Up to the boundary gradient estimates for viscosity solutions to nonlinear free boundary problems with unbounded measurable ingredients

Abstract

In this paper, we prove up to the boundary gradient estimates for viscosity solutions to inhomogeneous nonlinear Free Boundary Problems (FBP) governed by fully nonlinear and quasilinear elliptic equations with unbounded measurable ingredients. Here, we build upon our previous results in [9] to construct Inhomogeneous Pucci Barriers (IPB) for the Pucci extremal equations with unbounded coefficients. Using these barriers, we obtain a version of a boundary growth type lemma for inhomogeneous nonlinear equations that may be of independent interest. In a certain way, this lemma detects the expansion of the level sets of supersolutions from the boundary to the interior. The use of this boundary growth type lemma together with the geometry of IPB bridge the interchanging information between the free boundary condition and Dirichlet boundary data on the free and fixed boundary respectively. This produces an estimate on the trace of solutions to FBP along the fixed boundary. This way, control of such solutions (up to the boundary) by the distance to the negative phase is obtained. Finally, this distance control combined with the PDE boundary gradient estimates render our final result.

Appendix: On The Comparison Principle and Additional Properties

1.1 Fully Nonlinear

Lemma 14.1

Let $U\subset {\mathbb {R}}^{n}$ be an open set and $f,g\in L^{n}(U)$. Suppose that ${\mathcal {P}}_{\gamma }^{\pm }[u]= f$ in U in the $L^{n}$-viscosity sense and that $v\in W_{loc}^{2,n}(U)$ is a strong solution to ${\mathcal {P}}_{\gamma }^{\pm }[v]= g$ in U. Then, $u-v\in S(\gamma , f-g)$ in U.

Proof

The proof is done in Lemma 7.2 in [8] although there the drift coefficient is bounded. It is easy to see that the argument does not depend on the boundedness of the drift coefficient and thus can be repeated here. $\square$

Proposition 14.1

(Comparison Principle) Suppose $U\subset {\mathbb {R}}^{n}$ is a bounded open set. Assume that

$$\begin{aligned} {\mathcal {P}}_{\gamma }^{-}(M-N, p-q,x) \le F(x,p,M) -F(x,q,N) \le {\mathcal {P}}_{\gamma }^{+}(M-N, p-q,x) \end{aligned}$$

(UE)

$$\begin{aligned} \forall p\in {\mathbb {R}}^{n}, \forall M\in {\mathcal {S}}^{n\times n},\ \text { and for } \ a.e.\ x\in U. \end{aligned}$$

Assume that $u,v\in C^{0}({\overline{U}})$ with $v\in W^{2,n}_{loc}(U)$ be such that

$$\begin{aligned} F(D^2u(x), \nabla v(x), x)\ge & {} f(x) \quad \text { in } \ L^{n}-\text {viscosity sense in } \ U \\ F(D^2v(x), \nabla v(x), x)\le & {} f(x) \quad a.e \ \text { in } \ U. \end{aligned}$$

This way

$$\begin{aligned} u\le v \quad \text { along } \ \partial U \Longrightarrow u\le v \quad \text { in } \ U. \end{aligned}$$

Moreover, the roles of $L^{n}-$viscosity and $L^{n}-$strong solutions played by u and v can be swapped.

Proof

The Proposition above is essentially a consequence of the ABP estimate and the fact the $L^{p}-$strong solutions are also viscosity solutions. Both of these facts also holds in the case $\gamma , f$ are in $L^{q}$ with $q>n$. The ABP estimate is Theorem 2.4 of [54] and the second one is Theorem 3.1 in [54]. Thus, under the possession of these facts, we can repeat the same proof of the bounded drift coefficient case done in Theorem 2.10 in [17]. $\square$

As a Corollary of the Comparison Principle, Proposition 14.1, we indicate the following Lipschitz type estimates in rings that was needed in the proof of Theorem 4.1. As before, we use the following notation

$$\begin{aligned} {\mathcal {A}}_{\frac{r}{2}, r}(x_0){:}{=} \Big \{x\in {\mathbb {R}}^{n}; \ \frac{r}{2}<|x - x_0|<r \Big \}. \end{aligned}$$

Proposition 14.2

Let us assume that $u\in C^{0}(\overline{{\mathcal {A}}_{\frac{r}{2}, r}(x_0)})\cap S^{*}(\gamma ;f)$ in ${\mathcal {A}}_{\frac{r}{2}, r}(x_0)$ where $\gamma \in L_{+}^{q_{0}}({\mathcal {A}}_{\frac{r}{2}, r}(x_0))$ and $f\in L^{q}({\mathcal {A}}_{\frac{r}{2}, r}(x_0))$ under (EC). Assume that $u=0$ along $\partial {\mathcal {A}}_{\frac{r}{2}, r}(x_0)$. Then, the following estimate holds

$$\begin{aligned} |u(x)|\le C \cdot r^{1-\frac{n}{q}}\cdot ||f||_{L^{q}({\mathcal {A}}_{\frac{r}{2}, r}(x_0))}\quad \forall x\in \overline{{\mathcal {A}}_{\frac{r}{2}, r}(x_0))} \end{aligned}$$

where

$$\begin{aligned} C=C(n,\lambda ,\Lambda , q, q_{0}, r^{1-\frac{n}{q_{0}}}\cdot ||\gamma ||_{L^{q_{0}}( {\mathcal {A}}_{\frac{r}{2},r})})>0. \end{aligned}$$

Proof

The proof goes essentially ipsis-literis to the proof of Lemma 7.3 in [8] that was originally hinted to us by Ovidiu Savin. The only difference is that now the drift coefficient $\gamma \in L^{q_{0}}$ with $q_0\ge q>n$ instead of $\gamma \in L^{\infty }.$ So, we argue as in Lemma 7.3 in [8] just by replacing the existence Theorem (to barriers in (7.54) in [8]) and the $C^{1,\alpha }$ up to the boundary estimate by the case where the drift coefficient $\gamma$ is now unbounded. These results are respectively Theorem 2.4 in [55] and Theorem 1.1 in [68]. $\square$

1.2 Quasilinear

We start by proving a technical approximation Lemma.

We define for an open set $U\subset {\mathbb {R}}^{n}$

$$\begin{aligned} W_{c}^{1,G}(U){:}{=}\Big \{u\in W^{1,G}(U); \ supp(u) \ \text { is compact } \Big \}, \quad \end{aligned}$$

Lemma 14.2

Let $V \subset {\mathbb {R}}^{n}$ be a bounded open set in ${\mathbb {R}}^{n}$. Assume that $0\le u\in W_{c}^{1,G}(V)$. Then,

$$\begin{aligned} \exists \big \{\phi _{\varepsilon }\big \}_{\varepsilon >0}\subset C_{0}^{\infty }(V), \quad \phi _{\varepsilon }\ge 0 \ \text { such that } \ \phi _{\varepsilon } \rightarrow u \ \text { in } \ W_{0}^{1,G}(V). \end{aligned}$$

Proof

Let $supp(u)\subset V_{0}\subset \subset V_{00}\subset \subset V$ for open sets $V_{0}, V_{00}$. Let $\eta$ be a smooth standard mollifier, i.e, $\eta \in C_{0}^{\infty }({\mathbb {R}}^{n}), \ supp(\eta )\subset B_{1}$, $0\le \eta \le 1$ and $\eta$ radially symmetric such that $\int _{{\mathbb {R}}^{n}}\eta (x)dx=1$. We then set

$$\begin{aligned} \phi _{\varepsilon }(x){:}{=}\int _{V}u(y)\eta _{\varepsilon }(x-y)dy \quad \text { for } \quad x\in U_{\varepsilon }{:}{=}\big \{x\in U; \ dist(x,\partial V)> \varepsilon \big \} \end{aligned}$$

where $\eta _{\varepsilon }(x){:}{=}\varepsilon ^{-n}\eta (x/\varepsilon ).$ For $p=1+\delta _{0}$, we have $u\in W^{1,G}(V_{0})\hookrightarrow W^{1,p}(V_{0})\hookrightarrow L^{1}(V_{0})$ by Theorem 2.2 in [64]. Then for $\varepsilon<<1$, we have $0\le \phi _{\varepsilon }\in C_{c}^{\infty }(V_{00})$ and also that $\nabla \phi _{\varepsilon }(x)= (\nabla u*\eta _{\varepsilon })(x)\ \forall x\in V_{00}$ (by Lemma 7.3 in [42]). The proof is finished by observing that $||\phi _{\varepsilon } - u||_{W^{1,G}(V)} = ||\phi _{\varepsilon } - u||_{W^{1,G}({\mathbb {R}}^{n})}\rightarrow 0$ by Lemma 2.1 in [35].$\square$

Lemma 14.3

Let $V\subset {\mathbb {R}}^{n}$ be a bounded open set and $\Upsilon :W_{c}^{1,G}(V)\rightarrow {\mathbb {R}}$ be a linear functional. Assume that

$$\begin{aligned} \Upsilon (\varphi ) \le 0 \quad \forall \varphi \in C_{0}^{\infty }(V), \ \varphi \ge 0. \end{aligned}$$

(14.1)

If $\Upsilon$ is bounded on the smooth compactly supported functions, i.e, if there exists $C>0$ such that

$$\begin{aligned} |\Upsilon (\varphi )|\le C\cdot ||\varphi ||_{W_{0}^{1,G}(V)}, \quad \forall \varphi \in C_{0}^{\infty }(V). \end{aligned}$$

Then, (14.1) holds for every $\varphi \in W_{c}^{1,G}(V), \ \varphi \ge 0.$

Proof

Let $0\le \xi \in W_{c}^{1,G}(V).$ By Lemma 14.2, there exists $0\le \phi _{\varepsilon } \in C_{0}^{\infty }(V)$ with $\phi _{\varepsilon }\rightarrow \xi$ in $W_{0}^{1,G}(V).$ Now,

$$\begin{aligned} |\Upsilon (\xi )-\Upsilon (\phi _{\varepsilon })| = |\Upsilon (\xi -\phi _{\varepsilon })| \le C\cdot ||\xi -\phi _{\varepsilon }||_{W_{0}^{1,G}(V)}. \end{aligned}$$

Since $\Upsilon (\phi _{\varepsilon })\le 0$ by (14.1), passing to the limit via inequality above we arrive to $\Upsilon (\xi )\le 0$. $\square$

Lemma 14.4

Let $V\subset {\mathbb {R}}^{n}$ be a open bounded set. Let $X\in L^{{\widetilde{G}}}(V;{\mathbb {R}}^{n})$ and define $\Lambda _{X}:W_{c}^{1,G}(V)\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} \Lambda _{X}(\varphi ){:}{=}\int _{V} X(x)\cdot \nabla \varphi (x) dx. \end{aligned}$$

Then, $\Lambda _{X}$ is linear and bounded, i.e,

$$\begin{aligned} |\Lambda _{X}(\varphi )|\le ||X||_{L^{{\widetilde{G}}}(V)} \cdot || \varphi ||_{W_{0}^{1,G}(V)}. \end{aligned}$$

Proof

Linearity is trivial. The boundedness is a direct consequence of Hölder inequality in the context of Orlicz spaces (Remark 9.1 item f). Indeed,

$$\begin{aligned} |\Lambda _{X}(\varphi )|\le \int _{V} |X(x)|\cdot |\nabla \varphi (x)| dx \le 2||X||_{L^{{\widetilde{G}}}(V)} ||\nabla \varphi ||_{L^{{G}}(V)} \le 2 ||X||_{L^{{\widetilde{G}}}(V)} || \varphi ||_{W_{0}^{1,G}(V)}. \end{aligned}$$

$\square$

Lemma 14.5

Let $V\subset {\mathbb {R}}^{n}$ be a bounded open set and $f\in L^{q}(V)$ with $q\ge n$. Consider the functional given by

$$\begin{aligned} \Lambda _{f}:W_{c}^{1,G}(V)\rightarrow {\mathbb {R}}, \quad \Lambda _{f}(\xi )=\int _{V}f(x)\xi (x)dx. \end{aligned}$$

Then, there exist a constant $C=C(||f||_{L^{q}(V)}, |V|,n , \delta _{0}, g_{0})>0$ such that

$$\begin{aligned} |\Lambda _{f}(\xi )|\le C\cdot ||\xi ||_{W_{0}^{1,G}(V)}. \end{aligned}$$

(14.2)

Proof

Let $\xi \in W_{c}^{1,G}(V).$ By regularization process, it is easy to see that $\xi \in W_{c}^{1,G}(V)\subset W_{0}^{1,G}(V)$. Now, we recall once more (by by Theorem 2.2 in [64]) that $W_{0}^{1,G}(V)\hookrightarrow W_{0}^{1,p}(V)$ where $p=1+\delta _{0}>1$. We study three possible cases:

Case 1: Suppose $p>n$. In this case, $W_{0}^{1,p}(V)\hookrightarrow L^{\infty }(V)$ and then if $q'=q/(q-1)$,

$$\begin{aligned} |\Lambda _{f}(\xi )|\le & {} ||f||_{L^{q}(V)}\cdot ||\xi ||_{L^{q'}(V)}\\\le & {} |V|^{\frac{1}{q'}}\cdot ||f||_{L^{q}(V)}\cdot ||\xi ||_{L^{\infty }(V)} \\\le & {} C_{1}\cdot |V|^{\frac{1}{q'}}\cdot ||f||_{L^{q}(V)}\cdot ||\xi ||_{W_{0}^{1,p}(V)}\\\le & {} C_{2}\cdot |V|^{\frac{1}{q'}} ||f||_{L^{q}(V)}\cdot ||\xi ||_{W_{0}^{1,G}(V)} \end{aligned}$$

for $C_{1}, C_{2}>0$ depending on $n,\delta _{0}, g_{0}$.

Case 2: Suppose $p=n$. In this case, $W_{0}^{1,p}(V)\hookrightarrow L^{r}(V)$ for all $r\in [1,\infty )$. Then, as before

$$\begin{aligned} |\Lambda _{f}(\xi )|\le & {} ||f||_{L^{q}(V)}\cdot ||\xi ||_{L^{q'}(V)}\\\le & {} C_{1} \cdot ||f||_{L^{q}(V)}\cdot ||\xi ||_{W_{0}^{1,p}(V)}\\\le & {} C_{2}\cdot ||f||_{L^{q}(V)}\cdot ||\xi ||_{W_{0}^{1,G}(V)} \end{aligned}$$

where $C_{1}, C_{2}>0$ depend on $n, \delta _{0}, g_{0}, q$.

Case 3: Suppose $1<p<n$. In this case, $W_{0}^{1,p}(V)\hookrightarrow L^{p*}(V)$ with $p^{*}=np/(n-p)$. Observe that $(p^{*})' = np/(np-n+p).$ Now, we have

$$\begin{aligned} p>1 \Longleftrightarrow n<np \Longleftrightarrow p< np-n+p \Longleftrightarrow (p^{*})'=\frac{np}{np-n+p} < n. \end{aligned}$$

This way, since we know that $p=1+\delta _{0}>1$ then $(p^{*})'<n\le q$. Thus,

$$\begin{aligned} |\Lambda _{f}(\xi )|\le & {} ||f||_{L^{(p^{*})'}(V)}\cdot ||\xi ||_{L^{p^{*}}(V)}\\\le & {} C_{1}\cdot |V|^{\frac{1}{(p^{*})'}-\frac{1}{q}}\cdot ||f||_{L^{q}(V)}\cdot ||\xi ||_{W_{0}^{1,p}(V)} \\\le & {} C_{2}\cdot |V|^{\frac{1}{(p^{*})'}-\frac{1}{q}}\cdot ||f||_{L^{q}(V)}\cdot ||\xi ||_{W_{0}^{1,G}(V)} \end{aligned}$$

where $C_{1}, C_{2}>0$ depend on $n, \delta _{0}, g_{0}$. Thus, in any case, there exists a constant $C>0$ with the dependence as indicated in the statement so that

$$\begin{aligned} |\Lambda _{f}(\varphi )|\le C\cdot ||\varphi ||_{W_{0}^{1,G}(V)}. \end{aligned}$$

(14.3)

$\square$

Lemma 14.6

In the definition of subsolution given in (3.5) we can take test functions $0\le \varphi \in W_{c}^{1,G}(\Omega )$ provided $f\in L_{loc}^{n}(\Omega )$.

Proof

Indeed, for each $V\subset \subset \Omega$, we set $\Upsilon _{V}:W_{c}^{1,G}(V)\rightarrow {\mathbb {R}}$ given by

$$\begin{aligned} \Upsilon _{V}(\varphi ){:}{=} \int _{V} H_{g}(\nabla u)\nabla u\cdot \nabla \varphi ~dx+ \int _{V} f(x)\varphi (x)dx = \Lambda _{X}(\varphi )+\Lambda _{f}(\varphi ) \end{aligned}$$

where $X(x)=H_{g}(\nabla u)\nabla u(x)$. In order to prove the result, it is enough to prove that

$$\begin{aligned} \Upsilon _{V}(\varphi )\le 0 \quad \forall \varphi \in W_{c}^{1,G}(V), \quad \varphi \ge 0. \end{aligned}$$

Since u is a subsolution, what we do know is that

$$\begin{aligned} \Upsilon _{V}(\varphi )\le 0 \quad \forall \varphi \in C_{0}^{\infty }(V), \quad \varphi \ge 0. \end{aligned}$$

We observe now that

$$\begin{aligned} \int _{V}{\widetilde{G}}(|X(x)|)dx= & {} \int _{V} {\widetilde{G}}\Big (\big |H_{g}(\nabla u)\nabla u\big |\Big )dx\le \int _{V} {\widetilde{G}}\Big (g(|\nabla u|)\Big )dx\\ {}\le & {} g_{0}\int _{V}G(|\nabla u|)dx =T<\infty \end{aligned}$$

Thus, $||X||_{L^{{\widetilde{G}}}(V)} <\infty$ by Remark 9.1 item e). By Lemma 14.4 and Lemma 14.5, we conclude that $\Upsilon _{V}$ is bounded in $W_{c}^{1,G}(V)$. The result follows now immediately from Lemma 14.3. $\square$

Now, we are in conditions to prove the Comparison Principle for quasilinear equations.

We consider equations of the form in $U\subset {\mathbb {R}}^{n}$ an open set in ${\mathbb {R}}^{n}$

$$\begin{aligned} Q_{{\mathcal {A}}}[u] {:}{=}div(A(x,\nabla u) = f \quad \text { in } \ \ U. \end{aligned}$$

Here, ${\mathcal {A}}:\Omega \times {\mathbb {R}}^{n}\rightarrow {\mathbb {R}}^{n}$ are continuous satisfying the following conditions

The concepts of solutions (sub and supersolutions) are defined as in (3.5).

Proposition 14.3

(Comparison principle for the quasilinear case with local integrabiltiy) Let $\Omega$ be an open bounded set. Assume $u,v\in W_{loc}^{1,G}(\Omega )$ and $f\in L^{1}(\Omega )$ be such that $Q_{{\mathcal {A}}}[u] \ge f$ and $Q_{{\mathcal {A}}}[v]\le f$ in the distributional sense in $\Omega$. Suppose the operator ${\mathcal {A}}$ satisfy the properties $Q_{1}), Q_{2}), Q_{3})$. This way, if $u,v\in C^{0}({\overline{\Omega }})$ with $u\le v$ on $\partial \Omega$ then $u\le v$ along $\Omega$.

Proof

Writing down the weak formulation of the equations, we have

$$\begin{aligned}&\int _{\Omega } {\mathcal {A}}(x, \nabla u(x))\cdot \nabla \varphi (x) dx \le -\int _{\Omega }f(x)\varphi (x)dx \quad \forall \varphi \in C_{0}^{\infty }(\Omega ), \quad \varphi \ge 0.\\&-\int _{\Omega } {\mathcal {A}}(x, \nabla v(x))\cdot \nabla \varphi (x) dx \le \int _{\Omega }f(x)\varphi (x)dx \quad \forall \varphi \in C_{0}^{\infty }(\Omega ), \quad \varphi \ge 0. \end{aligned}$$

Adding the inequalities above, we arrive to

$$\begin{aligned} \int _{\Omega } \Big ({\mathcal {A}}(x, \nabla u(x)) - {\mathcal {A}}(x, \nabla v(x)) \Big )\cdot \nabla \varphi (x) ~dx \le 0\quad \forall \varphi \in C_{0}^{\infty }(\Omega ), \quad \varphi \ge 0. \end{aligned}$$

(14.4)

We now set the vector fields

$$\begin{aligned} X_{u}(x){:}{=}{\mathcal {A}}(x,\nabla u(x)), \quad X_{v}{:}{=}{\mathcal {A}}(x,\nabla v(x)) \ \text { for } \ x\in \Omega . \end{aligned}$$

Observe that by ($Q_{2}$)

$$\begin{aligned} {\widetilde{G}}(|X_{u}(x)|)\le & {} {\widetilde{G}}\Big (\Lambda _{0}^{\star }\cdot g\Big (|\nabla u(x)|\Big )\Big )\\\le & {} {\widetilde{G}}\Big (\big (\Lambda _{0}^{\star }+1\big )\cdot g\Big (|\nabla u(x)|\Big )\Big ) \\\le & {} \frac{\delta _{0}}{\delta _{0}+1}\cdot \big (\Lambda _{0}^{\star }+1\big )^{1+\frac{1}{\delta _{0}}}\cdot {\widetilde{G}}\Big (g\Big (|\nabla u(x)|\Big )\Big ) \quad \big (\text {by Remark} \ 9.1) \ c)\big ) \\\le & {} g_{0} \frac{\delta _{0}}{\delta _{0}+1}\cdot \big (\Lambda _{0}^{\star }+1\big )^{1+\frac{1}{\delta _{0}}}\cdot G\Big (\big |\nabla u(x)\big |\Big ). \quad \big (\text {by Remark} \ 9.1) \ d)\big ) \end{aligned}$$

Thus, let $V\subset \subset \Omega$ be an open set. Then,

$$\begin{aligned} \int _{V} {\widetilde{G}}(|X_{u}(x)|) dx \le g_{0} \cdot \big (\Lambda _{0}^{\star }+1\big )^{1+\frac{1}{\delta _{0}}}\cdot \int _{V} G\Big (\big |\nabla u(x)\big |\Big )dx<\infty . \end{aligned}$$

Analogously,

$$\begin{aligned} \int _{V} {\widetilde{G}}(|X_{v}(x)|) dx \le g_{0}\cdot \big (\Lambda _{0}^{\star }+1\big )^{1+\frac{1}{\delta _{0}}}\cdot \int _{V} G\Big (\big |\nabla v(x)\big |\Big )dx<\infty . \end{aligned}$$

Thus

$$\begin{aligned} \max \Big \{||X_{u}||_{L^{{\widetilde{G}}}(V)},||X_{v}||_{L^{{\widetilde{G}}}(V)} \Big \}<\infty . \end{aligned}$$

(14.5)

Thus, if we define $\Upsilon :W_{c}^{1,G}(V)\rightarrow {\mathbb {R}}$ given by $(\varphi \in W_{c}^{1,G}(V))$

$$\begin{aligned} \Upsilon (\varphi )= & {} \int _{\Omega } \Big ({\mathcal {A}}(x, \nabla u(x)) - {\mathcal {A}}(x, \nabla v(x)) \Big )\cdot \nabla \varphi (x) ~dx\\= & {} \int _{V} \Big ({\mathcal {A}}(x, \nabla u(x)) - {\mathcal {A}}(x, \nabla v(x)) \Big )\cdot \nabla \varphi (x) ~dx \\= & {} \Lambda _{X_{u}}(\varphi ) - \Lambda _{X_{v}}(\varphi ). \\ \end{aligned}$$

Now, by (14.5) and Lemma 14.4

$$\begin{aligned} \Upsilon :W_{c}^{1,G}(V)\rightarrow {\mathbb {R}}\quad \text {is linear and satisfies} \quad |\Upsilon (\varphi )|\le C\cdot || \varphi ||_{W_{0}^{1,G}(V)}. \end{aligned}$$

Observe that $\Upsilon (\varphi ) \le 0$ forall $\varphi \in C_{0}^{\infty }(V)$ such that $\varphi \ge 0$ by (14.4). This way, we conclude by Lemma 14.3 that (14.4) holds in fact for any $0\le \varphi \in W_{c}^{1,G}(V)$. Since $V\subset \subset \Omega$ was taken arbitrary, we actually proved that

$$\begin{aligned} \int _{\Omega } \Big ({\mathcal {A}}(x, \nabla u(x)) - {\mathcal {A}}(x, \nabla v(x)) \Big )\cdot \nabla \varphi (x) ~dx \le 0\quad \forall \varphi \in W_{c}^{1,G}(\Omega ), \quad \varphi \ge 0.\qquad \end{aligned}$$

(14.6)

We set $w{:}{=}u-v\in C^{0}({\overline{\Omega }})$ and $w\le 0$ along $\partial \Omega$ pointwise. Since $\Omega$ is bounded, w is uniformly continuous in ${\overline{\Omega }}$. This way, for any $\varepsilon >0$ there exists a $\delta >0$ such that $w_{\varepsilon }(x){:}{=}(w-\varepsilon )^{+}(x) =0$ for all $x\in \Omega _{\delta }{:}{=}\{y\in {\overline{\Omega }}; \ dist(y,\partial \Omega ) <\delta \} \subset \subset \Omega .$ In particular, this proves that $w_{\varepsilon }\in W_{c}^{1,G}(\Omega )\subset W_{0}^{1,G}(\Omega )$ and thus, $0\le w_{\varepsilon }$ becomes an admissible test function, i.e, by (14.6) we find

$$\begin{aligned} 0\ge & {} \int _{\Omega } \Big ({\mathcal {A}}(x, \nabla u(x)) - {\mathcal {A}}(x, \nabla v(x)) \Big )\cdot \nabla w_{\varepsilon }(x) ~dx\\= & {} \int _{\{ u>v+\varepsilon \} } \Big ({\mathcal {A}}(x, \nabla u(x)) - {\mathcal {A}}(x, \nabla v(x)) \Big )\cdot \big (\nabla u(x) -\nabla v(x)\big )~dx. \end{aligned}$$

By property $Q_{1}$,

$$\begin{aligned} I(x){:}{=} \Big ({\mathcal {A}}(x, \nabla u(x)) - {\mathcal {A}}(x, \nabla v(x)) \Big )\cdot \big (\nabla u(x) -\nabla v(x)\big )\ge 0 \quad a.e. \ \text { in }\ \Omega . \end{aligned}$$

This implies, in particular, that $I(x)=0$ for almost every $x\in \{ u>v+\varepsilon \}$. As a matter of fact, we can deduce from $Q_{1}$ that $\nabla u = \nabla v \ a.e.$ in $\{ u>v+\varepsilon \}.$ This implies $\nabla w_{\varepsilon } = 0 \ a.e.$ in $\Omega .$ Since $W_{0}^{1,G}(\Omega )\hookrightarrow W_{0}^{1,1+\delta }(\Omega )$ (see Theorem 2.2 in [64]), we conclude by Poincaré inequality that $||w_{\varepsilon }||_{L^{1+\delta }(\Omega )} \le C\cdot ||\nabla w_{\varepsilon }||_{L^{1+\delta }(\Omega )}=0.$ Thus $w_{\varepsilon }=0 \ a.e.$ in $\Omega$. Since $w_{\varepsilon } \in C^{0}({\overline{\Omega }})$, we actually have $w_{\varepsilon }\equiv 0$ in $\Omega$. Moreover, since $g(t)=t^{+}$ is Lipschitz continuous with $[g]_{C^{0,1}({\mathbb {R}})} = 1$, then $\forall x\in \Omega$ and $\forall \varepsilon >0$

$$\begin{aligned} |(u-v)^{+}(x)| = |w_{\varepsilon }(x) -(u-v)^{+}(x)| = | (u(x)-v(x)-\varepsilon )^{+}- (u-v)^{+}(x) | \le \varepsilon \end{aligned}$$

Letting $\varepsilon \rightarrow 0$, we conclude that $(u-v)^{+}\equiv 0$ in $\Omega$, i.e, $u\le v$ in $\Omega$. $\square$

Remark 14.1

(Inhomogeneous Hopf-Oleĭnik Lemma for quasilinear equations in [9]) The version of the Comparison Principle for quasilinear equations above is needed in the proof of the Inhomogeneous Hopf-Oleĭnik Lemma, Theorem 3.2 in [9]. Theorem 3.2 in [9] is correct and as it was stated for functions in $C^{0}({\overline{\Omega }})\cap W_{loc}^{1,G}(\Omega )$. However, in our proof there, we used a weaker version of the Comparison Principle for quasilinear equations, namely, Proposition 5.2 in [9]. This version assumes integrability of the gradients all the way up to the boundary, i.e, $u,v\in C^{0}({\overline{\Omega }})\cap W^{1,G}(\Omega ).$ The new version of the Comparison Principle above (Proposition 14.3) requires only interior local integrability of the gradients. Thus, by using it (instead of Proposition 5.2 in [9]), we fix this small shortcoming into the proof of Theorem 3.2 in [9].