I am studying an article of Berestychi-Caffarelli-Niremberg - Monotonicity for elliptic equations in unbounded Lipschitz domains, and I don't understand a convergence in the demonstration of the lemma 3.4.

Suppose that $u\in C^2(\Omega)\cap C(\overline\Omega)$, such that $$ \left\{ \begin{array}{rl} \Delta u+f(u)=0, & in \ \Omega,\\ u=0, & on \ \partial\Omega,\\ 0<u<1, & in \ \Omega, \end{array} \right. $$ where $\Omega=\{x=(x',x_n)\in\mathbb{R}^n\times\mathbb{R};x_n>\varphi(x')\}$, with $\varphi$ is a Lipschitz function.

LEMMA 3.4: For any $h>0$, the solution $u$ is bounded away from $1$ in $\Omega_h=\{x\in\Omega;\varphi(x')<x_n<\varphi(x')+h\}$.

Sketch of the proof: Suppose by contradiction that exists a sequence $(x'^j,x_n^j)_j=(x^j)_j\subset\Omega_h$ such that $u(x^j)\rightarrow1$. By the translation $T^j(x)=x-x^j$ we move the set $\Omega$ to $\Omega^j$, given by $$\Omega^j=\{z=(z',z_n)\in\mathbb{R}^n;z_n>\varphi^j(z')=\varphi(z'+x'^j)-x_n^j\}.$$ Is easy to verify that the functions $\varphi^j$ is Lipschitz continuous and uniformly bounded in compact sets, so by the Arzela-Ascoli Theorem, for a subsequence $\varphi^j$ tend to a function $\widehat\varphi$. For each set $\Omega^j$ you have a shifted solution $$u^j(z',z_n)=u(z'+x'^j,z_n+x_n^j),$$ satisfying $$ \left\{ \begin{array}{rl} \Delta u^j+f(u^j)=0, & in \ \Omega^j,\\ u^j=0, & on \ \partial\Omega^j,\\ 0<u^j<1, & in \ \Omega^j, \end{array} \right. $$

FINALLY, the doubt: In the article, he says that the shifted solutions converge uniformly in compact subsets of $$\widehat\Omega=\{x\in\mathbb{R}^n;x_n>\widehat\varphi(x')\},$$ to a solution $\widehat u$ that satisfies $$ \left\{ \begin{array}{rl} \Delta \widehat u+f(\widehat u)=0, & in \ \widehat\Omega,\\ \widehat u=0, & on \ \partial\widehat\Omega,\\ \end{array} \right. $$

If the initial sequence $(x^j)_j$ is bounded, I can to argument this implies (Because in compact sets, the shifted solutions and the first and second derivatives, would be uniformly continuous and uniformly bounded, then you could apply the Arzela-Ascoli theorem). But I think that the sequence could be unbounded, I don't know. Someone can help me in this argument?

Thank you.


First of all, you should think of the sequence $(x^j)$ as unbounded, because if it had a finite limit, we'd immediately hit a contradiction: $x^j\to x\in \partial \Omega$, $u\in C(\overline{\Omega})$, and $u(x)=0$.

The reason for convergence of shifted solutions on compact sets is also Arzela-Ascoli, but we need uniform continuity on an unbounded set, which does not come for free.

Claim: For any $C>1$ the solution $u$ is uniformly Lipschitz on the set $\Omega_C=\{x: C^{-1}\le d(x)\le C\}$ where $d(x)=\operatorname{dist}(x,\partial\Omega)$.

Proof. The function $u$ solves the Poisson equation $\Delta u = g$ where $g=f\circ u$. Notice that both $u$ and $g$ are bounded in $\Omega$. The standard inner regularity result for the Poisson equation (Theorem 3.9 in Gilbarg-Trudinger) implies that $\nabla u$ is uniformly bounded in $\Omega_C$. (Why? Because any point $x\in \Omega_C$ is the center of a ball of radius $C^{-1}$ contained in $\Omega$. Apply the theorem on this ball, and recall that $u$ and $\Delta u$ are uniformly bounded.) It follows that $u$ is Lipschitz on $\Omega_C$. $\Box$

When we fix a compactly contained subdomain $G\Subset \widehat{\Omega}$ and consider the restrictions $u^j$ to $G$, we are actually looking at the restriction of $u$ to a subset of $\Omega_C$. Hence $u^j$ are uniformly Lipschitz on $G$. Since $f$ is Lipschitz and $\Delta u^j=f(u^j)$, it follows that the sequence $\Delta u^j$ is uniformly Lipschitz as well. By Theorem 4.6 in Gilbarg-Trudinger, this implies uniform Hölder continuity of all second-order derivatives of $u^j$.

Therefore, we can choose a subsequence, still denoted $u^j$, such that both $u^j$ and $D^2 u^j $ converge uniformly on $G$. Say, $u^j\to \widehat{u}$ and $\Delta u^j \to g$. We should check that $\Delta \widehat{u}=g$. To that end, pick a test function $\psi$ supported in $G$, and observe that $$\int u^j\Delta\psi\to \int \widehat{u}\,\Delta\psi\tag{1}$$ $$\int \Delta u^j \,\psi\to \int g\,\psi\tag{2}$$ Since $\int u^j\Delta\psi = \int \Delta u^j \,\psi$ (integration by parts), the combination of (1) and (2) implies $\Delta \widehat{u}=g$.

  • $\begingroup$ Why the sequence $x^j$ can not converge to a point in the interior of $\Omega$? $\endgroup$
    – Tomás
    Jan 23 '13 at 0:28
  • $\begingroup$ @Tomás Good point. If $x^j\to x\in\Omega$ then $u(x)=1$, contradicting an earlier result in the paper, Theorem 1.2 (a). $\endgroup$
    – user53153
    Jan 23 '13 at 0:35
  • $\begingroup$ Why the laplacian converges in $\widehat\Omega$? What is the explanation to $\Delta\widehat u+f(\widehat u)=0$ in $\widehat\Omega$? $\endgroup$ Jan 24 '13 at 18:59
  • 1
    $\begingroup$ @JoséCarlos Since $\Delta u_j = -f(u^j)$, the functions $\Delta u_j $ also form a uniformly convergent sequence. Uniform convergence of $u_j$ and $\Delta u_j$ implies uniform convergence of all derivatives of 1st and 2nd order. Indeed, take $v=u^j-u^k$ and apply Poisson equation estimates from Chapter 3 of GT to $u$. $\endgroup$
    – user53153
    Jan 24 '13 at 19:34
  • 1
    $\begingroup$ I have two doubts yet. First of all, before to read this article, I thought that to define a sequence of functions, the functions would need to be defined in the same set, but in this case, we have one different set to each $u^j$. Is this not a problem? In the end, for each compact set $K$, we have to consider the restriction of the sequence in the compact, ok? My second doubt, is about your second answer. Could you explain better, why the first and the second derivative are uniformly convergent? Thank you very much, you are helping me a lot. $\endgroup$ Jan 25 '13 at 15:43

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