\nabla v,\,\,\,\nabla(\lambda
u)=\lambda\nabla u\,\,\,{\mathcal{L}}^{d}\text{-$a.e.$ in }\Omega,$ (2.8)
$\displaystyle\displaystyle D^{c}(u+v)=D^{c}u+D^{c}v,\,\,\,D^{c}(\lambda
u)=\lambda D^{c}u\,\,\,\text{ on }\Omega,$ (2.9) $\displaystyle\displaystyle
D^{\Gamma}(u+v)=D^{\Gamma}u+D^{\Gamma}v,\,\,D^{\Gamma}(\lambda u)=\lambda
D^{\Gamma}u\,\,\,\text{ on }\Omega\setminus\Gamma.$ (2.10)

##  3\. The space $GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$

We begin this section by defining the analogue of $GBV_{\star}(\Omega)$ in the
vectorial case.

######  Definition 3.1.

Let $u\in L^{0}(\Omega,{{\mathbb{R}}}^{k})$. We say $u\in
GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ if and only if $u^{i}\in
GBV_{\star}(\Omega)$ for every $i=1,...,k$.

It follows immediately from Proposition 2.5 that
$GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ is a vector space.

######  Proposition 3.2.

$GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ is a vector space.

If $m>0$ and $u\in L^{0}(\Omega,{{\mathbb{R}}}^{k})$, we set
$u^{(m)}:=((u^{1}\land m)\lor(-m),...,(u^{k}\land m)\lor(-m))$.

######  Definition 3.3.

Let $u\in GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$. We define the measure
$D^{c}u\in\mathcal{M}_{b}(\Omega,{\mathbb{R}}^{k\times d})$ as the matrix-
valued measure whose $i$-th row is defined for every $B\in\mathcal{B}(\Omega)$
as

$(D^{c}u(B))_{i}=D^{c}u^{i}(B).$

######  Lemma 3.4.

Let $u\in GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$. Then $D^{c}u^{(m)}$
converges pointwise to $D^{c}u$ and

$|D^{c}u|(B)=\sup_{m>0}|D^{c}u^{(m)}|(B)\quad\text{for every
}B\in\mathcal{B}(\Omega).$ (3.1)

###### Proof.

The pointwise convergence of $D^{c}u^{(m)}$ to $D^{c}u$ is a consequence of
(2.5) applied to $D^{c}(u^{i})^{(m)}$ for $i=1,...,k$.

As for (3.1), one inequality is trivial. To prove the other inequality, we set
for $m>0$

$A_{m}:=\bigcap_{i=1}^{k}\\{|\tilde{u}^{i}|\leq
m\\},\,\,A:=\bigcup_{m>0}A_{m}.$

Thanks to the elementary inequality

$|D^{c}u|(B)\leq\sum_{i=1}^{k}|(D^{c}u)_{i}|(B)=\sum_{i=1}^{k}|D^{c}u^{i}|(B)\,\,\,\,\text{for
every $B\in\mathcal{B}(\Omega)$}$

and Proposition 2.4(c), we have that $|D^{c}u|(\Omega\setminus A)=0$.

Thus every $B\in\mathcal{B}(\Omega)$ can be rewritten as $B=D\cup N$, where
$D\subset A_{m}$ for $m>0$ large enough and $|D^{c}u|(N)=0$. Thanks to 2.4(e),
we have that we have that
$D^{c}u(B)=D^{c}u(D)=D^{c}u^{(m)}(D)=D^{c}u^{(m)}(B)$.

Let $B\in\mathcal{B}(\Omega)$. For every $\varepsilon>0$ there exists
$n\in\mathbb{N}$ and pairwise disjoint Borel sets $B_{i}\subset\subset\Omega$
such that

$|D^{c}u|(B)\leq\sum_{i=1}^{n}|D^{c}u(B_{i})|+\varepsilon.$

Thanks to the previous remark, this last inequality implies that if $m$ is
large enough

$|D^{c}u|(B)\leq\sum_{i=1}^{n}|D^{c}u^{(m)}(B^{i})|+\varepsilon\leq|D^{c}u^{(m)}|(B)+\varepsilon.$

From the arbitrariness of $\varepsilon$, identity (3.1) follows. ∎

######  Remark 3.5.

In [3, Lemma 2.10], Alicandro and Focardi are able to define $D^{c}u$ for
every $u\in GBV(\Omega)^{k}$ such that $|D^{c}u|(\Omega)<+\infty$ with a
slightly different approach with respect to ours.

Instead of considering the Cantor derivatives of the truncated functions
$D^{c}u^{(m)}$, they consider an increasing diverging sequence $R_{n}>0$ and
smooth functions $\psi_{n}\colon{{\mathbb{R}}}^{k}\to{{\mathbb{R}}}^{k}$
satisfying (3.23a), (3.23b), and (3.23c) below. Set $u_{n}:=\psi_{n}(u)$.

They prove that there exists a measure
$\lambda\in\mathcal{M}_{b}(\Omega,{\mathbb{R}}^{k\times d})$ such that for
every $B\in\mathcal{B}(\Omega)$ one has

$\displaystyle\displaystyle\lambda(B)=\lim_{n\to\infty}D^{c}u_{n}(B),$ (3.2)
$\displaystyle\displaystyle|\lambda|(B)=\sup_{n>0}|D^{c}u_{n}|(B)$ (3.3)

It is easy to check that when $u\in GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$
then $\lambda=D^{c}u$.

If $u\in GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ and
$\Gamma\in\mathcal{B}(\Omega)$ satisfies (2.7), we define
$D^{\Gamma}u\in\mathcal{M}_{b}(\Omega\setminus\Gamma,{\mathbb{R}}^{k\times
d})$ the matrix-valued measure

$D^{\Gamma}u:=\nabla
u{\mathcal{L}}^{d}+D^{c}u+[u]\otimes\nu_{u}{\mathcal{H}}^{d-1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(\mathcal{J}_{u}\setminus\Gamma).$

As a consequence of Proposition 2.5, we see that
$GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ is well-behaved under linear changes
of coordinates of the target.

######  Proposition 3.6.

Let $u\in GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$, $C\in{\mathbb{R}}^{n\times
k}$ and $\Gamma\in\mathcal{B}(\Omega)$ such that (2.7) holds. Then $v:=Cu\in
GBV_{\star}(\Omega,{\mathbb{R}}^{n})$ and

$\displaystyle\displaystyle\nabla v=C\nabla
u\,\,\text{${\mathcal{L}}^{d}$-a.e. in }\Omega,\,\,$ (3.4)
$\displaystyle\displaystyle D^{c}v(B)=CD^{c}v(B)\text{ for every
$B\in\mathcal{B}(\Omega)$}$ (3.5) $\displaystyle
D^{\Gamma}(Cu)(B)=CD^{\Gamma}u(B)\,\,\,\text{for every
$B\in\mathcal{B}(\Omega\setminus\Gamma)$}.$ (3.6)

###### Proof.

The fact that $v\in GBV(\Omega,{\mathbb{R}}^{n})$ is a simple consequence of
Proposition 3.2, while (3.4) (3.5), (3.6), are a consequence of (2.8), (2.9),
and (2.10) . ∎

The following result, which generalises [15, Proposition 3.4], characterises
$GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ as a subspace of $GBV(\Omega)^{k}$.

######  Proposition 3.7.

The space $GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ conincides with the
subspace of $GBV(\Omega)^{k}$ of vector-valued funtions $u$ satisfying

$\displaystyle\displaystyle\nabla u\in L^{1}(\Omega,{\mathbb{R}}^{k\times
d}),$ (3.7) $\displaystyle\displaystyle
D^{c}u\in\mathcal{M}_{b}(\Omega,{\mathbb{R}}^{k\times d}),$ (3.8)
$\displaystyle\displaystyle\int_{\mathcal{J}_{u}\setminus\mathcal{J}_{u}^{1}}|[u]|\,d{\mathcal{H}}^{d-1}<+\infty\text{
and }{\mathcal{H}}^{d-1}(\mathcal{J}_{u}^{1})<+\infty,$ (3.9)

where $\mathcal{J}_{u}^{1}:=\\{x\in\mathcal{J}_{u}:|[u]|\geq 1\\}$ .

###### Proof.

The only if part is a consequence of Proposition 2.4(b), (d), (e) and of the
trivial identity

$\int_{\mathcal{J}_{u}}|[u]|\land
1\,d{\mathcal{H}}^{d-1}=\int_{\mathcal{J}_{u}\setminus\mathcal{J}_{u}^{1}}|[u]|\,d{\mathcal{H}}^{d-1}+{\mathcal{H}}^{d-1}(\mathcal{J}_{u}^{1}).$
(3.10)

To prove the converse, let $u\in GBV(\Omega)^{k}$ be a function satisfying
(3.7), (3.8) and (3.9). Fix $i=1,...,k$. We show that for every $m>0$ we have
$(u^{i})^{(m)}\in BV(\Omega)$. To prove this, simply note that

$\displaystyle\displaystyle\int_{\mathcal{J}_{(u^{i})^{(m)}}}|[(u^{i})^{(m)}]|\,d{\mathcal{H}}^{d-1}$
$\displaystyle=$
$\displaystyle\int_{\mathcal{J}_{(u^{i})^{(m)}}\setminus\mathcal{J}_{(u^{i})^{(m)}}^{1}}\\!\\!\\!\\!\\!\\!\\!\\!\\!|[(u^{i}){(m)}]|\,d{\mathcal{H}}^{d-1}+\int_{\mathcal{J}_{(u^{i})^{(m)}}^{1}}|[(u^{i})^{(m)}]|\,d{\mathcal{H}}^{d-1}$
$\displaystyle\leq$
$\displaystyle\int_{\mathcal{J}_{u}\setminus\mathcal{J}_{u}^{1}}|[u]|+(2m+1){\mathcal{H}}^{d-1}(\mathcal{J}_{u}^{1})<+\infty.$

Using again the identity (3.10), together with (3.9), we conclude that

$\sup_{m>0}\int_{\mathcal{J}_{(u^{i})^{(m)}}\setminus\mathcal{J}_{(u^{i})^{(m)}}^{1}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!|[(u^{i})^{(m)}]|\land
1\,d{\mathcal{H}}^{d-1}\leq\int_{\mathcal{J}_{u}\setminus\mathcal{J}_{u}^{1}}|[u]|\,d{\mathcal{H}}^{d-1}+{\mathcal{H}}^{d-1}(\mathcal{J}_{u}^{1})<+\infty.$

This last inequality, together with (3.7) and (3.8), finally implies $u^{i}\in
GBV_{\star}(\Omega)$, so that $u\in GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$. ∎

In analogy with the definition of the space $GBV(\Omega,{{\mathbb{R}}}^{k})$
(see [5, Section 1], [23, Section 3]), we prove a result which shows the
relation between functions in $GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ and
composition on the left with smooth functions with compact support.

######  Proposition 3.8.

The two following conditions are equivalent:

  1. (1)

$u\in GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$;

  2. (2)

for every $\varphi\in C^{1}_{c}({{\mathbb{R}}}^{k})$, setting $v:=\varphi\circ
u$, we have $v\in BV(\Omega)$ and

$\sup_{\varphi\in C^{1}_{c}({{\mathbb{R}}}^{k})}\int_{\Omega}|\nabla
v|\,dx+|D^{c}v|(\Omega)+\int_{\mathcal{J}_{v}}|[v]|\land
1\,d\mathcal{H}^{d-1}\leq M\textup{Lip}(\varphi),$ (3.11)

for some $M>0$ independent of $\varphi$.

###### Proof.

We begin by proving that $(1)$ implies $(2)$. Let $u\in
GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$, $\varphi\in
C^{1}_{c}({{\mathbb{R}}}^{k})$ and set $v=\varphi\circ u$,
$K:=\text{supp}(\varphi)$. If $m>\max_{y\in K}\lvert y\rvert$, then
$v=\varphi(u^{(m)})$ ${\mathcal{L}}^{d}$-a.e. in $\Omega$. Since $u\in
GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$, $u^{(m)}\in
BV(\Omega,{{\mathbb{R}}}^{k})$. By the chain rule (2.3) we then infer that
$v\in BV(\Omega)$. Using the chain rule once again, as well as Proposition
2.4(b), 2.4(d), 2.4(e), we see that

$\displaystyle\int_{\Omega}|\nabla
v|\,dx+|D^{c}v|(\Omega)+\int_{\mathcal{J}_{v}}|[v]|\land
1\,d\mathcal{H}^{d-1}$ $\displaystyle=$
$\displaystyle\int_{\Omega}|\nabla\varphi(u^{(m)})\cdot\nabla
u^{(m)}|\,dx+\int_{\Omega}|\nabla\varphi(\widetilde{u^{(m)}})|\,d|D^{c}u^{(m)}|+\int_{\mathcal{J}_{\varphi(u^{(m)})}}|[\varphi(u^{{(m)}})]|\,d\mathcal{H}^{d-1}$
$\displaystyle\leq$
$\displaystyle\textup{Lip}(\varphi)\Bigg{(}\int_{\Omega}|\nabla
u^{(m)}|\,dx+|D^{c}u^{(m)}|(\Omega)+\int_{\mathcal{J}_{u^{(m)}}}|[u^{(m)}]|\land
1\,d\mathcal{H}^{d-1}\Bigg{)}$ $\displaystyle\leq$ $\displaystyle
k\textup{Lip}(\varphi)\Bigg{(}\max_{i\in\\{1,..,k\\}}\int_{\Omega}|\nabla(u^{i})^{{(m)}}|\,dx++\max_{i\in\\{1,..,k\\}}|D^{c}((u^{i})^{{(m)}})|(\Omega)$
$\displaystyle+\max_{i\in\\{1,..,k\\}}\int_{\mathcal{J}_{(u^{i}){{(m)}}}}|[(u^{i})^{{(m)}}]|\land
1\,d\mathcal{H}^{d-1}\Bigg{)}.$

Since for every $i=1,...,k$, the function $u^{i}$ satisfies (2.4) the last
estimate implies (3.11).

We now prove that $(2)$ implies $(1)$. It is enough to show that $u_{1}\in
GBV_{\star}(\Omega)$. To this aim, we fix $(R_{n})_{n}\subset(0,+\infty)$
increasing and diverging. For every $n\in\mathbb{N}$ we consider a smooth
function $\varphi_{n}:{{\mathbb{R}}}^{k}\rightarrow{\mathbb{R}}$ such that

$\begin{cases}\varphi_{n}(y)=y_{1}\quad&\text{for }y\in\\{|y|\leq R_{n}\\},\\\
\varphi_{n}(y)=0\quad&\text{for }y\in\\{|y|\geq R_{n+1}\\},\\\
\text{Lip}(\varphi_{n})\leq 1.\end{cases}$

For $m>0$ and $n\in\mathbb{N}$, we set $v^{m}_{n}:=\varphi_{n}(u)^{(m)}$. For
$m>0$ fixed, it follows immediately that

$\lim_{n\to+\infty}v^{m}_{n}(x)=(u^{1})^{(m)}(x)\quad\quad\text{for
}\mathcal{L}^{d}\text{-a.e. }x\in{{\mathbb{R}}}^{d}.$ (3.12)

Since $\|v^{m}_{n}\|_{\infty}\leq m$, it follows from (3.12) that

$v^{m}_{n}\text{ converges to }(u^{1})^{(m)}\text{ in }L^{1}(\Omega).$ (3.13)

Recalling that $C^{1}_{c}({{\mathbb{R}}}^{k})$ and
$\text{Lip}(\varphi_{n})\leq 1$, it follows from (3.11) that for every $m>0$
we have

$\int_{\Omega}|\nabla
v^{m}_{n}|\,dx+|D^{c}v^{m}_{n}|(\Omega)+\int_{\mathcal{J}_{v^{m}_{n}}}\\!\\!\\!\\!|[v^{m}_{n}]|\land
1\,d\mathcal{H}^{d-1}\leq M,$ (3.14)

so that $\varphi_{n}(u)\in GBV_{\star}(\Omega)$ for every $n\in\mathbb{N}$.

We claim that there exists a constant $C=C(m,M)>0$ such that

$\sup_{h}|Dv^{m}_{n}|(\Omega)<C.$ (3.15)

To prove this, it is sufficient to prove that

$\int_{\mathcal{J}_{v^{m}_{n}}}|[v^{m}_{n}]|\,d\mathcal{H}^{d-1}<C.$

By Proposition 2.4(d) we have that
$\mathcal{J}_{v^{m}_{n}}\subset\mathcal{J}_{\varphi_{n}(u)}$ up to a
$\mathcal{H}^{d-1}$-negligible set and that
$|[v^{m}_{n}]|\leq|[\varphi_{n}(u)]|$ $\mathcal{H}^{d-1}$ a.e. in
$\mathcal{J}_{v^{m}_{n}}$, so that

$\displaystyle\int_{\mathcal{J}_{v^{m}_{n}}}\\!\\!\\!\\!|[v^{m}_{n}]|\,d\mathcal{H}^{d-1}$
$\displaystyle\leq$
$\displaystyle\int_{\mathcal{J}_{v^{m}_{n}}\setminus\mathcal{J}^{1}_{v^{m}_{n}}}|[v^{m}_{n}]|\,d\mathcal{H}^{d-1}+2m\mathcal{H}^{d-1}(\mathcal{J}^{1}_{\varphi_{n}(u)})$
$\displaystyle\leq$
$\displaystyle\int_{\mathcal{J}_{\varphi_{n}(u)}\setminus\mathcal{J}^{1}_{\varphi_{n}(u)}}|[\varphi_{n}(u)]|\,d\mathcal{H}^{d-1}+(2m+1)\mathcal{H}^{d-1}(\mathcal{J}^{1}_{\varphi_{n}(u)})<+\infty,$

where the last inequality follows from the fact that $\varphi_{n}(u)$
satisfies (3.9) being in $GBV_{\star}(\Omega)$. This proves (3.15) and that
$v^{m}_{n}\in BV(\Omega)$ for any $n\in\mathbb{N}$.

Thanks to (3.13) and to (3.15), the lower semicontinuity of the Total
Variation with respect to the $L^{1}(\Omega)$-convergence implies that
$u^{(m)}\in BV(\Omega)$ for every $m>0$.

Finally, using [9, Theorem 2.1], together with (3.11), (3.13) and the fact
that Lip$(\varphi_{n})\leq 1$, we are able to conclude that

$\displaystyle\displaystyle\int_{\Omega}|\nabla(u^{1})^{(m)}|\,dx+\int_{\mathcal{J}_{(u^{1})^{(m)}}}|[(u^{1})^{(m)}]|\land
1\,d\mathcal{H}^{d-1}+|D^{c}(u^{1})^{(m)}|(\Omega)$ $\displaystyle\leq$
$\displaystyle\displaystyle\liminf_{n\to\infty}\Big{(}\int_{\Omega}|\nabla
v_{n}^{m}|\,dx+\int_{\mathcal{J}_{v_{n}^{m}}}\\!\\!\\!\\!\\!\\!\\!\\!|[v_{n}^{m}]|\land
1\,d{\mathcal{H}}^{d-1}+|D^{c}v^{m}_{n}|(\Omega)\Big{)}\leq M,$

which implies $u^{1}\in GBV_{\star}(\Omega)$. ∎

######  Remark 3.9.

Note that while for $k>1$ it is easily shown that

$\varphi\circ u\in BV_{\text{loc}}(\Omega,{{\mathbb{R}}}^{k})\text{ for every
$\varphi\in C^{1}_{c}({{\mathbb{R}}}^{k})$ }\implies u\in
GBV(\Omega,{{\mathbb{R}}}^{k}),$ (3.16)

an example due to Pallara ([23, Remark 4.9]) shows that the same implication
does not hold for $k=1$.

Thanks to estimate (3.11), Proposition 3.8 holds also for $k=1$.

######  Remark 3.10.

If $n\in\mathbb{N}$, $n\geq 2$, following the lines of the proof of
Proposition 3.8, it can be proved that $u\in
GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ if and only if (2) of Proposition 3.8
holds for every $\varphi\in C^{1}_{c}({{\mathbb{R}}}^{k},{\mathbb{R}}^{n})$.

It is straightforward to prove that an analogue of Alberti’s rank-one Theorem
(see [1, Corollary 4.6] or [6, Theorem 3.94]) holds in
$GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$.

######  Proposition 3.11.

Let $u\in GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$. Then $D^{c}u/|D^{c}u|$ has
rank one $|D^{c}u|$-a.e. in $\Omega$.

###### Proof.

Being $u\in GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$, it follows that
$u^{(m)}\in BV(\Omega,R^{k})$ for every $m>0$. Thanks to Proposition 2.4(e)
and Besicovitch derivation Theorem (see [6, Theorem 2.22]), we have that

$\frac{D^{c}u^{(m)}}{|D^{c}u|}=\frac{D^{c}u^{(m)}}{|D^{c}u^{(m)}|}\quad|D^{c}u^{(m)}|\text{-}a.e.\text{
in }\Omega.$

By Alberti’s rank-one Theorem we infer that $D^{c}u^{(m)}/|D^{c}u|$ has rank
one $|D^{c}u^{(m)}|$-a.e. in $\Omega$. Since for every $m>0$ and every
$B\subset\\{|\tilde{u}|\leq m\\}$ we have $D^{c}u^{(m)}(B)=D^{c}u(B)$, we have

$\frac{D^{c}u^{(m)}}{|D^{c}u|}=\frac{D^{c}u}{|D^{c}u|}\quad\quad|D^{c}u^{(m)}|\text{-a.e.
in }\Omega.$

Recalling (3.1) and that

$|D^{c}u|(\Omega\setminus\bigcup_{m>0}\\{|\tilde{u}|\leq m\\})=0,$

it follows that $D^{c}u^{(m)}/|D^{c}u|$ converges $|D^{c}u|$-a.e. to
$D^{c}u/|D^{c}u|$ for $m\to\infty$. The claim then follows from the lower
semicontinuity of the rank. ∎

Following the notation of [16], we introduce a relevant family of functionals
on $L^{0}(\Omega,{{\mathbb{R}}}^{k})$, whose domain is the space
$GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$.

In what follows, $\sigma\colon[0,+\infty)\to[0,+\infty)$ is a bounded
continuous function and $c_{1},....,c_{6}>0$ are positive constants such that

$0<c_{1}\leq 1\leq c_{3}\leq c_{5},\,\,\,c_{6}\geq 1,$

and

$\sigma(0)=0\text{ and }\sigma(t)\geq c_{3}(t\land 1).$

######  Definition 3.12.

We denote by $\mathcal{F}$ the space of all functions
$f:{{\mathbb{R}}}^{d}\times{\mathbb{R}}^{k\times d}\to[0,+\infty)$ satisfying
the following

  * (f1)

$f$ is a Borel measurable function;

  * (f2)

$c_{1}|\xi|_{\rm op}-c_{2}\leq f(x,\xi)$ for every $x\in{{\mathbb{R}}}^{d}$
and every $\xi\in{\mathbb{R}}^{k\times d}$;

  * (f3)

$f(x,\xi)\leq c_{3}|\xi|_{\rm op}+c_{4}$ for every $x\in{{\mathbb{R}}}^{d}$
and every $\xi\in{\mathbb{R}}^{k\times d}$;

  * (f4)

$|f(x,\xi_{1})-f(x,\xi_{2})|\leq c_{5}|\xi_{1}-\xi_{2}|_{\rm op}$ for every
$x\in{{\mathbb{R}}}^{d}$ and every $\xi_{1},\xi_{2}\in{\mathbb{R}}^{k\times
d}$.

######  Definition 3.13.

Let $\mathcal{G}$ be the set of functions
$g:{{\mathbb{R}}}^{k}\times{{\mathbb{R}}}^{k}\times\mathbb{S}^{d-1}\to[0,+\infty)$
satisfying the following conditions:

  * (g1)

$g$ is a Borel measurable function;

  * (g2)

$c_{1}(|\zeta|\land 1)\leq g(x,\zeta,\nu)$ for every $x\in{{\mathbb{R}}}^{d}$,
$\zeta\in{{\mathbb{R}}}^{k}$, $\nu\in\mathbb{S}^{d-1}$;

  * (g3)

$g(x,\zeta,\nu)\leq c_{5}(|\zeta|\land 1)$ for every $x\in{{\mathbb{R}}}^{d}$,
$\zeta\in{{\mathbb{R}}}^{k}$, $\nu\in\mathbb{S}^{d-1}$;

  * (g4)

$|g(x,\zeta_{1},\nu)-g(x,\zeta_{2},\nu)|\leq\sigma(|\zeta_{1}-\zeta_{2}|)$ for
every $x\in{{\mathbb{R}}}^{d}$, $\zeta_{1},\zeta_{2}\in{{\mathbb{R}}}^{k}$,
$\nu\in\mathbb{S}^{d-1}$;

  * (g5)

$g(x,-\zeta,-\nu)=g(x,\zeta,\nu)$ for every $x\in{{\mathbb{R}}}^{d}$,
$\zeta\in{{\mathbb{R}}}^{k}$, $\nu\in\mathbb{S}^{d-1}$;

  * (g6)

for every $\zeta_{1},\zeta_{2}\in{{\mathbb{R}}}^{k}$ with
$c_{6}|\zeta_{1}|\leq|\zeta_{2}|$ it holds $g(x,\zeta_{1},\nu)\leq
g(x,\zeta_{2},\nu)$ for every $x\in{{\mathbb{R}}}^{d}$,
$\nu\in\mathbb{S}^{d-1}$.

######  Definition 3.14.

For every $f\in{{\mathbb{R}}}^{d}\times{\mathbb{R}}^{k\times
d}\rightarrow[0,+\infty)$, the recession function
$f^{\infty}:{{\mathbb{R}}}^{d}\times{\mathbb{R}}^{k\times
d}\rightarrow[0,+\infty]$ is the function defined as

$f^{\infty}(x,\xi):=\limsup_{t\to+\infty}\frac{f(x,t\xi)}{t}$ (3.17)

for every $x\in\Omega$ and for every $\xi\in{\mathbb{R}}^{k\times d}$.

######  Remark 3.15.

The function $\xi\to f(x,\xi)$ is positively $1$-homogeneous. If the function
$\xi\to f(x,\xi)$ is convex for every $x\in\Omega$, the $\limsup$ in (3.17) is
actually a limit. If $f$ satisfies (f2) and (f3), then $f^{\infty}$ satisfies

$c_{1}|\xi|_{\rm op}\leq f^{\infty}(x,\xi)\leq c_{3}|\xi|_{\rm
op}\quad\text{for every $x\in{{\mathbb{R}}}^{d}$, $\xi\in{\mathbb{R}}^{k\times
d}$}.$

If $f$ satisfies (f4), then $f^{\infty}$ satisfies

$|f^{\infty}(x,\xi_{1})-f^{\infty}(x,\xi_{2})|\leq c_{5}|\xi_{1}-\xi_{2}|_{\rm
op}\quad\text{for every $x\in{{\mathbb{R}}}^{d}$,
$\xi_{1},\xi_{2}\in{\mathbb{R}}^{k\times d}$}.$

######  Definition 3.16.

Given $f\in\mathcal{F}$ and $g\in\mathcal{G}$ we set
$E^{f,g}:L^{0}(\Omega,{{\mathbb{R}}}^{k})\times\mathcal{A}(\Omega)\to[0,+\infty]$
the functional defined for $A\in\mathcal{A}(\Omega)$ as

$E^{f,g}(u,A):=\int_{A}f(x,\nabla
u)\,dx+\int_{A}f^{\infty}(x,\frac{D^{c}u}{|D^{c}u|})\,d|D^{c}u|+\int_{\mathcal{J}_{u}\cap
A}g(x,[u],\nu_{u})\,{\mathcal{H}}^{d-1}$ (3.18)

if $u|_{A}\in GBV_{\star}(A,{{\mathbb{R}}}^{k})$ and as $E^{f,g}(u,A)=+\infty$
otherwise.

######  Remark 3.17.

Let $A\in\mathcal{A}(\Omega)$, $f\in\mathcal{F}$, and $g\in\mathcal{G}$. It is
immediate to check that

$E^{f,g}(u,A)<+\infty\Longleftrightarrow u|_{A}\in
GBV_{\star}(A,{{\mathbb{R}}}^{k}).$

Let $f(x,\xi)=|\xi|_{\rm op}$ and $g(x,\zeta,\nu)=|\zeta|\land 1$. We denote
by $V$ the functional $E^{f,g}$ obtained with these choices of $f$ and $g$.
Note that in this case $f^{\infty}(x,\xi)=|\xi|_{\rm op}$, so that recalling
(2), we may rewrite

$V(u,A)=\int_{A}|\nabla u|_{\rm op}\,dx+|D^{c}u|_{\rm
op}(\Omega)+\int_{\mathcal{J}_{u}\cap A}|[u]|\land 1\,d{\mathcal{H}}^{d-1}.$

In [9, Theorem 2.1], the authors prove through a slicing argument a lower
semicontinuity result for functionals on the space $BV(\Omega)$. By carefully
revisiting their proof, we are able to prove the lower semicontinuity in
$L^{0}(\Omega,{{\mathbb{R}}}^{k})$ of a subclass of functionals of type
$E^{f,g}$ which includes the functional $V$.

######  Lemma 3.18.

Let $\hat{f}\colon[0,+\infty)\rightarrow[0,+\infty)$ be a convex, lower
semicontinuous, non-decreasing function such that

$\hat{f}^{\infty}(1)=\lim_{t\to+\infty}\frac{\hat{f}(t)}{t}=C.$

Let $\hat{g}\colon[0,+\infty)\rightarrow[0,+\infty)$ be a lower semicontinuous
function satisfying the subadditivity condition

$\hat{g}(|t_{1}+t_{2}|)\leq\hat{g}(|t_{1}|)+\hat{g}(|t_{2}|)\quad\text{for
every $t_{1},t_{2}\in{{\mathbb{R}}}^{k}$},$

and suppose in addition that

$\hat{g}^{0}(1)=\lim_{t\to 0}\frac{\hat{g}(t)}{t}=C.$

Set $f:=\hat{f}(|\cdot|_{\rm op})$ and $g:=\hat{g}(|\cdot|)$. Then the
functional defined for $u\in BV(\Omega,{{\mathbb{R}}}^{k})$ as

$E^{f,g}(u):=\int_{\Omega}\hat{f}(|\nabla u|_{\rm op})\,dx+C|D^{c}u|_{\rm
op}(\Omega)+\int_{\mathcal{J}_{u}}\hat{g}(|[u]|)\,d{\mathcal{H}}^{d-1}$ (3.19)

is weakly∗ lower semicontinuous on $BV(\Omega,{{\mathbb{R}}}^{k})$.

###### Proof.

For every $\nu\in\mathbb{S}^{d-1}$, we introduce the functional
$E^{f,g}_{\nu}\colon
BV(\Omega,{{\mathbb{R}}}^{k})\times\mathcal{B}(\Omega)\to[0,+\infty)$ defined
for every $B\in\mathcal{B}(\Omega)$ and $u\in BV(\Omega,{{\mathbb{R}}}^{k})$
as

$E^{f,g}_{\nu}(u,B)=\int_{\Omega}\hat{f}(|\nabla
u\cdot\nu|)\,dx+C|D^{c}u\cdot\nu|(\Omega)+\int_{\mathcal{J}_{u}}\hat{g}(|[u]|)|\nu_{u}\cdot\nu|{\mathcal{H}}^{d-1}.$

We want to prove that for every $B\in\mathcal{B}(\Omega)$, the functional
$E^{f,g}_{\nu}(\cdot,B)$ is lower semicontinuous with respect to the weak∗
convergence in $BV(\Omega,{{\mathbb{R}}}^{k})$.

Since for every $u\in BV(\Omega,{{\mathbb{R}}}^{k})$ the set function
$E^{f,g}_{\nu}(u,\cdot)$ is a non-negative Borel measure, it is enough to
check that $E^{f,g}_{\nu}(\cdot,A)$ is lower semicontinuous for every
$A\in\mathcal{A}(\Omega)$. For every $u\in BV(\Omega,{{\mathbb{R}}}^{k})$ the
measure $E^{f,g}_{\nu}(u,\cdot)$ does not charge sets
$B\in\mathcal{B}(\Omega)$ such that

${\mathcal{L}}^{d}(B)+|D^{c}u|(B)+|D^{j}u|(B)=0.$ (3.20)

Every set $A\in\mathcal{A}(\Omega)$ can be written, up to a
${\mathcal{L}}^{d}$-negligible set, as a countable union of pairwise disjoint
open sets $Q_{i}=\Sigma_{i}\times I_{i}$, with $\Sigma_{i}\subset\Pi_{\nu}$
and $I_{i}\subset{\mathbb{R}}$, such that $\partial Q_{i}$ satisfies (3.20).
Thus, it is enough to show that $E^{f,g}_{\nu}(\cdot,Q)$ is lower
semicontinuous for every cylinder $Q=\Sigma\times I$, with
$\Sigma\subset\Pi_{\nu}$ and $I\subset{\mathbb{R}}$.

Thanks to Proposition 2.1 and arguing exactly as in [9, Lemma 2.3], we deduce
that, for every open $\Sigma\subset\Pi_{\nu}$, $I\subset{\mathbb{R}}$, setting
$Q=\Sigma\times I$, we have

$E^{f,g}_{\nu}(u,Q)=\int_{\Sigma}\Phi(u_{y}^{\nu},I)\,d{\mathcal{H}}^{d-1}(y)\,\,\,\text{
for every $u\in BV(\Omega,{{\mathbb{R}}}^{k})$,}$ (3.21)

where $\Phi\colon
BV({\mathbb{R}},{{\mathbb{R}}}^{k})\times\mathcal{B}({\mathbb{R}})\to[0,+\infty]$
is the functional defined for every $I\in\mathcal{B}({\mathbb{R}})$ as

$\Phi(v,I)=\int_{I}\hat{f}(|\nabla v|_{\rm
op})\,dx+C|D^{c}v|(I)+\int_{\mathcal{J}_{v}\cap
I}\hat{g}(|[v]|)\,d\mathcal{H}^{0}\,\,\,\text{ for every $v\in
BV({\mathbb{R}},{{\mathbb{R}}}^{k}).$}$

We fix now $u\in BV(\Omega,{{\mathbb{R}}}^{k})$ and $(u_{n})_{n}\subset
BV(\Omega,{{\mathbb{R}}}^{k})$ such that $u_{n}\rightharpoonup u$ weakly ∗ in
$BV(\Omega,{{\mathbb{R}}}^{k})$. Thanks to Proposition 2.1(c), we have that
for ${\mathcal{H}}^{d-1}$-a.e. $y\in\Sigma$

$(u_{n})_{y}^{\nu}\rightharpoonup u_{y}^{\nu}\text{ weakly${}^{*}$ in
$BV(I,{{\mathbb{R}}}^{k})$. }$

Thanks to [10, Theorem 3.3], we know that $\Phi$ is weakly∗ lower
semicontinuous on $BV({\mathbb{R}},{{\mathbb{R}}}^{k})$, so that Fatou’s Lemma
and (3.21) imply that

$E^{f,g}_{\nu}(u,Q)\leq\liminf_{n\to\infty}E^{f,g}_{\nu}(u_{n},Q).$

Fix now $(\nu_{i})_{i=1}^{\infty}$ a family of dense directions in the unit
spere $\mathbb{S}^{d-1}$. To prove that $E^{f,g}$ is weakly∗ lower
semicontinuous in $BV(\Omega,{{\mathbb{R}}}^{k})$, we show that it can be
written as the supremum of finite sums of functionals of the form
$E^{f,g}_{\nu_{i}}$, which we have just shown to be lower semicontinuous. More
precisely, repeating the proof of [9, Lemma 2.4] replacing the function $\psi$
by

$\psi(x)=\begin{cases}\hat{f}(|\nabla u(x)|_{\rm
op})&{\mathcal{L}}^{d}\text{-}a.e\,\,\text{on }\Omega,\\\
C&|D^{c}u|\text{-}a.e.\text{ on }\Omega\setminus\mathcal{J}_{u},\\\
\hat{g}(|[u]|){\mathcal{H}}^{d-1}\text{-}.a.e.&\text{ on
}\mathcal{J}_{u},\end{cases}\\\ $

and the function $\psi_{h}$ by

$\psi_{i}(x)=\begin{cases}\hat{f}(|\nabla
u(x)\cdot\nu_{i}|)&{\mathcal{L}}^{d}\text{-}a.e\,\,\text{on }\Omega,\\\
C|\nu_{u}\cdot\nu_{i}|&|D^{c}u|\text{-}a.e.\text{ on
}\Omega\setminus\mathcal{J}_{u},\\\
\hat{g}(|[u]|)|\nu_{u}\cdot\nu_{i}|&{\mathcal{H}}^{d-1}\text{-}.a.e.\text{ on
}\mathcal{J}_{u},\end{cases}$

we get

$E^{f,g}(u)=\sup\Big{\\{}\sum_{i=1}^{n}E^{f,g}_{\nu_{i}}(u,A_{i}):n\in\mathbb{N},\,(A_{i})_{i=1}^{n}\,\,\text{
pairwise disjoint open subsets of $\Omega$}\Big{\\}}.$

∎

######  Proposition 3.19.

Let $\hat{f},\hat{g}$ be as in the statement of Lemma 3.18. Suppose in
addition that $f:=\hat{f}(|\cdot|_{\rm op})$ satisfies (f2),(f3) of Definition
(3.12), $g:=\hat{g}(|\cdot|)$ satisfies (g2),(g3) of Definition (3.13), and
that $\hat{g}$ is non-decreasing. Then for every $A\in\mathcal{A}(\Omega)$,
the functional $E^{f,g}(\cdot,A)$ defined as

$E^{f,g}(u,A)=\begin{cases}\displaystyle\int_{A}\hat{f}(|\nabla
u|_{op})\,dx+C|D^{c}u|_{\rm op}(A)+\int_{\mathcal{J}_{u}\cap
A}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\hat{g}(|[u]|)\,d{\mathcal{H}}^{d-1}\,\,&\text{
if }u|_{A}\in GBV_{\star}(A,{{\mathbb{R}}}^{k}),\\\\[10.0pt] +\infty&\text{
otherwise}\end{cases}$ (3.22)

is lower semicontinuous in $L^{0}(A,{{\mathbb{R}}}^{k})$.

###### Proof.

Without loss of generality, we may assume $A=\Omega$. Let $u\in
L^{0}(\Omega,{{\mathbb{R}}}^{k})$ and $(u_{n})_{n}\subset
L^{0}(\Omega,{{\mathbb{R}}}^{k})$ be a sequence converging to $u$ in
$L^{0}(\Omega,{{\mathbb{R}}}^{k})$. If
$\liminf_{n}E^{f,g}(u_{n},\Omega)=+\infty$ there is nothing to prove. Suppose
now that $E^{f,g}(u_{n},\Omega)\leq M$. Since $f$ and $g$ satisfy (f2) and
(g2), we have $(u_{n})_{n}\subset GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ and
that $V(u_{n},\Omega)\leq C$, for $C$ a constant depending only on $M$.

We fix $(R_{h})_{h}\subset(0,+\infty)$ a strictly increasing and diverging
sequence, and for each $h\in\mathbb{N}$ we consider a smooth function
$\psi_{h}\colon{{\mathbb{R}}}^{k}\to{{\mathbb{R}}}^{k}$ satisfying

$\displaystyle\psi_{h}(y)=y$ $\displaystyle\text{for }y\in\\{|y|\leq
R_{h}\\},$ (3.23a) $\displaystyle\psi_{h}(y)=0$ $\displaystyle\text{for
}y\in\\{|y|\geq R_{h+1}\\},$ (3.23b) $\displaystyle\text{Lip}(\psi_{h})\leq
1.$ (3.23c)

Fix now $h\in\mathbb{N}$ and consider the sequence $v_{n,h}:=\psi_{h}(u_{n})$.
By Remark 3.10 and the fact that $\psi_{h}$ is smooth with compact support, we
have that $(v_{n,h})_{n}\subset BV(\Omega,{{\mathbb{R}}}^{k})\cap
L^{\infty}(\Omega,{{\mathbb{R}}}^{k})$. We claim that

$|Dv_{n,h}|(\Omega)\leq K,$ (3.24)

where $K=K(h,M)>0$ is a constant depending only on $h$ and on $M$.

To prove this, we use the chain rule (2.3) to get

$\displaystyle|Dv_{n,h}|(\Omega)$ $\displaystyle=$
$\displaystyle\int_{\Omega}|\nabla\psi_{h}(u_{n})\nabla
u_{n}|\,dx+\int_{\Omega}\Big{|}\nabla\psi_{h}(\tilde{u}_{n})\frac{D^{c}u}{|D^{c}u|}\Big{|}\,d|D^{c}u|+\int_{\mathcal{J}_{v_{n,h}}}|[v_{n,h}]|\,d{\mathcal{H}}^{d-1}.$
$\displaystyle\leq$ $\displaystyle\sqrt{k}\Big{(}\int_{\Omega}|\nabla
u_{n}|\,dx+|D^{c}u|(\Omega)\Big{)}+\int_{\mathcal{J}_{v_{n,h}}}|[v_{n,h}]|\,d{\mathcal{H}}^{d-1}.$

By assumption $u_{n}\in GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ and
$V(u_{n},\Omega)\leq C$, so that to prove (3.24) it is sufficient to prove
that

$\int_{\mathcal{J}_{v_{n,h}}\setminus\mathcal{J}_{v_{n,h}}^{1}}|[v_{n,h}]|\,d{\mathcal{H}}^{d-1}+\int_{\mathcal{J}_{v_{n,h}}^{1}}|[v_{n,h}]|\,{\mathcal{H}}^{d-1}=\int_{\mathcal{J}_{v_{n,h}}}|[v_{n,h}]|\,d{\mathcal{H}}^{d-1}\leq
K.$

To prove this last inequality, we first note

$\displaystyle\int_{\mathcal{J}_{v_{n,h}}\setminus\mathcal{J}_{v_{n,h}}^{1}}|[v_{n,h}]|\leq\int_{\mathcal{J}_{u_{n}}\setminus\mathcal{J}_{u_{n}}^{1}}|[u_{n}]|\,d{\mathcal{H}}^{d-1}+{\mathcal{H}}^{d-1}(\mathcal{J}_{u_{n}}^{1}\setminus\mathcal{J}_{v_{n,h}}^{1})\leq
C,$

where the last inequality follows from the fact that $V(u_{n},\Omega)\leq C$
and (3.9).

Finally, we note that thanks to (3.9), the fact that $V(u_{n},\Omega)\leq C$
and the fact that $\psi_{h}$ is smooth with compact support, we have

$\int_{\mathcal{J}_{v_{n,h}}^{1}}|[v_{n,h}]|\,d{\mathcal{H}}^{d-1}\leq
2\sqrt{k}\max_{y\in{{\mathbb{R}}}^{k}}|\psi_{h}(y)|{\mathcal{H}}^{d-1}(\mathcal{J}_{u_{n}}^{1})\leq
2\sqrt{k}\max_{y\in{{\mathbb{R}}}^{k}}|\psi_{h}(y)|C,$

so that (3.24) is proved.

The sequence $v_{n,h}$ converges to $\psi_{h}(u)$ in
$L^{1}(\Omega,{{\mathbb{R}}}^{k})$, so that from (3) it follows that $v_{n,h}$
converges to $\psi_{h}(u)$ weakly∗ $BV(\Omega,{{\mathbb{R}}}^{k})$ as well.
Therefore we can use Lemma 3.18 to obtain

$E^{f,g}(\psi_{h}(u),\Omega)\leq\liminf_{n\to+\infty}E^{f,g}(v_{n,h},\Omega).$
(3.25)

Using the chain rule (2.3), we see that

$\displaystyle\displaystyle E^{f,g}(v_{n,h},\Omega)$ $\displaystyle=$
$\displaystyle\int_{\Omega}\hat{f}(|\nabla\psi_{h}(u_{n})\nabla u_{n}|_{\rm
op})\,dx+C\int_{\Omega}\Big{|}\nabla\psi_{h}(\tilde{u}_{n})\frac{D^{c}u_{n}}{|D^{c}u_{n}|}\Big{|}_{\rm
op}\,d|D^{c}u_{n}|$ $\displaystyle+$
$\displaystyle\int_{\mathcal{J}_{u}}\hat{g}(|[\psi_{h}(u_{n})]|)\,d{\mathcal{H}}^{d-1}$
$\displaystyle\leq$ $\displaystyle\int_{\Omega}\hat{f}(|\nabla u_{n}|_{\rm
op})\,dx+C|D^{c}u_{n}|_{\rm
op}(\Omega)\displaystyle+\int_{\mathcal{J}_{u}}\hat{g}(|[u_{n}]|)\,d{\mathcal{H}}^{d-1},$

where in the last inequality we have used the trivial inequality $|AB|_{\rm
op}\leq|A|_{\rm op}|B|_{\rm op}$, the fact that $\hat{f}$ and $\hat{g}$ are
increasing and (3.23c). From this last estimate and (3.25), we get

$E^{f,g}(\psi_{h}(u),\Omega)\leq\liminf_{n\to+\infty}E^{f,g}(u_{n},\Omega).$
(3.26)

Recalling Proposition 2.4(b), (d) and taking advantage of the lower
semicontinuity of $\hat{f}$ and of $\hat{g}$, we get

$\displaystyle\displaystyle\hat{f}(|\nabla u|_{\rm
op})\leq\liminf_{h\to\infty}\hat{f}(|\nabla\psi_{h}(u)|_{\rm op})\quad$
$\displaystyle\text{for }{\mathcal{L}}^{d}\text{-}a.e\,\,x\in\Omega,$ (3.27)
$\displaystyle\displaystyle\hat{g}(|[u]|)\leq\liminf_{h\to\infty}\hat{g}(|[\psi_{h}(u)]|)\quad$
$\displaystyle\text{for
}{\mathcal{H}}^{d-1}\text{-}a.e\,\,x\in\mathcal{J}_{u}.$ (3.28)

We have seen in the proof of Proposition 3.11 that
$D^{c}u^{(m)}/|D^{c}u^{(m)}|$ converges $|D^{c}u|$-a.e. in $\Omega$ to
$D^{c}u/|D^{c}u$. A similar argument shows that
$D^{c}\psi_{h}(u)/|D^{c}\psi_{h}(u)|$ converges $|D^{c}u|$-a.e. in $\Omega$ to
$D^{c}u/|D^{c}u|$ for $h\to\infty$, so that the continuity of $|\cdot|_{\rm
op}$ and the dominated convergence Theorem imply

$|D^{c}u|_{\rm op}(\Omega)=\lim_{h\to\infty}|D^{c}\psi_{h}(u)|_{\rm
op}(\Omega).$ (3.29)

Finally, (3.27), (3.28), and (3.29), together with Fatou’s Lemma and (3.26),
imply

$E^{f,g}(u,\Omega)\leq\liminf_{n\to+\infty}E^{f,g}(u_{n},\Omega),$

concluding the proof. ∎

######  Corollary 3.20.

Let $A\in\mathcal{A}(\Omega)$. The functional $V(\cdot,A)$ is lower
semicontinuous with respect to the convergence in
$L^{0}(A,{{\mathbb{R}}}^{k})$.

One of the main reasons motivating the introduction of the space
$GBV_{\star}(\Omega)$ in [15] is a useful (for variational purposes)
compactness result. Such a result easily adapts to
$GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$.

######  Theorem 3.21.

Let $(u_{n})_{n}$ be a sequence in $GBV_{\star}(\Omega)$. Suppose that there
exist a constant $M>0$ and an increasing continuous function
$\psi:[0,+\infty)\rightarrow[0,+\infty)$ with $\psi(t)\to+\infty$ as
$t\to+\infty$, such that

$\displaystyle\displaystyle\sup_{n}V(u_{n},\Omega)\leq M,$
$\displaystyle\displaystyle\sup_{n}\int_{\Omega}\psi(|u_{n}|)\,dx<+\infty.$
(3.30)

Then there exists a subsequence, not relabelled, and a function $u\in
GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ such that $u_{n}\to u$
$\mathcal{L}^{d}$-a.e. in $\Omega$.

###### Proof.

It is enough to apply Theorem [15, Theorem 3.11] component-wise. ∎

We now present a result showing that under suitable hypotheses on a sequence
$(u_{n})_{n}\subset GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$, it is possible to
produce a modification $y_{n}$ of $u_{n}$ (which can be performed preserving
eventual boundary conditions), satisfying the hypotheses of Theorem 3.21. This
result is a direct adaptation of [15, Theorem 5.5], whose proof is obtained by
adapting the arguments of [19].

In the following theorem $\Omega$ has a Lipschitz boundary and
$\Omega\subset\subset\Omega^{\prime}\subset{{\mathbb{R}}}^{d}$ is a domain
with Lipschitz boundary.

######  Theorem 3.22.

Let $h\in W^{1,1}(\Omega^{\prime},{{\mathbb{R}}}^{k})$, $f\in\mathcal{F}$,
$g\in\mathcal{G}$, and $E^{f,g}$ the functional defined by (3.18). Let
$(u_{n})_{n}\subset GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ with $u_{n}=h$
${\mathcal{L}}^{d}$-a.e. on $\Omega^{\prime}\\!\setminus\\!\overline{\Omega}$,
and $V(u_{n},\Omega^{\prime})\leq M$ for every $n\in\mathbb{N}$. Then there
exists a subsequence of $(u_{n})_{n}$, not relabelled, modifications $y_{n}\in
GBV_{\star}(\Omega^{\prime},{{\mathbb{R}}}^{k})$ of $u_{n}$, with $y_{n}=h$ on
$\Omega^{\prime}\\!\setminus\\!\overline{\Omega}$, and a continuous increasing
function $\psi:[0,+\infty)\rightarrow[0,+\infty)$ satisfying
$\psi(t)\to+\infty$ for $t\to+\infty$, such that

$\displaystyle\displaystyle E^{f,g}(y_{n},\Omega^{\prime})\leq
E^{f,g}(u_{n},\Omega)+\frac{1}{n},$ (3.31)
$\displaystyle\displaystyle\sup_{n}\int_{\Omega^{\prime}}\psi(|y_{n}|)\,dx<+\infty.$
(3.32)

###### Proof.

The theorem follows from the same lines of proof of [15, Theorem 5.5],
replacing [15, Lemma 5.2] by Lemma 3.23 below. ∎

We recall that given $\Omega$ a ${\mathcal{L}}^{d}$-measurable set a
collection $(P_{j})_{j}$ of ${\mathcal{L}}^{d}$-measurable subsets of $\Omega$
is said to be a Caccioppoli partition of $\Omega$ if

$\displaystyle\displaystyle{\mathcal{L}}^{d}(\Omega\setminus\bigcup_{j=1}^{\infty}P_{j})=0,$
$\displaystyle\displaystyle\sum_{j=1}^{\infty}{\mathcal{H}}^{d-1}(\partial^{*}P_{j}\cap\Omega)<+\infty.$

######  Lemma 3.23.

For every $M>0$ and $u\in GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ satisfying

$V(u,\Omega)\leq M,$

there exists a Caccioppoli partition $(P_{j})$ of $\Omega$ and a family of
translations $(t_{j})_{j}\subset{{\mathbb{R}}}^{k}$ such that the function

$v=u-\sum_{j=1}^{\infty}t_{j}\chi_{P_{j}}$

is in $BV(\Omega,{{\mathbb{R}}}^{k})\cap
L^{\infty}(\Omega,{{\mathbb{R}}}^{k})$ and the following estimates hold:

$\displaystyle\displaystyle\sum_{j=1}^{\infty}\mathcal{H}^{d-1}(\partial^{*}P_{j})\leq
k(2+2M+\mathcal{H}^{d-1}(\partial\Omega)),$ (3.33)
$\displaystyle\displaystyle\|v\|_{L^{\infty}(\Omega,{{\mathbb{R}}}^{k})}\leq
2M.$ (3.34)

###### Proof.

Thanks to [15, Lemma 5.1], for every $i=1,...,k$ we find a Caccioppoli
Partition $(P^{i}_{j})_{j}$ and a family of translations
$(t^{i}_{j})_{j}\subset{\mathbb{R}}$ satisfying

$\displaystyle\displaystyle\sum_{j=1}^{\infty}\mathcal{H}^{d-1}(\partial^{*}P^{i}_{j})\leq
2+2M+\mathcal{H}^{d-1}(\partial\Omega),$ (3.35)
$\displaystyle\displaystyle\|u^{i}-\sum_{j=1}^{\infty}t^{i}_{j}\chi_{P^{i}_{j}}\|_{L^{\infty}(\Omega)}\leq
2M.$ (3.36)

The family

$\displaystyle\mathcal{P}:=\\{E\subset\Omega\\!:P=\cap_{i=1}^{k}P_{j_{i}}^{i}\text{
for $(j_{1},...,j_{k})\in\mathbb{N}^{k}$}\\},$ (3.37)

is countable and is still a partition of $\Omega$. We order $\mathcal{P}$ and
denote every element of $\mathcal{P}$ as $P_{l}$ for some $l\in\mathbb{N}$.
Thanks to (3.35) and to the standard inequality

$\mathcal{H}^{d-1}(\partial^{*}(A\cap
B)\cap\Omega)\leq\mathcal{H}^{d-1}(\partial^{*}A\cap\Omega)+\mathcal{H}^{d-1}(\partial^{*}B\cap\Omega),$

we get that $(P_{l})_{l}$ satisfies (3.33).

Note that for every $i=1,...,k$ there exists a unique $j^{i}_{l}$ such that
$P_{l}\subset P^{i}_{j^{i}_{l}}$. For every $l\in\mathbb{N}$, we set
$t_{l}=(t^{1}_{j^{1}_{l}},...,t^{k}_{j^{k}_{l}})$ and

$v:=u-\sum_{l=1}^{+\infty}t_{l}\chi_{P_{l}}.$

Recalling (3.36), we infer that $v$ satisfies (3.34), so that Lemma 3.23 is
finally proved. ∎

######  Remark 3.24.

Let $f\in\mathcal{F},g\in\mathcal{G}$, and $h\in W^{1,1}(\Omega^{\prime})$.
Consider $(u_{n})_{n}\subset GBV_{\star}(\Omega,{{\mathbb{R}}}^{k})$ a
minimising sequence for the problem

$\inf\\{E^{f,g}(u,\Omega^{\prime})\colon u\in
GBV_{\star}(\Omega^{\prime},{{\mathbb{R}}}^{k}),\,\,u=h\,\,{\mathcal{L}}^{d}\text{-}a.e.\text{
on }\Omega^{\prime}\setminus\overline{\Omega}\\}.$

Thanks to (f2), (f3), (g2), (g3) and (3.15), it follows that
$V(u_{n},\Omega^{\prime})\leq M$ for some $M>0$, so that by Theorem 3.22 and
(3.31), there exist a minimising sequence $(y_{n})_{n}$, with $y_{n}=h$ on
$\Omega^{\prime}\setminus\overline{\Omega}$ and satisfying (3.32). In
particular, $(y_{n})_{n}$ satisfies the hypotheses of Theorem (3.21), so that
there exists a subsequence of $(y_{n})_{n}$, not relabelled, converging
pointwise a.e. in $\Omega^{\prime}$ to a function $u$, with $u=h$
${\mathcal{L}}^{d}$-a.e. in $\Omega^{\prime}\setminus\overline{\Omega}$. If
the functional $E^{f,g}(u,\Omega)$ were to be lower semicontinuous with
respect to the convergence of $L^{0}(\Omega^{\prime},{{\mathbb{R}}}^{k})$, we
would have concluded that $u$ minimises (3.24).

Acknowledgements. The author wishes to thank Professor Gianni Dal Maso for all
the useful and fruitful discussions on the topic. This paper is based on work
supported by the National Research Project (PRIN 2022J4FYNJ) ”Variational
methods for stationary and evolution problems with singularities and
interfaces”, funded by the Italian Ministry of University and Research. The
author is a member of the Gruppo Nazionale per l’Analisi Matematica, la
Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta
Matematica (INdAM).

## References

  * [1] Giovanni Alberti  “Rank one property for derivatives of functions with bounded variation”  In _Proceedings of the Royal Society of Edinburgh Section A: Mathematics_ 123.2 Royal Society of Edinburgh Scotland Foundation, 1993, pp. 239–274
  * [2] Roberto Alicandro, Andrea Braides and Jayant Shah  “Free-discontinuity problems via functionals involving the $L^{1}$-norm of the gradient and their approximations”  In _Interfaces Free Bound._ 1.1, 1999, pp. 17–37
  * [3] Roberto Alicandro and Matteo Focardi  “Variational approximation of free-discontinuity energies with linear growth”  In _Commun. Contemp. Math._ 4.4, 2002, pp. 685–723
  * [4] L. Ambrosio  “A compactness theorem for a new class of functions of bounded variation”  In _Boll. Un. Mat. Ital. B (7)_ 3.4, 1989, pp. 857–881
  * [5] Luigi Ambrosio  “Existence theory for a new class of variational problems”  In _Archive for Rational Mechanics and Analysis_ 111, 1990, pp. 291–322
  * [6] Luigi Ambrosio, Nicola Fusco and Diego Pallara  “Functions of bounded variation and free discontinuity problems”, Oxford Mathematical Monographs  The Clarendon Press, Oxford University Press, New York, 2000
  * [7] Marco Bonacini, Sergio Conti and Flaviana Iurlano  “Cohesive fracture in 1D: quasi-static evolution and derivation from static phase-field models”  In _Arch. Ration. Mech. Anal._ 239.3, 2021, pp. 1501–1576
  * [8] Marco Bonacini and Flaviana Iurlano  “Convergence of critical points for a phase-field approximation of 1D cohesive fracture energies”, 2023  arXiv:2309.17064
  * [9] Guy Bouchitté, Andrea Braides and Giuseppe Buttazzo  “Relaxation results for some free discontinuity problems”  In _Journal für die reine und angewandte Mathematik_ 458, 1995, pp. 1–18
  * [10] Guy Bouchitté and Giuseppe Buttazzo  “New lower semicontinuity results for nonconvex functionals defined on measures”  In _Nonlinear Analysis_ 15, 1990
  * [11] Blaise Bourdin, Gilles A. Francfort and Jean-Jacques Marigo  “The variational approach to fracture”  In _J. Elasticity_ 91.1-3, 2008, pp. 5–148
  * [12] S. Conti, M. Focardi and F. Iurlano  “Phase field approximation of cohesive fracture models”  In _Ann. Inst. H. Poincaré C Anal. Non Linéaire_ 33.4, 2016, pp. 1033–1067
  * [13] Sergio Conti, Matteo Focardi and Flaviana Iurlano  “Phase-field approximation of a vectorial, geometrically nonlinear cohesive fracture energy”, 2022  arXiv:2205.06541
  * [14] Gianni Dal Maso, Gianluca Orlando and Rodica Toader  “Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case”  In _Calc. Var. Partial Differential Equations_ 55.3, 2016, pp. Art. 4539
  * [15] Gianni Dal Maso and Rodica Toader  “A new space of generalised functions with bounded variation motivated by fracture mechanics” cvgmt preprint  In _NoDEA Nonlinear Differential Equations Appl._ 29.Paper No. 63., 2022  URL: http://cvgmt.sns.it/paper/5415/
  * [16] Gianni Dal Maso and Rodica Toader  “Gamma-convergence and integral representation for a class of free discontinuity functionals” cvgmt preprint, 2023  URL: http://cvgmt.sns.it/paper/6027/
  * [17] Gianni Dal Maso and Rodica Toader  “Homogenisation problems for free discontinuity functionals with bounded cohesive surface term” cvgmt preprint, 2023  URL: http://cvgmt.sns.it/paper/6125/
  * [18] Lawrence C. Evans and Ronald F. Gariepy  “Measure theory and fine properties of functions, revised edition”  In _Measure Theory and Fine Properties of Functions, Revised Edition_ CRC Press, 2015
  * [19] Manuel Friedrich  “A compactness result in $GSBV^{p}$ and applications to $\Gamma$-convergence for free discontinuity problems”  In _Calculus of Variations and Partial Differential Equations_ 58 Springer, 2019, pp. 1–31
  * [20] A.. Griffith  “The Phenomena of Rupture and Flow in Solids”  In _Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character_ 221 The Royal Society, 1921, pp. 163–198
  * [21] Francesco Maggi  “Sets of finite perimeter and geometric variational problems: an introduction to Geometric Measure Theory”, Cambridge Studies in Advanced Mathematics 135  Cambridge University Press, 2012
  * [22] J-J Marigo  “Revisiting brittle fracture as an energy minimization problem”  In _Journal of the Mechanics and Physics of Solids_ 46.8 Elsevier, 1998, pp. 1319–1342
  * [23] Diego Pallara  “Some new results on functions of bounded variation”  In _Rend. Accad. Naz. Sci. XL Mem. Mat. (5)_ 14.1, 1990, pp. 295–321


uniformly in $\eps$, for any ball $B$ with radius $r$, and for some integer $m$ large enough. We shall prove (<ref>) with $p_0=2+\epsilon_1$. Then by Lemma <ref>,
$\norm{\mathcal{G}_{\eps}(f)}_{L^p}\lesssim\norm{f}_{L^p}$ for all $2<p<p_0$, uniformly in $\eps$. Letting $\eps\to 0$, one obtains $\norm{\mathcal{G}f}_{L^p}\lesssim\norm{f}_{L^p}$ for all $2<p<p_0$.

Proof of $\mathcal{G}_{\eps}(f)\in L^p$ for $f\in L^p\cap L^2$.

We rewrite $\mathcal{G}_{\eps}(f)$ to be

\[
\mathcal{G}_{\eps}(f)=\br{\int_{{\eps}^2}^{\infty}\abs{t\nabla L^{1/2}e^{-tL}f}^2\frac{dt}{t}}^{1/2}.
\]

By Minkowski's inequality,

\begin{align*}
\norm{\mathcal{G}_{\eps}(f)}_{L^p}&\le\Big\{ \int_{\eps^2}^{\infty}\br{\int_{\Rn}\br{t\abs{\nabla L^{1/2}e^{-tL}f}^2}^{p/2}dx}^{2/p}dt\Big\}^{1/2}\\
&=\Big\{ \int_{\eps^2}^{\infty}\br{\int_{\Rn}\abs{\sqrt{t}\nabla L^{1/2}e^{-tL}f}^pdx}^{2/p}dt\Big\}^{1/2}\\
&=\Big\{ \int_{\eps^2}^{\infty}\br{\int_{\Rn}\abs{\sqrt{t}\nabla e^{-tL/2}L^{1/2}e^{-tL/2}f}^pdx}^{2/p}dt\Big\}^{1/2}.
\end{align*}

We first use the $L^2-L^p$ bounds for $(\sqrt{t}\nabla e^{-tL})_{t>0}$, then (<ref>), and finally the $L^2$ bounds for $(\sqrt{t}\nabla e^{-tL})_{t>0}$ to obtain

\begin{align*}
&\quad\norm{\mathcal{G}_{\eps}(f)}_{L^p}\lesssim \br{\int_{\eps^2}^{\infty}t^{-\gamma_p}\int_{\Rn}\abs{L^{1/2}e^{-tL/2}f}^2dxdt}^{1/2}\\
&\lesssim\br{\int_{\eps^2}^{\infty}t^{-\gamma_p}\int_{\Rn}\abs{\nabla e^{-tL/2}f}^2dxdt}^{1/2}
\lesssim\br{\int_{\eps^2}^{\infty}t^{-\gamma_p-1}dt}^{1/2}\norm{f}_{L^2}
\le C_{\eps, p}\norm{f}_{L^2}.
\end{align*}

Proof of (<ref>)

Since the domain of $e^{-tL}$ is $L^2$, the operators commute for $f\in L^2$:

\begin{equation}
\nabla L^{1/2}e^{-t^2L}(e^{-r^2L}f)=\nabla e^{-r^2L}L^{1/2}e^{-t^2L}f.
\end{equation}

By Minkowski inequality,

\begin{align}\label{G_2(2.4)Mink}
&\quad\Big\{\frac{1}{\abs{B}}\int_B\Big(\int_{\eps}^{\infty}\abs{t^2\nabla L^{1/2}e^{-t^2L}(e^{-r^2L}f)}^2\frac{dt}{t}\Big)^{\frac{p_0}{2}}dx\Big\}^{\frac{2}{p_0}}\nonumber\\
&\le\int_{\eps}^{\infty}\Big(\frac{1}{\abs{B}}\int_B\abs{t^2\nabla L^{1/2}e^{-t^2L}(e^{-r^2L}f)}^{p_0}dx\Big)^{\frac{2}{p_0}}\frac{dt}{t}\nonumber\\
&=\int_{\eps}^{\infty}\Big(\frac{1}{\abs{B}}\int_B\abs{t^2\nabla e^{-r^2L}(L^{1/2}e^{-t^2L}f)}^{p_0}dx\Big)^{\frac{2}{p_0}}\frac{dt}{t}
\end{align}

Using the $L^2-L^{p_0}$ off-diagonal estimates for $(\sqrt{r}\nabla e^{-rL})_{r>0}$, as well as Poincaré inequality, one can show

\begin{equation*}
\Big(\frac{1}{\abs{B}}\int_B\abs{\nabla e^{-r^2L}f}^{p_0}\Big)^{1/{p_0}}\le\sum_{j\ge1}g(j)\Big(\frac{1}{\abs{2^{j+1}B}}
\int_{2^{j+1}B}\abs{\nabla f}^2\Big)^{1/2}
\end{equation*}

with $\sum_{j\ge1}g(j)<\infty$. By this and Hölder inequality we can bound (<ref>) by

\begin{align}
\Big(\frac{1}{\abs{2^{j+1}B}}\int_{2^{j+1}B}\abs{t^2\nabla L^{1/2}e^{-t^2L}f}^2dx\Big)^{1/2}\Big\}^2\frac{dt}{t}\nonumber\\
&\le C\int_{\eps}^{\infty}\sum_{j\ge1}g(j)\frac{1}{\abs{2^{j+1}B}}\int_{2^{j+1}B}\abs{t^2\nabla L^{1/2}e^{-t^2L}f}^2dx\frac{dt}{t}\nonumber\\
&\le C\sup_{j\ge1}\frac{1}{\abs{2^{j+1}B}}\int_{2^{j+1}B}\int_{\eps}^{\infty}\abs{t^2\nabla L^{1/2}e^{-t^2L}f}^2\frac{dt}{t}dx\nonumber\\
&\le C(\mathcal{M}\abs{\mathcal{G}_{\eps}f}^2)^{1/2}(y)\qquad\forall\, y\in B.
\end{align}

Proof of (<ref>)
Now write $f=\sum_{j\ge1}f_j$, where

\begin{cases}
f_j=(f-(f)_{4B})\mathbbm{1}_{2^{j+1}B\setminus 2^jB} \quad j\ge2,\\
\end{cases}

\br{\frac{1}{\abs{B}}\int_B\abs{\mathcal{G}_{\eps}(I-e^{-r^2L})^mf}^2}^{1/2}\le\sum_{j\ge1}\br{\frac{1}{\abs{B}}\int_B\abs{\mathcal{G}_{\eps}(I-e^{-r^2L})^mf_j}^2}^{1/2}.

For $f_1$, the $L^2$ bound of $\mathcal{G}$ and that of $(I-e^{-r^2L})^m$ (the latter is a consequence of the holomorphic functional calculus on $L^2$) imply

\begin{align}
&\quad\br{\frac{1}{\abs{B}}\int_B\abs{\mathcal{G}_{\eps}(I-e^{-r^2L})^mf_1}^2)}^{1/2}\le C(\frac{1}{\abs{B}}\int_{\Rn}\abs{f_1}^2)^{1/2}\nonumber\\
&\le C\br{\frac{1}{\abs{B}}\int_{4B}\abs{f}^2}^{1/2}\le C(\mathcal{M}\abs{f}^2)^{1/2}(y) \quad\forall\, y\in B.
\end{align}

For $f_j$ with $j\ge2$, let $\vp(z)=tz^{1/2}e^{-t^2z}(1-e^{-r^2z})^m$. Then (see e.g. [2] section 3.2)

\begin{equation}
\end{equation}

where $\Gamma_{\pm}$ is the half-ray $\Real^+e^{\pm i(\frac{\pi}{2}-\theta)}$,

\begin{equation}
\eta_{\pm}(z)=\frac{1}{2\pi i}\int_{\gamma_{\pm}}e^{\zeta z}\vp(\zeta)d\zeta, \quad z\in\Gamma_{\pm},
\end{equation}

with $\gamma_{\pm}$ being the half-ray $\Real^{+}e^{\pm i\nu}$, and
$0<\omega_0<\theta<\nu<\frac{\pi}{2}$, where $\omega_0$ is as in Proposition (<ref>).
One can show

\begin{equation}\label{eta+_}
\abs{\eta_{\pm}(z)}\le\frac{Ct}{(\abs{z}+t^2)^{3/2}}\inf(1,\frac{r^{2m}}{(\abs{z}+t^2)^m}), \quad z\in\Gamma_{\pm},
\end{equation}

whose proof is postponed to the end.

\begin{align}
&=\int_{\eps}^{\infty}\frac{t}{\abs{B}}\int_B\abs{\int_{\Gamma_+}\nabla e^{-zL}\eta_+(z)dzf_j+\int_{\Gamma_-}\nabla e^{-zL}\eta_-(z)dzf_j}^2dxdt\nonumber\\
&\le C\int_0^{\infty}\frac{t}{\abs{B}}\int_B\abs{\int_{\Gamma_+}\nabla e^{-zL}\eta_+(z)dzf_j}^2dxdt\nonumber\\ &\quad+\int_0^{\infty}\frac{t}{\abs{B}}\int_B\abs{\int_{\Gamma_-}\nabla e^{-zL}\eta_-(z)dzf_j}^2dxdt\nonumber\\
&\doteq I_++I_-.
\end{align}

By Minkowski inequality and (<ref>),

\begin{align*}
&I_+\le\int_0^{\infty}\frac{t}{\abs{B}}\Big\{\int_{\Gamma_+}\Big(\int_B\abs{\nabla e^{-zL}\eta_+(z)f_j}^2dx\Big)^{1/2}\abs{dz}\Big\}^2dt\\
&\lesssim\int_0^{\infty}\frac{t}{\abs{B}}\Big\{\int_{\Gamma_+}\Big(\int_B\abs{\sqrt{z}\nabla e^{-zL}\frac{t}{\abs{z}^{1/2}(\abs{z}+t^2)^{3/2}}\frac{r^{2m}}{(\abs{z}+t^2)^m}f_j}^2dx\Big)^{1/2}\abs{dz}\Big\}^2dt.
\end{align*}

Since $z\in\Gamma_+=\Real^+e^{i(\frac{\pi}{2}-\theta)}$ and $\theta<\omega_0$, we can apply the $L^2-L^2$ off-diagonal estimates for $\br{\sqrt{z}\nabla e^{-zL}}_{z\in\Sigma_{\frac{\pi}{2}-\theta}}$, and bound the expression above by

\begin{equation}\label{G_2 5.4}
\int_0^{\infty}\frac{t}{\abs{B}}\Big(\int_{\Gamma_+}e^{-\frac{c4^jr^2}{\abs{z}}}\frac{t}{\abs{z}^{1/2}(\abs{z}+t^2)^{3/2}}\frac{r^{2m}}{(\abs{z}+t^2)^m}\abs{dz}\norm{f_j}_{L^2}\Big)^2dt
\end{equation}

We use the following lemma to estimate (<ref>)

Let $\gamma,\alpha\ge0$, $m>0$ be fixed parameters, and $c$ a positive constant. For some $C$ independent of $j\in\mathbb{N}$, $r,t>0$, the integral

satisfies the estimate

\begin{equation}
I\le \frac{C}{4^{jm}(2^jr)^{\gamma}}\inf\Big((\frac{t}{4^jr^2})^{\alpha},(\frac{4^jr^2}{t})^m\Big).
\end{equation}

Letting $\gamma=1$, $\alpha=\frac{1}{2}$, and $t$ replaced by $t^2$ in the lemma, we obtain

\begin{align*}
\eqref{G_2 5.4}&\lesssim\int_0^{\infty}\frac{t}{(4^{jm}(2^jr))^2}\inf\Big((\frac{t^2}{4^jr^2}),(\frac{4^jr^2}{t^2})^{2m}\Big)dt\frac{1}{\abs{B}}\int_{2^{j+1}B}\abs{f}^2dx\\
&\lesssim2^{j(n-4m)}\mathcal{M}\abs{f}^2(y)\quad\forall\, y\in B.
\end{align*}

$I_-$ can be estimated similarly. Choose $4m>n$, we get

\frac{1}{\abs{B}}\int_B\abs{\mathcal{G}_{\eps}(I-e^{-r^2L})^mf}^2\lesssim\mathcal{M}\abs{f}^2(y)\quad\forall\, y\in B,

which proves (<ref>).

Case 3: $1<p<2$. We use Lemma <ref> to prove (<ref>) holds for $1<p<2$. Define $\mathcal{G}$ as before. Then by letting $T=\mathcal{G}$ and $A_r=I-(I-e^{-r^2L})^m$ in Lemma <ref>, it suffices to show the following: let $1<p_0<2$,

\begin{equation}\label{2.1}
\br{\frac{1}{\abs{2^{j+1}B}}\int_{C_j(B)}\abs{\mathcal{G}(I-e^{-r^2L})^mf}^2}^{1/2}\le g(j)\br{\frac{1}{\abs{B}}\int_B\abs{f}^{p_0}}^{1/{p_0}}\qquad\text{for }j\ge2
\end{equation}

and for $j\ge1$

\begin{equation}\label{2.2}
\br{\frac{1}{\abs{2^{j+1}B}}\int_{C_j(B)}\abs{(e^{-kr^2L}f)}^2}^{1/2}\le g(j)\br{\frac{1}{\abs{B}}\int_B\abs{f}^{p_0}}^{1/{p_0}}
\end{equation}

for any ball $B$ with radius $r$, for all $f$ supported in $B$, for some integer $m$ sufficiently large, $1\le k\le m$, and $\sum_jg(j)2^{nj}<\infty$.

(<ref>) follows directly from the $L^{p_0}-L^2$ off-diagonal estimate of $(e^{-tL})_{t>0}$. We now turn to (<ref>).
As in the proof of (<ref>), we have

\begin{align}
&\le C\int_0^{\infty}\frac{t}{\abs{2^{j+1}B}}\int_{C_j(B)}\abs{\int_{\Gamma_+}\nabla e^{-zL}\eta_+(z)dz\,f}^2dxdt\nonumber\\ &\quad+\int_0^{\infty}\frac{t}{\abs{2^{j+1}B}}\int_{C_j(B)}\abs{\int_{\Gamma_-}\nabla e^{-zL}\eta_-(z)dz\,f}^2dxdt\nonumber\\
&\doteq I_++I_-\label{G2p<2 I}
\end{align}

where $\eta_{\pm}$ and $\Gamma_{\pm}$ are same as before. We only estimate $I_+$, as $I_-$ can be estimated in a similar manner.
By Minkowski inequality and (<ref>), $I_+$ is bounded up to a constant by

\begin{align*}
\int_0^{\infty}\frac{t}{\abs{2^{j+1}B}}\Big\{\int_{\Gamma_+}\Big(\int_{C_j(B)}\abs{\sqrt{z}\nabla e^{-zL}\frac{t}{\abs{z}^{1/2}(\abs{z}+t^2)^{3/2}}\frac{r^{2m}}{(\abs{z}+t^2)^m}\,f}^2dx\Big)^{1/2}\abs{dz}\Big\}^2dt.
\end{align*}

The $L^{p_0}-L^2$ off-diagonal estimates for $\br{\sqrt{z}\nabla e^{-zL}}_{z\in\Sigma_{\frac{\pi}{2}-\theta}}$ implies that the expression above is bounded up to a constant by

\begin{equation*}
\int_0^{\infty}\frac{t}{\abs{2^{j+1}B}}\Big(\int_{\Gamma_+}e^{-\frac{c4^jr^2}{\abs{z}}}\frac{t}{\abs{z}^{(1+\gamma_{p_0})/2}(\abs{z}+t^2)^{3/2}}\frac{r^{2m}}{(\abs{z}+t^2)^m}\abs{dz}\norm{f}_{L^{p_0}(B)}\Big)^2dt.
\end{equation*}

Applying Lemma <ref> with $\gamma=1+\gamma_{p_0}$ and $\alpha=\frac{1}{2}$, this is bounded up to a constant by

\begin{align*}
&\quad 4^{-2jm}(2^jr)^{-2\gamma_{p_0}}\frac{\norm{f}^2_{L^{p_0}(B)}}{\abs{2^{j+1}B}}\int_0^\infty\frac{t}{(2^jr)^2}\inf\br{\frac{t^2}{4^jr^2},\br{\frac{4^jr^2}{t^2}}^{2m}}dt\\
&\lesssim 2^{-j(4m+n+2\gamma_{p_0})}\br{\frac{1}{\abs{B}}\int_B\abs{f}^{p_0}}^{\frac{2}{p_0}}.
\end{align*}

Combining this with (<ref>) gives (<ref>) with $g(j)=2^{-j(2m+\frac{n}{2}+\gamma_{p_0})}$. And thus by choosing $m$ to be an integer such that $2m+\gamma_{p_0}>\frac{n}{2}$, we obtain the desired result.

Proof of (<ref>)We only show the estimate for $\eta_+$. The proof for $\eta_-$ is similar. Write $\zeta=\rho e^{i\nu}$, and $z=\abs{z}e^{i(\frac{\pi}{2}-\theta)}$.
Then $\abs{e^{\zeta z}}=e^{-\rho\abs{z}\sin{(\nu-\theta)}}$, $\abs{e^{-t^2\zeta}}=e^{-t^2\rho\cos{\nu}}$. Since $0<\theta<\nu<\frac{\pi}{2}$,
$\abs{e^{\zeta z}e^{-t^2\zeta}}\le e^{-c\rho(\abs{z}+t^2)}$ for some $0<c<1$. So

\abs{\eta_+(z)}\lesssim t\int_0^{\infty} \rho^{1/2}e^{-c\rho(\abs{z}+t^2)}H(\rho)^md\rho,

where $H(\rho)=\abs{1-\exp(-r^2\rho e^{i\nu})}$. Observe that $H(0)=0$, $H(\rho)\le 2$, and that $H$ is a Lipschitz function with $[H]_{C^{0,1}}\le r^2$. So we have

H(\rho)\le C\inf(1,r^2\rho).

Using this estimate of $H$, we can bound $\abs{\eta_+(z)}$ by

\begin{equation}
\end{equation}

and by

\begin{equation}
\end{equation}

Combining the two bounds we obtain

\abs{\eta_+(z)}\le \frac{Ct}{(\abs{z}+t^2)^{3/2}}\inf(1,\frac{r^{2m}}{(\abs{z}+t^2)^m}).

We also have a similar estimate when the derivative falls on $t$. But in this case, the $L^p$ estimates hold for any $1<p<\infty$:

\begin{equation}\label{Lp_G2t}
\norm{\Big(\int_0^{\infty}\abs{t^2\Dt L e^{-t^2L}F}^2\frac{dt}{t}\Big)^{1/2}}_{L^p(\Rn)}\le C_p\norm{\nabla F}_{L^p(\Rn)}
\end{equation}

for all $1<p<\infty$, and all $F\in W^{1,2}\cap W^{1,p}$.

\begin{equation}\label{Gdef G2t}
\mathcal{G}(f)=\Big(\int_0^{\infty}\abs{t^2\Dt L^{1/2}e^{-t^2L}f}^2\frac{dt}{t}\Big)^{1/2}.
\end{equation}

Then by the $L^p$ bounds of the square root of $L$, it suffices to show

\begin{equation}\label{G2t_Gt_Lp}
\norm{\mathcal{G}f}_{L^p}\le C_p\norm{f}_{L^p}.
\end{equation}

Case 1: $p=2$. The argument can be copied almost verbatim from the proof in Proposition <ref>. The only difference is that we would use the uniform $L^2$ boundedness of $(t\Dt e^{-t^2L/2})_{t>0}$, rather than that of $(t\nabla e^{-t^2L/2})_{t>0}$.

Case 2: $p>2$. We shall apply Lemma <ref> again. And as in the proof of Proposition <ref>, we should derive the analog of (<ref>) and (<ref>) for the approximation operator $\mathcal{G}_{\eps}$. We omit this limiting process here for simplicity, and only derive the analogous estimates for $\mathcal{G}$.

Let $p_0>2$. We wish to prove

\begin{equation*}
\br{\frac{1}{\abs{B}}\int_B\abs{\mathcal{G}(e^{-r^2L}f)}^{p_0}}^{1/{p_0}}\lesssim(M(\abs{\mathcal{G}f}^2))^{1/2}(y) \qquad\forall\, y\in B.
\end{equation*}

To this end, we first claim that

\begin{equation}\label{commute}
t^2\Dt L^{1/2}e^{-t^2L}(e^{-r^2L}f)=-2t^3e^{-r^2L}L^{1/2}Le^{-t^2L}f, \qquad\forall\, f\in L^2.
\end{equation}

Then by this and Minkowski inequality,

\begin{align}
&\quad\Big\{\frac{1}{\abs{B}}\int_B\Big(\int_0^{\infty}\abs{t^2\Dt L^{1/2}e^{-t^2L}(e^{-r^2L}f)}^2\frac{dt}{t}\Big)^{\frac{p_0}{2}}dx\Big\}^{\frac{2}{p_0}}\nonumber\\
&\le\int_0^{\infty}\Big(\frac{1}{\abs{B}}\int_B\abs{t^2\Dt L^{1/2}e^{-t^2L}(e^{-r^2L}f)}^{p_0}dx\Big)^{\frac{2}{p_0}}\frac{dt}{t}\nonumber\\
&=4\int_0^{\infty}\Big(\frac{1}{\abs{B}}\int_B\abs{t^3 e^{-r^2L}L^{1/2}Le^{-t^2L}f}^{p_0}dx\Big)^{\frac{2}{p_0}}\frac{dt}{t}
\end{align}

Using the $L^2-L^{p_0}$ off-diagonal estimates for $(e^{-r^2L})_{r>0}$, one can show

\begin{equation}
\Big(\frac{1}{\abs{B}}\int_B\abs{e^{-r^2L}f}^{p_0}\Big)^{2/{p_0}}\le\sum_{j\ge1}\frac{C_j}{\abs{2^{j+1}B}}\int_{2^{j+1}B}\abs{f}^2
\end{equation}

with $C_j=Ce^{-c4^j}$. So

\begin{align}
&\quad\Big\{\frac{1}{\abs{B}}\int_B\Big(\int_0^{\infty}\abs{t^2\Dt L^{1/2}e^{-t^2L}(e^{-r^2L}f)}^2\frac{dt}{t}\Big)^{\frac{p_0}{2}}dx\Big\}^{\frac{2}{p_0}}\nonumber\\
&\le 4\int_0^{\infty}\sum_{j\ge1}\frac{C_j}{\abs{2^{j+1}B}}\int_{2^{j+1}B}\abs{t^3L^{1/2}Le^{-t^2L}f}^2dx\frac{dt}{t}\nonumber\\
&\lesssim\sup_j\frac{1}{\abs{2^{j+1}B}}\int_{2^{j+1}B}\int_0^{\infty}\abs{t^2\Dt L^{1/2}e^{-t^2L}f}^2\frac{dt}{t}dx\nonumber\\
&\lesssim M(\abs{\mathcal{G}(f)}^2)(y)\qquad\forall\, y\in B.
\end{align}

Now we prove

\begin{equation}\label{G2t 2.1 p>2}
\br{\frac{1}{\abs{B}}\int_B\abs{\mathcal{G}(I-e^{-r^2L})^mf}^2}^{1/2}\lesssim(M(\abs{f}^2))^{1/2}(y) \qquad\forall\, y\in B.
\end{equation}

Define $\{f_j\}_{j\ge1}$ as in the proof of Proposition <ref>. The estimates for $f_1$ again follows from the $L^2$ bound of $\mathcal{G}$ and that of $(I-e^{-r^2L})^m$. For $f_j$ with $j\ge2$, we let $\vp(z)=t^3z^{3/2}e^{-t^2z}(1-e^{-r^2z})^m$. Then

\begin{align*}
&\quad-\frac{1}{2}t^2\Dt L^{1/2}e^{-t^2L}(I-e^{-r^2L})^m=t^3L^{1/2}Le^{-t^2L}(I-e^{-r^2L})^m\\
\end{align*}

where $\eta_{\pm}$ and $\Gamma_{\pm}$ are defined as in the proof of Proposition <ref>. By a similar argument as in the proof of (<ref>), one can show

\begin{equation}\label{eta_t}
\abs{\eta_{\pm}(z)}\lesssim t^3\int_0^{\infty}\rho^{3/2}e^{-c\rho(\abs{z}+t^2)}H(\rho)^md\rho
\le \frac{Ct^3}{(\abs{z}+t^2)^{5/2}}\inf(1,\frac{r^{2m}}{(\abs{z}+t^2)^m}).
\end{equation}

\begin{align}
&\lesssim\int_0^{\infty}\frac{1}{\abs{B}}\int_B\abs{\int_{\Gamma_+} e^{-zL}\eta_+(z)dzf_j}^2dx\frac{dt}{t} +\int_0^{\infty}\frac{1}{\abs{B}}\int_B\abs{\int_{\Gamma_-} e^{-zL}\eta_-(z)dzf_j}^2dx\frac{dt}{t}\nonumber\\
&\doteq I_++I_-.
\end{align}

By Minkowski inequality and (<ref>),

\begin{equation*}
\frac{1}{\abs{B}}\Big\{\int_{\Gamma_+}\Big(\int_B\abs{t^3e^{-zL}\frac{r^{2m}}{(\abs{z}+t^2)^{\frac{5}{2}+m}}f_j}^2dx\Big)^{1/2}\abs{dz}\Big\}^2\frac{dt}{t}.
\end{equation*}

We use the $L^2$ off-diagonal estimate for $(e^{zL})_{z\in\Sigma_{\frac{\pi}{2}-\theta}}$ to bound the above expression by

\int_0^{\infty}\frac{1}{\abs{B}}
\Big(\int_{\Gamma_+}e^{-\frac{c4^jr^2}{\abs{z}}}\frac{t^3r^{2m}}{(\abs{z}+t^2)^{5/2+m}}\abs{dz}\norm{f_j}_{L^2}\Big)^2\frac{dt}{t}.

Applying Lemma <ref> and letting $\alpha=\frac{3}{2}$, $\gamma=0$ and $t=t^2$ there, we obtain

\begin{align*}
\frac{1}{\abs{B}}\norm{f}^2_{L^2(2^{j+1}B)}\frac{dt}{t}\\
&\lesssim \frac{2^{jn-4jm}}{\abs{2^{j+1}B}}\norm{f}_{L^2(2^{j+1}B)}^2
\Big(\int_0^{2^jr}\frac{t^5}{(4^jr^2)^3}dt+\int_{2^jr}^{\infty}\frac{(4^jr^2)^{2m}}{t^{4m+1}}dt\Big)\\
&\lesssim 2^{jn-4jm}M(\abs{f}^2)(y) \qquad\forall\, y\in B.
\end{align*}

$I_-$ can be estimated similarly. Choose $4m>n$, we get

\frac{1}{\abs{B}}\int_B\abs{\mathcal{G}(I-e^{-r^2L})^mf}^2\lesssim\mathcal{M}\abs{f}^2(y)\quad\forall\, y\in B.

Therefore, (<ref>) holds for all $2\le p<p_0$. And since $p_0>2$ is arbitrary, (<ref>) holds for all $2\le p<\infty$.

Case 3: $1<p<2$. Applying Lemma <ref> and letting $1<p_0<2$, it suffices to show (<ref>) and (<ref>), where $\mathcal{G}$ is defined in (<ref>). Note that (<ref>) is independent of the operator $\mathcal{G}$ and is verified in the proof of Proposition <ref>. To see (<ref>), we proceed similarly as the proof of (<ref>). Using the $L^{p_0}-L^2$ off-diagonal estimate of $(e^{zL})_{z\in\Sigma_{\frac{\pi}{2}-\theta}}$, we obtain

\begin{align*}
\frac{t^3r^{2m}}{(\abs{z}+t^2)^{\frac{5}{2}+m}}\norm{f}_{L^{p_0}(B)}\abs{dz}\Big\}^2\frac{dt}{t}
\end{align*}

plus an integral over $\Gamma_-$ with the same integrand.
Applying Lemma <ref> with $\gamma=\gamma_{p_0}$ and $\alpha=\frac{3}{2}$, we have

\begin{align*}
&\lesssim 4^{-2jm}(2^jr)^{-2\gamma_{p_0}}\frac{\norm{f}_{L^{p_0}(B)}^2}{\abs{2^{j+1}B}}\int_0^\infty\inf\br{\br{\frac{t^2}{4^jr^2}}^3,\br{\frac{4^jr^2}{t^2}}^{2m}}\frac{dt}{t}\\
&\lesssim 2^{-j(n+4m+2\gamma_{p_0})}r^{-2\gamma_{p_0}}\frac{\norm{f}^2_{L^{p_0}(B)}}{\abs{B}}\\
&\lesssim 2^{-j(n+4m+2\gamma_{p_0})}\br{\frac{1}{\abs{B}}\int_B\abs{f}^{p_0}}^{\frac{2}{p_0}}.
\end{align*}

Choosing $m$ to be an integer such that $2m+\gamma_{p_0}>\frac{n}{2}$ gives (<ref>). And this shows that (<ref>) holds for all $1<p<2$.

Proof of (<ref>).We know that for any $g\in L^2$, $e^{-t^2L}g\in D(L)$. And $\Dt e^{-t^2L}g\in W^{1,2}=D(L^{1/2})$. The latter follows from analyticity of the semigroup

$$\partial_je^{-t^2L}g=\frac{1}{2\pi i}\int_{\Gamma}e^{\lambda t^2}\partial_j(\lambda I+L)^{-1}(g)d\lambda$$

\Dt\partial_je^{-t^2L}g=\frac{2t}{2\pi i}\int_{\Gamma}e^{\lambda t^2}\lambda\partial_j(\lambda I+L)^{-1}(g)d\lambda\in L^2.

We then use the following lemma

For any bounded linear operator $B$ on $L^2$, if $BL=LB$ in $D(L)$, then $L^{1/2}B=BL^{1/2}$ in $D(L^{1/2})$.

Since $e^{-r^2L}f\in D(L)\subset D(L^{1/2})$ and that $e^{-t^2L}$ is a bounded, linear operator on $L^2$, the lemma implies

\begin{equation}\label{commute1}
t^2\Dt L^{1/2}e^{-t^2L}(e^{-r^2L}f)=t^2 \Dt e^{-t^2L}L^{1/2}(e^{-r^2L}f).
\end{equation}

\begin{align}
&\quad t^2\Dt L^{1/2}e^{-t^2L}(e^{-r^2L}f)=-2t^3Le^{-t^2L}(L^{1/2}e^{-r^2L}f)\nonumber\\
\end{align}

where the last equality follows from Lemma <ref> and $L^{1/2}L^{1/2}=L$.

\begin{align}
&=-2t^3L^{1/2}e^{-r^2L}Le^{-t^2L}f=t^2L^{1/2}e^{-r^2L}\Dt e^{-t^2L}f\nonumber\\
\end{align}

where in the last step we have used $-2tLe^{-t^2L}f=\Dt e^{-t^2L}f\in W^{1,2}=D(L^{1/2})$ and Lemma <ref>. Combining (<ref>)-(<ref>), we have proved (<ref>).

We include some frequently used results in this appendix for reader's convenience.

Let $Q$ be a cube in $\Rn$. Let $g\in L^q(Q)$, $q>1$, and $f\in L^s(Q)$, $s>q$, be two nonnegative functions. Suppose

\[
\fint_{Q_R(x_0)}g^qdx\le b\br{\fint_{Q_{2R}(x_0)}gdx}^q+\fint_{Q_{2R}(x_0)}f^qdx+\theta\fint_{Q_{2R}(x_0)}g^qdx
\]

for each $x_0\in Q$ and each $R<\min\set{\frac{1}{2}\dist (x_0,\bdy Q),R_0}$, where $R_0$, $b$, $\theta$ are constants with $b>1$, $R_0>0$, $0\le\theta<1$. Then $g\in L_{\loc}^p(Q)$ for $p\in[q,q+\epsilon)$ and

\[
\br{\fint_{Q_R}g^pdx}^{1/p}\le c\Big\{\br{\fint_{Q_{2R}}g^qdx}^{1/q}+\br{\fint_{Q_{2R}}f^pdx}^{1/p}\Big\}
\]

for $Q_{2R}\subset Q$, $R<R_0$, where $c$ and $\epsilon$ are positive constants depending only on $b$, $\theta$, $q$, $n$ (and $s$).

Let $\phi(t)$ be a nonnegative and nondecreasing function. Suppose that

\[
\phi(\rho)\le A\br{\br{\frac{\rho}{R}}^{\alpha}+\eps}\phi(R)+BR^\beta
\]

for all $\rho\le R\le R_0$, with $A$, $\alpha$, $\beta$ nonnegative constants, $\beta<\alpha$. Then there exists a constant $\eps_0=\eps_0(A,\alpha,\beta)$ such that if $\eps<\eps_0$, for all $\rho\le R\le R_0$ we have

\[
\phi(\rho)\le c\br{\br{\frac{\rho}{R}}^{\beta}\phi(R)+B\rho^\beta}
\]

where $c=c(\alpha,\beta,A)$.

Suppose $u,v\in L^2\br{(0,T),W^{1,2}(\Rn)}$ with $\Dt u, \Dt v \in L^2\br{(0,T),\wt{W}^{-1,2}(\Rn)}$. Then

* $u\in C\br{[0,T], L^2(\Rn)}$;

* The mapping $t\mapsto\norm{u(\cdot,t)}_{L^2(\Rn)}$ is absolutely continuous, with

\[\frac{d}{dt}\norm{u(\cdot,t)}_{L^2(\Rn)}^2=2\Re\act{\Dt u(\cdot,t),u(\cdot,t)} \quad\text{for a.e. }t\in[0,T].\]

As a consequence,

\[
\frac{d}{dt}\br{u(\cdot,t),v(\cdot,t)}_{L^2(\Rn)}=\act{\Dt u(\cdot,t),v(\cdot,t)}+\overline{\langle{\Dt v(\cdot,t),u(\cdot,t)\rangle}}_{\wt{W}^{-1,2},W^{1,2}}\quad\text{a.e.}.
\]

For its proof see e.g. [7] Section 5.9.2 Theorem 3.

[1]
Pascal Auscher.
Regularity theorems and heat kernel for elliptic operators.
Journal of the London Mathematical Society, 54(2):284–296,

[2]
Pascal Auscher.
On necessary and sufficient conditions for $L^p$-estimates of
Riesz transforms associated to elliptic operators on $R^n$ and related
American Mathematical Soc., 2007.

[3]
Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Philippe
The solution of the Kato square root problem for second order
elliptic operators on $\mathbb{R}^n$.
Annals of mathematics, 156(2):633–654, 2002.

[4]
Pascal Auscher and Philippe Tchamitchian.
Square root problem for divergence operators and related
Number 249 in Astérisque. Société mathématique de France,

[5]
Hongjie Dong and Seick Kim.
Fundamental solutions for second-order parabolic systems with drift
Proceedings of the American Mathematical Society,
146(7):3019–3029, 2018.

[6]
Luis Escauriaza and Steve Hofmann.
Kato square root problem with unbounded leading coefficients.
Proceedings of the American Mathematical Society,
146(12):5295–5310, 2018.

[7]
Lawrence C. Evans.
Partial Differential Equations, volume 19.
American Mathematical Society, 1998.

[8]
Mariano Giaquinta.
Multiple Integrals in the Calculus of Variations and Nonlinear
Elliptic Systems.
Number 105. Princeton University Press, 1983.

[9]
Steve Hofmann, Carlos Kenig, Svitlana Mayboroda, and Jill Pipher.
Square function/non-tangential maximal function estimates and the
Dirichlet problem for non-symmetric elliptic operators.
Journal of the American Mathematical Society, 28(2):483–529,

[10]
Steve Hofmann, Michael Lacey, and Alan McIntosh.
The solution of the Kato problem for divergence form elliptic
operators with Gaussian heat kernel bounds.
Annals of Mathematics, 156:623–631, 2002.

[11]
Steve Hofmann, Linhan Li, Svitlana Mayboroda, and Jill Pipher.
The Dirichlet problem for elliptic operators having a BMO
anti-symmetric part.

[12]
Steve Hofmann and Alan McIntosh.
The solution of the Kato problem in two dimensions.
Publicacions Matemàtiques, pages 143–160, 2002.

[13]
David S Jerison and Carlos E Kenig.
The Neumann problem on Lipschitz domains.
Bulletin of the American Mathematical Society, 4(2):203–207,

[14]
Tosio Kato.
Fractional powers of dissipative operators.
Journal of the Mathematical Society of Japan, 13(3):246–274,

[15]
Tosio Kato.
Perturbation Theory for Linear Operators.
Springer-Verlag, New York, 1966.

[16]
Tosio Kato.
Perturbation theory for linear operators, volume 132.
Springer, second edition, 1976.

[17]
Carlos E Kenig and Jill Pipher.
The Neumann problem for elliptic equations with non-smooth
Inventiones mathematicae, 113(1):447–509, 1993.

[18]
Linhan Li and Jill Pipher.
Boundary behavior of solutions of elliptic operators in divergence
form with a BMO anti-symmetric part.
Communications in Partial Differential Equations, pages 1–49,

[19]
Alan McIntosh.
On the comparability of $A^{1/2}$ and $A^{*1/2}$.
Proceedings of the American Mathematical Society,
32(2):430–434, 1972.

[20]
Alan McIntosh and Atsushi Yagi.
Operators of type $\omega$ without a bounded $H_{\infty}$
functional calculus.
In Miniconference on operators in analysis, pages 159–179.
Centre for Mathematics and its Applications, Mathematical Sciences
Institute …, 1990.

[21]
Milan Miklavčič.
Applied functional analysis and partial differential equations.
World Scientific, 1998.

[22]
Amnon Pazy.
Semigroups of linear operators and applications to partial
differential equations, volume 44.
Springer Applied Mathematical Sciences, 1983.

[23]
Zhongmin Qian and Guangyu Xi.
Parabolic equations with singular divergence-free drift vector
Journal of the London Mathematical Society, 2018.

[24]
Gregory Seregin, Luis Silvestre, Vladimír Šverák, and Andrej
On divergence-free drifts.
Journal of Differential Equations, 252(1):505–540, 2012.

[25]
Gregory Verchota.
Layer potentials and regularity for the Dirichlet problem for
Laplace's equation in Lipschitz domains.
Journal of functional analysis, 59(3):572–611, 1984.