$\operatorname{div}\left(\frac{|x|A(x)\nabla|x|}{\mu(x)}-m\right)=\frac{|x|\operatorname{div}(A(x)\nabla|x|)}{\mu(x)}+1-m-\frac{|x|\langle
A(x)\nabla|x|,\nabla\mu(x)\rangle}{\mu^{2}(x)}\\\
=\frac{|x|}{\mu(x)}\left(\operatorname{div}(A(x)\nabla|x|)-\frac{m-1}{|x|}\right)+(m-1)\left(\frac{1}{\mu(x)}-1\right)-\frac{|x|\langle
A(x)\nabla|x|,\nabla\mu(x)\rangle}{\mu^{2}(x)}.$

Since

$|\langle
A(x)\nabla|x|,\nabla\mu(x)\rangle|\leq\left(|A(x)-Id|+1\right)|\nabla\mu(x)|\leq
C.$

(by item _5._), we have that _7._ follows from _3., 4._ and _6._ ∎

Set $B_{r}:=B_{r}(0)$, $u=(u_{1},\ldots,u_{\ell})$,
$|u|^{2}:=\sum_{i=1}^{\ell}u_{i}^{2}$, $\langle A(x)\nabla u,\nabla
u\rangle:=\sum_{i=1}^{\ell}\langle A(x)\nabla u_{i},\nabla u_{i}\rangle$ and
$f(x,u):=(f_{1}(x,u_{1}),\ldots,f_{\ell}(x,u_{\ell}))$. Define

$\displaystyle E(r)$
$\displaystyle:=\frac{1}{r^{m-2}}\int_{B_{r}}\left(\langle A(x)\nabla u,\nabla
u\rangle-\langle
f(x,u),u\rangle\right)dx=\frac{1}{r^{m-2}}\sum_{i=1}^{\ell}\int_{B_{r}}\left(\langle
A(x)\nabla u_{i},\nabla u_{i}\rangle-f_{i}(x,u_{i})u_{i}\right)dx,$
$\displaystyle H(r)$ $\displaystyle:=\frac{1}{r^{m-1}}\int_{\partial
B_{r}}\mu(x)|u|^{2}\,d\sigma=\frac{1}{r^{m-1}}\sum_{i=1}^{\ell}\int_{\partial
B_{r}}\mu(x)u_{i}^{2}\,d\sigma,$

and the Almgren quotient

$N(r):=\frac{E(r)}{H(r)}\qquad\text{ whenever }H(r)\neq 0.$

The main purpose of this section is to prove the following result.

######  Theorem C.3 (Monotonicity formula, case $A(0)=Id$).

Take $\omega\Subset\Omega$ such that $0\in\omega$. There exist $C,\bar{r}>0$
(depending only on the dimension $m$, the ellipticity constant $\theta$, the
domain $\omega$, and on an upper bound for $\|DA\|_{\infty}$ and
$\|u\|_{\infty}$) such that, whenever $r\in(0,\bar{r})$, we have that
$H(r)\neq 0$, the function $N$ is absolutely continuous, and

$N^{\prime}(r)\geq-C(N(r)+1).$

In particular $e^{Cr}(N(r)+1)$ is a non decreasing function and the limit
$N(0^{+}):=\lim_{r\to 0^{+}}N(r)$ exists and is finite. Moreover,

$\left|\left(\log H(r)\right)^{\prime}-\frac{2}{r}N(r)\right|\leq
C\qquad\text{ for every }r\in(0,\bar{r}).$ (C.1)

We present here a sketch of the proof of this result, which is based in ideas
from [55, 41, 47], with adaptations obtained from [37, 31, 30, 50] allowing us
to deal with variable coefficients. Our main goal in carrying out this proof
and in repeating some computations is to focus on the dependence of the
constants $C$ and $\bar{r}$, specially on the matrix $A$, something that was
not needed in previous papers.

######  Lemma C.4.

We have

$E(r)=\frac{1}{r^{m-2}}\sum_{i=1}^{\ell}\int_{\partial B_{r}}u_{i}\langle
A(x)\nabla u_{i},\nu(x)\rangle\ d\sigma.$

###### Proof.

For $\lambda<0$, we have

$\displaystyle E_{\lambda}(r):=$
$\displaystyle\frac{1}{r^{m-2}}\int_{B_{r}}\left(\langle A(x)\nabla
u_{\lambda},\nabla u_{\lambda}\rangle-\langle
f(x,u_{\lambda}),u_{\lambda}\rangle\right)dx-\frac{1}{r^{m-2}}\int_{B_{r}}a(x)\mathop{\sum_{i,j=1}^{\ell}}_{i\neq
j}\lambda|u_{\lambda,j}|^{\gamma+1}|u_{\lambda,i}|^{\gamma+1}dx$
$\displaystyle=$
$\displaystyle\frac{1}{r^{m-2}}\sum_{i=1}^{\ell}\int_{\partial
B_{r}}u_{\lambda,i}\langle A(x)\nabla u_{\lambda,i},\nu(x)\rangle\ d\sigma,$

where the last identity is a consequence of testing the $i$–th equation in
(B.2) by $u_{\lambda,i}$, integrating by parts, and taking the sum over $i$.
Passing to the limit as $\lambda\to-\infty$ and using assumption $(H4)$,
yields the claim. ∎

######  Lemma C.5.

Let $\omega\Subset\Omega$ be such that $0\in\omega$. There exist constants
$C,\bar{r}>0$, depending only on the dimension $m$, on $\omega$ and on an
upper bound for $\|DA\|_{\infty}$, such that

$\left|H^{\prime}(r)-\frac{2}{r}E(r)\right|\leq CH(r)\qquad\text{ for every
}r\in(0,\bar{r}).$

In particular, we get (C.1).

###### Proof.

We combine the proof of [50, Lemma 3.3] with the estimates from Lemma C.2. By
[50, eq. (3.3)],

$\displaystyle\frac{d}{dr}\int_{\partial B_{r}}\mu(x)u_{i}^{2}=2\int_{\partial
B_{r}}u_{i}\langle A(x)\nabla u_{i},\nabla|x|\rangle+\int_{\partial
B_{r}}u_{i}^{2}\operatorname{div}(A(x)\nabla|x|),$

which, together with Lemma C.4, yields

$\displaystyle H^{\prime}(r)=$
$\displaystyle\frac{1-m}{r}H(r)+\frac{2}{r^{m-1}}\sum_{i=1}^{\ell}\int_{\partial
B_{r}}u_{i}\langle A(x)\nabla
u_{i},\nu(x)\rangle+\frac{2}{r^{m-1}}\sum_{i=1}^{\ell}\int_{\partial
B_{r}}u_{i}^{2}\operatorname{div}(A(x)\nabla|x|)$ $\displaystyle=$
$\displaystyle\frac{1-m}{r}H(r)+\frac{2}{r}E(r)+\frac{1}{r^{m-1}}\sum_{i=1}^{\ell}\int_{\partial
B_{r}}u_{i}^{2}\operatorname{div}(A(x)\nabla|x|)$

The conclusion now follows from the estimate

$\left|\frac{1-m}{r}H(r)+\frac{1}{r^{m-1}}\sum_{i=1}^{\ell}\int_{\partial
B_{r}}u_{i}^{2}\operatorname{div}(A(x)\nabla|x|)\right|\\\
\leq\frac{1}{r^{m-1}}\sum_{i=1}^{\ell}\int_{\partial
B_{r}}\mu(x)u_{i}^{2}\left(\frac{\operatorname{div}(A(x)\nabla|x|)}{\mu(x)}-\frac{1-m}{r}\right)\leq
CH(r),$

where the constant $C>0$ arises from items _3._ and _6._ in Lemma C.2. ∎

Define

$Z(x):=\frac{A(x)x}{\mu(x)}\sim x\text{ as }x\to 0.$

From now on we use the summation convention for repeated indices, unless
stated otherwise.

######  Lemma C.6 (Local Pohozaev-type identities).

For every $r>0$ such that $B_{r}\subset\Omega$, we have the following identity
(where $A=(a_{ij})$)

$\displaystyle r\int_{\partial B_{r}}\langle A\nabla u_{i},\nabla
u_{i}\rangle$ $\displaystyle=\int_{B_{r}}\operatorname{div}Z\langle A\nabla
u_{i},\nabla u_{i}\rangle+2\int_{B_{r}}f_{i}(x,u_{i})\langle\nabla
u_{i},Z\rangle$ $\displaystyle+2\int_{\partial B_{r}}\langle Z,\nabla
u_{i}\rangle\langle A\nabla u_{i},\nu\rangle+\int_{B_{r}}\langle Z,\nabla
a_{hl}\rangle\frac{\partial u_{i}}{\partial x_{h}}\frac{\partial
u_{i}}{\partial x_{l}}-2\int_{B_{r}}a_{hl}\frac{\partial Z_{j}}{\partial
x_{h}}\frac{\partial u_{i}}{\partial x_{j}}\frac{\partial u_{i}}{\partial
x_{l}}.$ (C.2)

###### Proof.

From system (B.2) we derive an identity for the $u_{\lambda,i}$’s, and then
pass to the limit as $\lambda\to-\infty$. For each $i$, from the divergence
theorem and the definition of $\mu(x)$ and $Z(x)$, we derive

$\displaystyle r\int_{\partial B_{r}}\langle A\nabla u_{\lambda,i},\nabla
u_{\lambda,i}\rangle$ $\displaystyle=\int_{\partial B_{r}}\langle A\nabla
u_{\lambda,i},\nabla u_{\lambda,i}\rangle\langle
Z,\nu\rangle=\int_{B_{r}}\operatorname{div}(\langle A\nabla
u_{\lambda,i},\nabla u_{\lambda,i}\rangle Z)$
$\displaystyle=\int_{B_{r}}\operatorname{div}Z\langle A\nabla
u_{\lambda,i},\nabla u_{\lambda,i}\rangle+\int_{B_{r}}\langle Z,\nabla(\langle
A\nabla u_{\lambda,i},\nabla u_{\lambda,i}\rangle)\rangle.$

Following now [50, Lemma A.1] (see also [31, Lemma A.9]), we obtain

$\int_{B_{r}}\langle Z,\nabla(\langle A\nabla u_{\lambda,i},\nabla
u_{\lambda,i}\rangle)\rangle=\int_{B_{r}}\langle Z,\nabla
a_{hl}\rangle\frac{\partial u_{\lambda,i}}{\partial x_{h}}\frac{\partial
u_{\lambda,i}}{\partial x_{l}}+2\int_{\partial B_{r}}\langle Z,\nabla
u_{\lambda,i}\rangle\langle A\nabla u_{\lambda,i},\nu\rangle\\\
-2\int_{B_{r}}a_{hl}\frac{\partial Z_{j}}{\partial x_{h}}\frac{\partial
u_{\lambda,i}}{\partial x_{j}}\frac{\partial u_{\lambda,i}}{\partial
x_{l}}+2\int_{B_{r}}\langle Z,\nabla
u_{i}\rangle\Big{(}f_{i}(x,u_{\lambda,i})+a(x)\lambda\mathop{\sum_{j=1}^{\ell}}_{j\neq
i}|u_{\lambda,j}|^{\gamma+1}|u_{\lambda,i}|^{\gamma-1}u_{\lambda,i}\Big{)}.$

Passing to limit as $\lambda\to-\infty$, the conclusion will follow once we
prove the following claim

$\sum_{i=1}^{\ell}\int_{B_{r}}\langle Z,\nabla u_{\lambda,i}\rangle
a(x)\lambda\mathop{\sum_{j=1}^{\ell}}_{j\neq
i}|u_{\lambda,j}|^{\gamma+1}|u_{\lambda,i}|^{\gamma-1}u_{\lambda,i}\to
0\qquad\text{ as $\lambda\to-\infty$.}$ (C.3)

This statement is true due to the variational character of the coupling term;
in fact, as $\lambda\to-\infty$,

$\sum_{i=1}^{\ell}\int_{B_{r}}\langle Z,\nabla u_{\lambda,i}\rangle
a(x)\lambda\mathop{\sum_{j=1}^{\ell}}_{j\neq
i}|u_{\lambda,j}|^{\gamma+1}|u_{\lambda,i}|^{\gamma-1}u_{\lambda,i}=\sum_{i<j}\int_{B_{r}}\langle
Z,\nabla(|u_{\lambda,i}|^{\gamma+1}|u_{\lambda,j}|^{\gamma+1})\rangle\frac{a(x)\lambda}{\gamma+1}\\\
=-\sum_{i<j}\int_{B_{r}}\operatorname{div}Z\frac{a(x)\lambda}{\gamma+1}|u_{\lambda,i}|^{\gamma+1}|u_{\lambda,j}|^{\gamma+1}-\sum_{i<j}\int_{B_{r}}\langle
Z,\nabla
a(x)\rangle\lambda|u_{\lambda,i}|^{\gamma+1}|u_{\lambda,j}|^{\gamma+1}\\\
+\sum_{i<j}\int_{\partial B_{r}}\langle Z,\nu\rangle
a(x)\lambda|u_{\lambda,i}|^{\gamma+1}|u_{\lambda,j}|^{\gamma+1}\to 0$

by assumption $(H5)$ and because $\operatorname{div}Z=m+O(r)$ (see item _7._
in Lemma C.2). This proves the claim (C.3) and completes the proof of Lemma
C.6. ∎

Let

$\widetilde{E}(r):=\frac{1}{r^{m-2}}\int_{B_{r}}\langle A(x)\nabla u,\nabla
u\rangle=\frac{1}{r^{m-2}}\int_{B_{r}}\langle A(x)\nabla u_{i},\nabla
u_{i}\rangle.$

######  Lemma C.7.

We have

$\displaystyle\widetilde{E}^{\prime}(r)=$
$\displaystyle\frac{2}{r^{m-2}}\sum_{i=1}^{\ell}\int_{\partial
B_{r}}\frac{\langle A\nabla
u_{i},\nu\rangle^{2}}{\mu}+\frac{2}{r^{m-1}}\int_{B_{r}}f_{i}(x,u_{i})\langle
Z,\nabla u_{i}\rangle$ $\displaystyle+\frac{1}{r^{m-1}}\int_{B_{r}}\langle
Z,\nabla a_{hl}\rangle\frac{\partial u_{i}}{\partial x_{h}}\frac{\partial
u_{i}}{\partial
x_{l}}+\frac{1}{r^{m-1}}\int_{B_{r}}\operatorname{div}(Z-x)\langle A\nabla
u_{i},\nabla u_{i}\rangle$
$\displaystyle-\frac{2}{r^{m-1}}\int_{B_{r}}a_{hl}\frac{\partial(Z_{j}-x_{j})}{\partial
x_{h}}\frac{\partial u_{i}}{\partial x_{j}}\frac{\partial u_{i}}{\partial
x_{l}}.$ (C.4)

In particular, given $\omega\Subset\Omega$ with $0\in\omega$, there exist
constants $C,\bar{r}>0$ (depending only on $m$, $\theta$, $\omega$ and on an
upper bound for $\|DA\|_{\infty}$) such that, for every $r\in(0,\bar{r})$,

$\left|\widetilde{E}^{\prime}(r)-\frac{2}{r^{m-2}}\sum_{i=1}^{\ell}\int_{\partial
B_{r}}\frac{\langle A\nabla
u_{i},\nu\rangle^{2}}{\mu}-\frac{2}{r^{m-1}}\int_{B_{r}}f_{i}(x,u_{i})\langle
Z,\nabla u_{i}\rangle\right|\leq C\widetilde{E}(r).$

###### Proof.

From Lemma C.6 and since $\displaystyle\frac{2}{r}\int_{\partial B_{r}}\langle
Z,\nabla u_{i}\rangle\langle A\nabla u_{i},\nu\rangle=2\int_{\partial
B_{r}}\frac{\langle A\nabla u_{i},\nu\rangle^{2}}{\mu}$, we have

$\left(\int_{B_{r}}\langle A(x)\nabla u_{i},\nabla
u_{i}\rangle\right)^{\prime}=\frac{m-2}{r}\int_{B_{r}}\langle A\nabla
u_{i},\nabla u_{i}\rangle+\frac{2}{r}\int_{B_{r}}f_{i}(x,u_{i})\langle\nabla
u_{i},Z\rangle+2\int_{\partial B_{r}}\frac{\langle A\nabla
u_{i},\nu\rangle^{2}}{\mu}\\\
+\frac{1}{r}\int_{B_{r}}\operatorname{div}(Z-x)\langle A\nabla u_{i},\nabla
u_{i}\rangle+\frac{1}{r}\int_{B_{r}}\langle Z,\nabla
a_{hl}\rangle\frac{\partial u_{i}}{\partial x_{h}}\frac{\partial
u_{i}}{\partial
x_{l}}-\frac{2}{r}\int_{B_{r}}a_{hl}\frac{\partial(Z_{j}-x_{j})}{\partial
x_{h}}\frac{\partial u_{i}}{\partial x_{j}}\frac{\partial u_{i}}{\partial
x_{l}}.$

As

$\widetilde{E}^{\prime}(r)=\frac{2-m}{r}\widetilde{E}(r)+\frac{1}{r^{m-2}}\left(\int_{B_{r}}\langle
A\nabla u_{i},\nabla u_{i}\rangle\right)^{\prime},$

we conclude that identity (C.4) is true.

Now, by Lemma C.2-_3.,4.,7._ , we have

$\Big{\|}\frac{1}{\mu}-1\Big{\|}_{L^{\infty}(B_{r})},\;\Big{\|}\frac{1}{\mu^{2}}-1\Big{\|}_{L^{\infty}(B_{r})},\;\|\operatorname{div}(Z-x)\|_{L^{\infty}(B_{r})}\,\leq
Cr$

for some constant $C>0$ depending only on the dimension $m$, on $\omega$ and
on an upper bound for $\|DA\|_{\infty}$. Then, using also $(H2)$, we obtain

$\left|\frac{1}{r^{m-1}}\int_{B_{r}}\langle Z,\nabla
a_{hl}\rangle\frac{\partial u_{i}}{\partial x_{h}}\frac{\partial
u_{i}}{\partial
x_{l}}+\frac{1}{r^{m-1}}\int_{B_{r}}\operatorname{div}(Z-x)\langle A\nabla
u_{i},\nabla
u_{i}\rangle-\frac{2}{r^{m-1}}\int_{B_{r}}a_{hl}\frac{\partial(Z_{j}-x_{j})}{\partial
x_{h}}\frac{\partial u_{i}}{\partial x_{j}}\frac{\partial u_{i}}{\partial
x_{l}}\right|\\\ \leq C\frac{1}{r^{m-2}}\int_{B_{r}}\langle A\nabla
u_{i},\nabla u_{i}\rangle=C\widetilde{E}(r)$

(see equations (A.3)–(A.12) in [50] for more details). This completes the
proof. ∎

######  Remark C.8.

_Observe that identities ( C.2) and (C.7), which can be seen as local
Pohozaev-type identities, are equivalent. They correspond to the condition
(G3) for the Laplacian stated in [47] and [55] respectively._

###### Proof of Theorem C.3.

This result now follows from standard arguments. Here, as before, we mainly
verify the dependence of the constants. Within this proof, $O(1)$ will
represent a bounded function of $r$ depending only on $m$, $\theta$, $\omega$
and on an upper bound for $\|DA\|_{\infty}$ (but which is independent of
$\|u\|_{\infty}$). We have, by Lemma C.7,

$\displaystyle E^{\prime}(r)$
$\displaystyle=\widetilde{E}^{\prime}(r)-\frac{2-m}{r^{m-1}}\int_{B_{r}}f_{i}(x,u_{i})u_{i}-\frac{1}{r^{m-2}}\int_{\partial
B_{r}}f_{i}(x,u_{i})u_{i}$
$\displaystyle=\frac{2}{r^{m-2}}\sum_{i=1}^{\ell}\int_{\partial
B_{r}}\frac{\langle A\nabla u_{i},\nu\rangle^{2}}{\mu}+O(1)E(r)+R(r),$ (C.5)

where

$R(r):=\frac{2}{r^{m-1}}\int_{B_{r}}f_{i}(x,u_{i})\langle Z,\nabla
u_{i}\rangle+\frac{O(1)}{r^{m-1}}\int_{B_{r}}f_{i}(x,u_{i})u_{i}-\frac{1}{r^{m-2}}\int_{\partial
B_{r}}f_{i}(x,u_{i})u_{i}.$

By $(H3)$, there exists $\bar{d}$ depending on an upper bound for
$\|u\|_{\infty}$ such that $|f_{i}(x,u_{i})|\leq\bar{d}u_{i}$. This together
with assumption $(H2)$ and Lemma C.2-_2.-3._ , yields
$Z(x)=A(x)x/\mu(x)=O(1)|x|$ as $x\to 0$ and

$\displaystyle|R(r)|$ $\displaystyle\leq
O(1)\,\bar{d}\sum_{i=1}^{\ell}\left(\frac{1}{r^{m-2}}\int_{B_{r}}|u_{i}||\nabla
u_{i}|+\frac{1}{r^{m-1}}\int_{B_{r}}u_{i}^{2}+\frac{1}{r^{m-2}}\int_{\partial
B_{r}}u_{i}^{2}\right)$ $\displaystyle\leq
O(1)\,\bar{d}\sum_{i=1}^{\ell}\left(\frac{1}{r^{m-2}}\int_{B_{r}}\langle
A\nabla u_{i},\nabla
u_{i}\rangle+\frac{1}{r^{m}}\int_{B_{r}}u_{i}^{2}+\frac{1}{r^{m-1}}\int_{\partial
B_{r}}u_{i}^{2}\right)$ $\displaystyle\leq
O(1)\,\bar{d}\left(E(r)+H(r)+\frac{1}{r^{m}}\sum_{i=1}^{\ell}\int_{B_{r}}u_{i}^{2}\right).$
(C.6)

Using Poincaré’s inequality (see [55, pp. 279–280] for the details), we
conclude that

$\frac{1}{r^{m}}\sum_{i=1}^{\ell}\int_{B_{r}}u_{i}^{2}\leq
O(1)\,\bar{d}\,(E(r)+H(r))$ (C.7)

for every $r\in(0,\bar{r})$ sufficiently small. Combining (C.5)–(C.6)–(C.7),
we arrive at

$E^{\prime}(r)=\frac{2}{r^{m-2}}\sum_{i=1}^{\ell}\int_{\partial
B_{r}}\frac{\langle A\nabla
u_{i},\nu\rangle^{2}}{\mu}+O(1)\,\bar{d}\,(E(r)+H(r)).$

Recalling from Lemmas C.4 and C.5 that

$E(r)=\frac{1}{r^{m-2}}\sum_{i=1}^{\ell}\int_{\partial B_{r}}u_{i}\langle
A(x)\nabla u_{i},\nu(x)\rangle\qquad\text{ and }\qquad
H^{\prime}(r)=\frac{2}{r}E(r)+O(1)H(r),$

we finally deduce the existence of a constant $C$ with the required properties
such that

$\displaystyle N^{\prime}(r)$
$\displaystyle=\frac{E^{\prime}(r)H(r)-E(r)H^{\prime}(r)}{H^{2}(r)}$
$\displaystyle=\frac{2}{H^{2}(r)r^{2m-3}}\left(\sum_{i=1}^{\ell}\int_{B_{r}}\frac{\langle
A\nabla u_{i},\nu\rangle}{\mu}\sum_{j=1}^{\ell}\int_{\partial B_{r}}\mu
u_{j}^{2}-\Big{(}\sum_{i=1}^{\ell}\int_{\partial B_{r}}u_{i}\langle A\nabla
u_{i},\nu\rangle\Big{)}^{2}\right)$
$\displaystyle\qquad+\frac{1}{H^{2}(r)}\left(O(1)\,\bar{d}\,H(r)(E(r)+H(r))+O(1)E(r)H(r)\right)$
$\displaystyle\geq-C(N(r)+1),$

and $e^{Cr}(N(r)+1))$ is nondecreasing whenever $H(r)\neq 0$. Now observe that
$H$ solves $H^{\prime}(r)=a(r)H(r)$ with $a(r)=\frac{2}{r}N(r)+O(1)r$, and by
the existence and uniqueness theorem for this ODE we have that $H>0$ for
sufficiently small $r>0$. Finally, the validity of (C.1) is given by Lemma
C.5. ∎

###  C.2 Almgren’s monotonicity formula: the general case

We have proved a monotonicity formula under the assumption that $A(0)=Id$. The
general case can be reduced to this case in the following way: let
$A(x_{0})^{\frac{1}{2}}$ be the square root of the (positive definite) matrix
$A(x_{0})$, that is, the unique positive definite matrix whose square is
$A(x_{0})$. We recall that $A(x_{0})^{\frac{1}{2}}$ is also symmetric, it
commutes with $A(x_{0})$, it has real entries and that the map $x_{0}\mapsto
A(x_{0})^{\frac{1}{2}}$ is continuous (see for instance [35]). Following [30,
50], we set

$\displaystyle T_{x_{0}}x:=x_{0}+{A}(x_{0})^{\frac{1}{2}}x,$ $\displaystyle
A_{x_{0}}(x):=A(x_{0})^{-\frac{1}{2}}A(T_{x_{0}}x)A(x_{0})^{-\frac{1}{2}},$
$\displaystyle\mu_{x_{0}}(x):=\left\langle
A_{x_{0}}\frac{x}{|x|},\frac{x}{|x|}\right\rangle,$ $\displaystyle
f_{x_{0}}(x,s):=f(T_{x_{0}}x,s),$ $\displaystyle
v_{i,x_{0}}:=u_{i}(T_{x_{0}}x).$

Observe that $A_{x_{0}}(0)=Id$. Let now

$N(x_{0},u,r):=\frac{E(x_{0},u,r)}{H(x_{0},u,r)},$

where

$\displaystyle E(x_{0},u,r):=\frac{1}{r^{N-2}}\int_{B_{r}(0)}\Big{(}\langle
A_{x_{0}}\nabla v_{x_{0}},\nabla v_{x_{0}}\rangle-\langle
f_{x_{0}}(x,v_{x_{0}}),v_{x_{0}}\rangle\Big{)}dx$ $\displaystyle
H(x_{0},u,r):=\frac{1}{r^{N-1}}\int_{\partial
B_{r}(0)}\mu_{x_{0}}(x)|v_{x_{0}}|^{2}.$

These quantities can be expresed in terms of the original function $u$ in the
ellipsoidal set

$\mathcal{E}_{r}(x_{0}):=\\{x\in\mathbb{R}^{m}:\
|A(x_{0})^{-\frac{1}{2}}(x-x_{0})|<r\\}.$

Namely, by a change of variables one has

$\displaystyle\int_{B_{r}(0)}\langle A_{x_{0}}\nabla v_{x_{0}},\nabla
v_{x_{0}}\rangle=\det({A(x_{0})}^{-\frac{1}{2}})\int_{\mathcal{E}_{r}(x_{0})}\langle
A\nabla u,\nabla u\rangle,$ $\displaystyle\int_{B_{r}(0)}\langle
f_{x_{0}}(x,v_{x_{0}}),v_{x_{0}}\rangle=\det({A(x_{0})}^{-\frac{1}{2}})\int_{\mathcal{E}_{r}(x_{0})}\langle
f(x,u),u\rangle,$ $\displaystyle\int_{\partial
B_{r}(0)}\mu_{x_{0}}(x)|v_{x_{0}}(x)|^{2}\,d\sigma(x)=\int_{\partial\mathcal{E}_{r}(x_{0})}b_{x_{0}}(y)|u(y)|^{2}\,d\sigma(y),$

where $b_{x_{0}}(y):=c(x_{0},y)|A(x_{0})^{-\frac{1}{2}}(y-x_{0})|^{-2}\langle
A(x_{0})^{-1}A(y)A(x_{0})^{-1}y,y\rangle$, $c(x_{0},y)$ being the dilation
coefficient/tangential Jacobian (see for instance [40, Chapter 11]), which is
continuous and positive.

######  Theorem C.9 (Monotonicity formula, general case).

Take $\omega\Subset\Omega$ and let $u$ be as before. Then there exist
$C,\bar{r}>0$ (depending on the dimension $m$, the ellipticity constant
$\theta$ and the domain $\omega$, but _independent_ from $x_{0}$) such that,
whenever $r\in(0,\bar{r})$ and $x_{0}\in\omega$, we have that $v_{x_{0}}$
satisfies identities (C.2) and (C.4), $H(x_{0},u,r)\neq 0$, the function
$r\mapsto N(x_{0},u,r)$ is absolutely continuous, and

$\frac{\partial}{\partial r}N(x_{0},u,r)\geq-C(N(x_{0},u,r)+1).$

In particular, $e^{Cr}(N(x_{0},u,r)+1)$ is nondecreasing and the limit
$N(x_{0},u,0^{+}):=\lim_{r\to 0^{+}}N(x_{0},u,r)$ exists and is finite.
Moreover,

$\left|\frac{\partial}{\partial r}\log H(x_{0},u,r)-\frac{2}{r}N(r)\right|\leq
C\qquad\text{ for every }r\in(0,\bar{r}).$ (C.8)

###### Proof.

This is basically a direct consequence of Theorem C.3. The only thing left to
check is the dependence of the constants. But this is straightforward by
observing that $\|v_{i,x_{0}}\|_{\infty}=\|u_{i}\|_{\infty}$ for every
$i=1,\ldots,\ell$, and that

$\displaystyle\|DA_{x_{0}}(x)\|_{\infty}=\|DA(T_{x_{0}}x)A(x_{0})^{\frac{1}{2}}\|_{\infty},$

which is uniformly bounded for $x\in\omega$, because of $(H2)$ and the
continuity of the map $x_{0}\mapsto A(x_{0})^{\frac{1}{2}}$. This allows to
take $C$ and $\bar{r}$ which are independent of $x_{0}$. ∎

Now that we have shown an Almgren’s monotonicity formula with constants
independent of $x_{0}$ in any compactly contained subset of $\Omega$, we have
all tools required to conclude the proof of the main result of this appendix.

###  C.3 Proof of the regularity result

###### End of the proof of Theorem C.1.

$(a):$ To prove that the functions $u_{i}$ are Lipschitz continuous for any
$i=1,...,\ell$ we argue as in the proof of [47, Proposition 3.4], with minimal
adaptations at this point:

  * •

we have an elliptic divergence type operator instead of the pure Laplacian
operator, therefore the estimates will depend on the ellipticity constant
$\theta$;

  * •

the identity (C.2) plays the role of the identity in the last assumption of
[47, Proposition 3.4], while the monotonicity formula (Theorem C.9) plays the
role of the monotonicity formula [47, Theorem 3.3].

For related proofs of Lipschitz continuity in similar contexts, see also [41,
Section 4.1] or [54, Section 2.4], the latter being a more detailed version of
the former.

$(b):$ Regarding the regularity properties of $\Gamma:=\\{x\in\Omega:\
u_{i}(x)=0\ \forall i=1,\ldots,\ell\\}$, we argue as in the proof of [55,
Theorem 1.1]:

  * •

again, here we have an elliptic divergence type operator instead of the
Laplacian;

  * •

formula (C.4) plays the role of the expression for the derivative of
$\widetilde{E}(x_{0},U,r)$ in the statement of [55, Theorem 1.1] (see
condition (G3) therein), while our Theorem C.9 plays the role of the
monotonicity formula [55, Theorem 2.2].

At a regular point $x_{0}\in\Gamma$, identity (C.4) (or, equivalently, the
local Pohozaev identities (C.2)) together with the equations

$-\operatorname{div}(A(x)\nabla u_{i})=f_{i}(x,s)\quad\text{ in the open set
}\\{x\in\Omega:\ u_{i}(x)>0\\},\qquad i=1,\ldots,\ell,$

given by assumption $(H6)$, provide the free boundary condition

$\lim_{x\to x_{0}^{+}}\langle A(x)\nabla u_{i},\nabla u_{i}\rangle=\lim_{x\to
x_{0}^{-}}\langle A(x)\nabla u_{j},\nabla u_{j}\rangle\neq 0,$

where $x\to x_{0}^{\pm}$ are the limits taken from opposite sides of $\Gamma$;
see [55, Section 2] for the details.

For related proofs of regularity in similar contexts, see also [47, Theorem
1.7] or [54, Chapter 3]. ∎

######  Remark C.10.

_We remark that Theorem C.1 can be seen as a direct consequence of Theorem 7.1
in [55]. However, since the latter result is presented without proof, we have
decided to write this appendix and give all the necessary details._

### Acknowledgement

The authors would like to thank N. Soave for pointing out some references
regarding monotonicity formulas for equations with variable coefficients.

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Mónica Clapp
Instituto de Matemáticas

Universidad Nacional Autónoma de México

Circuito Exterior, Ciudad Universitaria

04510 Coyoacán, Ciudad de México, Mexico

<EMAIL_ADDRESS>
Angela Pistoia
Dipartimento di Metodi e Modelli Matematici

La Sapienza Università di Roma

Via Antonio Scarpa 16

00161 Roma, Italy

<EMAIL_ADDRESS>
Hugo Tavares
Departamento de Matemática do Instituto Superior Técnico

Universidade de Lisboa

Av. Rovisco Pais

1049-001 Lisboa, Portugal

<EMAIL_ADDRESS>