\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 09, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/09\hfil On the penalized obstacle problem] {On the penalized obstacle problem in the unit half ball} \author[E. Lindgren\hfil EJDE-2010/09\hfilneg] {Erik Lindgren} \address{Erik Lindgren \newline CERMICS - ENPC, 6 et 8 avenue Blaise Pascal, Cite Descartes Champs sur Marne, 77455 Marne la Vallee Cedex 2 France} \email{erik.lindgren@cermics.enpc.fr} \thanks{Submitted September 21, 2009. Published January 16, 2010.} \subjclass[2000]{35J70, 35J60, 35J85} \keywords{Obstacle problem; elliptic equation; regularity; penalization} \begin{abstract} We study the penalized obstacle problem in the unit half ball, i.e. an approximation of the obstacle problem in the unit half ball. The main result states that when the approximation parameter is small enough and when certain level sets are sufficiently close to the hyperplane $\{x_1=0\}$, then these level sets are uniformly $C^1$ regular graphs. As a by-product, we also recover some regularity of the free boundary for the limiting problem, i.e., for the obstacle problem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \section{Introduction} \subsection{Problem} Given some non-negative boundary data $g\in L^\infty(B_1^+)\cap H^1(B_1^+)$ we study the penalized obstacle problem in the unit half ball; that is \begin{gather*} \Delta u_\varepsilon=\beta_\varepsilon(u_\varepsilon) \quad\text{in }B_1^+=B_1\cap \{x_1>0\},\\ u_\varepsilon= g\quad\text{on }\partial B_1\cap \{ x_1>0\},\\ u_\varepsilon=0 \quad\text{on }B_1\cap \{x_1=0\}. \end{gather*} Here we assume that $\beta_\varepsilon$ is any function satisfying the same assumptions as in \cite{Red93a} and \cite{Red93b}, see section \ref{sec:main} for details. This would then imply by the maximum principle that $u_\varepsilon\geq 0$. We will refer to $B_1\cap \{x_1=0\}$ as the fixed boundary. \subsection{Known result} The penalized obstacle problem has been well studied, see \cite{KS80}, \cite{BK73}, \cite{Red93a} and \cite{Red93b}. Brezis and Kinderlehrer proved the uniform local $C^{1,1}$ regularity. Redondo proved, using the result of Kinderlehrer and Stampacchia, that the level sets of $u_\varepsilon$ are locally uniform $C^{1,\alpha}$-graphs in some direction. However, the behaviour of the level sets close to the fixed boundary is so far unknown for the penalized obstacle problem. The present paper contributes in this direction. In the case of the obstacle problem (the limiting case when $\varepsilon\to 0$), the behaviour of the free boundary has been studied in \cite{SU03}. There the authors prove that, close to the fixed boundary, the free boundary is a $C^1$-graph in directions close to $e_1$. In fact, they prove this in a more general setting, which includes the obstacle problem as a special case. \subsection{Main result}\label{sec:main} Before mentioning the main result of this paper we must carefully define which class of solutions we will work with. \begin{definition} \label{def1} \rm We say $\beta_\varepsilon \in \mathcal{C}_\varepsilon$ if \begin{enumerate} \item For any $\varepsilon>0$, $\beta_\varepsilon$ is uniformly Lipschitz on $\mathbb{R}$, \item $\beta_\varepsilon(t)\leq 0$ for all $t\leq 0$ and all $\varepsilon>0$, \item $\beta_\varepsilon(0)=0$ and $\beta_\varepsilon(t)\leq 1$, \item $\beta''_\varepsilon(t)\leq 0$, \item There holds $$0\leq \frac{1-\beta_\varepsilon(t)}{\varepsilon}\leq \beta_\varepsilon'(t) \leq \frac{1}{\varepsilon},$$ whenever $t\geq 0$. \end{enumerate} \end{definition} One can deduce that $\Psi_\varepsilon\leq \beta_\varepsilon\leq\Phi_\varepsilon$, where $$\Psi_\varepsilon(t)=\begin{cases} t/\varepsilon & t\leq 0,\\ 1-e^{-t/\varepsilon} & t>0, \end{cases}$$ and $$\Phi_\varepsilon(t)=\begin{cases} t/\varepsilon & t\leq \varepsilon,\\ 1 & t>\varepsilon. \end{cases}$$ In addition, one can see that if $\beta_\varepsilon(\cdot)\in \mathcal{C}_\varepsilon$ then $\beta_\varepsilon(s\cdot)\in \mathcal{C}_{\varepsilon/s}$. This scaling property will be used frequently further on. In what follows we use the notation $$B_r^+(x)=B_r(x)\cap \{x_1>0\}.$$ We define the local class of solutions as the following: \begin{definition} \label{def} \rm We say $u_\varepsilon\in P_r(M,\varepsilon)$ if \begin{enumerate} \item $\Delta u_\varepsilon=\beta_\varepsilon(u_\varepsilon)$ in $B_r^+$ where $\beta_\varepsilon\in\mathcal{C_\varepsilon}$. \item $u_\varepsilon\geq 0$ on $\partial B_r\cap \{x_1>0\}$. \item $u_\varepsilon=0$ on $B_r\cap \{x_1=0\}$, \item $\sup_{B_r^+}u_\varepsilon\leq M$. %\item $0\in\partial\{u_\varepsilon>0\}$. \end{enumerate} \end{definition} The main result of this paper is the $C^{1,1}$-estimates and the uniform (in $\varepsilon$) $C^1$ regularity of the $\varepsilon^\gamma$-level sets for any $0<\gamma<1$, as stated below in Theorem 1 and 2. A simple consequence of Theorem \ref{thm:C1} is Corollary \ref{cor:limit}. \begin{theorem}\label{thm:C11} Let $u_\varepsilon\in P_1(M,\varepsilon)$. Then there is a constant $C=C(n,M)$ such that $$\|u_\varepsilon\|_{C^{1,1}(B_{1/2}^+)}\leq C.$$ \end{theorem} \begin{theorem}\label{thm:C1} Let $u_\varepsilon\in P_1(M,\varepsilon)$. Assume moreover that $y\in \partial\{u_\varepsilon>\varepsilon^\gamma\}\cap B_{1/2}^+$ for some $0<\gamma<1$. Then there are constants $\varepsilon_0,\rho>0$ depending on $M$, $\gamma$ and the dimension such that $$\max(\varepsilon,y_1)<\varepsilon_0$$ implies that $B_\rho^+(y)\cap \partial\{u_\varepsilon>\varepsilon^\gamma\}$ is a $C^1$-graph with the $C^1$-norm bounded independently of $\varepsilon$ and $y_1$. Moreover, the normal of $\partial\{u_\varepsilon>\varepsilon^\gamma\}$ at $y$ converges to $e_1$ as $y_1\to 0$. \end{theorem} Sending $\varepsilon\to 0$ we obtain the following result for the obstacle problem, which is a slightly stronger result than the one in \cite{SU03} since it only requires the free boundary to be near the fixed one. \begin{corollary} \label{cor:limit} Let $u_0\in P_1(M,0)$, that is $u_0$ is a solution to the obstacle problem, and suppose $y\in \partial\{ u_0>0\}$. Then there is a constant $\varepsilon_0$ depending on $M$ such that $y_1<\varepsilon_0$ implies that $B_\rho^+(y)\cap \partial\{u_0>0\}$ is a $C^1$-graph with the $C^1$-norm bounded independently $y_1$. \end{corollary} \section{Non-degeneracy} Here we state the result that the solutions do not grow too slow. This is important when considering sequences as we will do later, since it implies the stability of the level sets under uniform convergence. For the sake of completeness, we give the standard proof. \begin{proposition}\label{prop:nondeg} Let $u_\varepsilon\in P_1(M,\varepsilon)$ and assume $u_\varepsilon(y)\geq \varepsilon$. Then there is $C=C(n)>0$ such that $$\sup_{B_r(y)\cap B_1^+}(u_\varepsilon-u_\varepsilon(y))\geq Cr^2$$ for $r<\mathop{\rm dist}(y,\partial B_1^+)$. \end{proposition} \begin{proof} Take $y\in \{u_\varepsilon=\delta\}$ for some $\delta\geq \varepsilon$ and let $$v=u_\varepsilon-\delta-\frac{|y-z|^2}{8n}.$$ Then $\Delta v\geq \beta_\varepsilon(u_\varepsilon)-1/4\geq 0$ in $\{u_\varepsilon> \delta/2\}\cap B_r^+(y)$ for any $r$ such that $B_r^+(y)\subset B_1^+$. Therefore, $v$ attains its maximum on the boundary of $\{u_\varepsilon> \delta/2\}\cap B_r^+(x_0)$. Furthermore, $v(y)=0$, $v\leq 0$ on $\{u_\varepsilon=\delta\}$ and $v\leq 0$ on $\{x_1=0\}$. Hence $v$ attains its positive maximum at some $z\in \partial B_r(y)\cap \{x_1>0\}$. This yields the desired result. \end{proof} \section{$C^{1,1}$-estimates in $B_{1/2}^+$} In this section we prove that the functions $u_\varepsilon$ are uniformly bounded in the space $C^{1,1}(B_{1/2}^+)$. The first step is to show that we have quadratic growth up to the fixed boundary. \begin{lemma}\label{lem:quadgrowth} Let $u_\varepsilon\in P_1(M,\varepsilon)$. Then there are constants $C,r_0>0$ depending on $M$ and the dimension such that $$\sup_{B_r^+}|u_\varepsilon(x)-x\cdot \nabla u_\varepsilon(0)|\leq Cr^2$$ for $r0$, since by the maximum principle $|\partial_eu_j(x)|\leq C|x|$, \item $|\Delta v_j|=C/j$ in $B_{1/r_j}^+$. \end{enumerate} Therefore, there is a subsequence, again labelled $v_j$, converging in $C^{1,\alpha}(\overline{B_R^+})$ for any $R$ to a function $v_0$ satisfying \begin{enumerate} \item $v_0(0)=|\nabla v_0(0)|=0$, \item $\sup_{B_1^+}|v_0|=1$, \item $\sup_{B_{2^k}^+}|v_0|\leq 4^k$ for all $k\in \mathbb{N}$, \item $D_e v_0=0$ for any $e\cdot e_1=0$ in ${\mathbb{R}^n_+}$, \item $\Delta v_0=0$ in ${\mathbb{R}^n_+}$. \end{enumerate} Clearly, (4) implies that $v_0$ is one-dimensional. Then (1) and (5) implies that $v_0=0$ which contradicts (2). \end{proof} This estimate implies, as we will now see, that $u_\varepsilon$ is uniformly $C^{1,1}$. The proof is standard. \begin{proof}[Proof of Theorem \ref{thm:C11}] Since interior estimates are already known (see Theorem 2 in \cite{BK73} even though this is stated in a slightly different form, this can also be proved with methods similar to the ones in the proof of Lemma \ref{lem:quadgrowth}), it suffices to obtain estimates near the fixed boundary. Take $y\in B_{1/2}^+\cap \{x_1t\}$ for some $t\geq 0$, \item $u=0$ on $x_1=0$, \item $\sup_{{B_r}^+}u\leq Cr^2$ for $r>1$, \item $\Delta u=\beta_k(u)$ for some $k\geq 0$, with $\beta_0(t)=\chi_{\{t>0\}}$ and $\beta_k\in \mathcal{C}_k$. \end{enumerate} Then $u$ is one-dimensional and monotone. \end{proposition} \begin{proof} First we extend $u$ to $x_1<0$ by odd reflection, that will give us $$\Delta u=\beta_k(u)\chi_{\{x_1>0\}}-\beta_k(-u)\chi_{\{x_1<0\}}.$$ Then we observe that $$\Delta (\partial_e u)=(\beta'_k(u)\chi_{\{x_1>0\}} +\beta_k'(-u)\chi_{\{x_1<0\}})\partial_e u+\beta_k(u)\partial_e(\chi_{\{x_1>0\}}) -\beta_k(-u)\partial_e(\chi_{\{x_1<0\}}).%=(\beta'_k(u)+\beta_k'(-u))\partial_e u$$ Since $\beta(0)=0$ and $\nabla \chi_{\{x_1>0\}}$ is a finite measure the last two terms vanish. This implies that $(\partial_e u)^\pm$ are both subharmonic and therefore Theorem \ref{thm:acf} applies. Let $$u_R(x)=\frac{u(Rx)}{R^2}.$$ Then \begin{enumerate} \item $\Delta u_R=\beta_k(R^2u_R)\chi_{\{x_1>0\}} -\beta_k(-R^2u_R)\chi_{\{x_1<0\}}$ where $\beta_k(R^2t)\in \mathcal{C}_{k/R^2}$, \item $|u_R(x)|\leq C|x|^2$ for $|x|>1/R$, \item $e_1/R\in \partial\{u_R>t/R^2\}$, \item $-e_1/R\in \partial\{u_R<-t/R^2\}$, \item $|\nabla u_R(0)|\leq C/R$. \end{enumerate} We see that $\Delta u_R$ will be uniformly bounded from above and from below. Moreover, $\sup_{B_1}|u_R|$ is uniformly bounded. Hence, there is by standard elliptic estimates, a subsequence $u_j=u_{R_j}$ with $R_j\to \infty$, converging in $W_\textup{loc}^{2,p}\cap C^{1,\alpha}_\textup{loc}$. Invoking Proposition \ref{prop:nondeg} we can conclude that $u_j\to u_\infty$ where $u_\infty$ satisfies \begin{enumerate} \item $\Delta u_\infty=\chi_{\{u_\infty>0\}}\chi_{\{x_1>0\}} -\chi_{\{u_\infty<0\}}\chi_{\{x_1<0\}} =\chi_{\{u_\infty>0\}}-\chi_{\{u_\infty<0\}}$, \item $|u_\infty(x)|\leq C|x|^2$ for all $x$, \item $0\in \partial\{u_\infty>0\}\cap \partial\{u_\infty<0\}\cap \{|\nabla u_\infty|=0\}$. \end{enumerate} By applying the classification of global solutions in \cite{SUW04}, we obtain that up to rotations $$u_\infty=\frac{(x_1^+)^2}{2}-\frac{(x_1^-)^2}{2}.$$ Again, Theorem \ref{thm:acf} applies to $\partial_e u_j^\pm$ for any direction $e$. Thus, $$0=\phi(\partial_e u_\infty,1)=\lim \phi(\partial_e u_j,1) =\lim \phi(\partial_e u,R_j) =\phi(\partial_e u,\infty).$$ Since $\phi$ is non-decreasing this implies that $\phi(\partial_e u,r)=0$ for all $r$. Hence, either $\partial_e u\geq 0$ or $\partial_e u\leq 0$, for any direction $e$. This implies that $u$ must be one-dimensional. \end{proof} \begin{remark} \label{rmk1} \rm It seems plausible that a proof of this can be given without the use of the ACF monotonicity formula. However, such a proof would be much longer and more technically involved. \end{remark} \section{The $C^1$ regularity of certain level sets close to the fixed boundary} In this section we will prove that if the $\varepsilon^\gamma$-level set is close to the fixed boundary then, due to the results in \cite{Red93a} and \cite{Red93b}, the $\varepsilon^\gamma$-level set is a uniform (with respect to $\varepsilon$) $C^1$-graph. The first step is to prove that the set $\{u_\varepsilon\leq \varepsilon^\gamma\}$ is not going to be too small in this case, see Figure \ref{fig:md}. This is needed in order to apply the results in \cite{Red93a} and \cite{Red93b}, which are of the type ''exterior flatness implies regularity''. In the rest of this paper $\gamma\in (0,1)$ is fixed, so all estimates and results will depend on $\gamma$. \begin{figure}[ht] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(0,0) \includegraphics[width=0.8\textwidth]{fig1} %\input{by1.pdf_t} \end{picture} \begin{picture}(100,48)(0,0) \put(45,41){$\partial\{u_\varepsilon>\varepsilon^\gamma\}$} \put(90,35){$B_1^+$} \put(38,20){$y$} \put(58,15){$B_{\sigma y_1}(y)$} \put(53,3){$B_{\sigma y_1}(y)\cap \{x_1\leq y_1-\sigma y_1/2\}$} \end{picture} \end{center} \caption{The situation when $\partial\{u_\varepsilon>\varepsilon^\gamma\}$ is close to the fixed boundary.} \label{fig:md} \end{figure} \begin{proposition}\label{prop:md} Let $u_\varepsilon\in P_1(M,\varepsilon)$ with $|\nabla u_\varepsilon(y)|0$ there is $\tau =\tau(\sigma,M,\gamma,n)$ such that $y\in \{u_\varepsilon>\varepsilon^\gamma\}$, $y\cdot e_1=y_1<\tau$ and $\varepsilon<\tau$ imply that $$\label{eq:cone} B_{y_1\sigma}(y)\cap \{x_1\leq y_1-\frac{y_1\sigma}{2}\}\subset \{u_\varepsilon\leq \varepsilon^\gamma\}.$$ \end{proposition} \begin{proof} If this is not true then there exist $\{u_{\varepsilon_j}>\varepsilon_j^\gamma\}\ni y_1^j\to 0$, $\varepsilon_j\to 0$, $u_{\varepsilon_j}\in P_1(M,\varepsilon_j)$ and $$x^j\in B_{y_1^j\sigma}(y)\cap \{x_1\leq y_1^j -\frac{y_1^j\sigma}{2}\}\cap \{u_{\varepsilon_j}> \varepsilon_j^\gamma\}, \quad\text{for j=1,2,\ldots}$$ For $$v_j(x)=\frac{u_{\varepsilon_j}(y_1^jx+y^j)}{(y_1^j)^2}$$ we have \begin{enumerate} \item $\Delta v_j=\beta_{\varepsilon_j}(v_j (y_1^j)^2)\in \mathcal{C}_{\varepsilon_j/(y_1^j)^2}$ for $x_1>-1$, \item $v_j\geq 0$, \item $\sup_{B_\rho\cap \{x_1>-1\}}v_j\leq C(M)\rho^2 +C|\nabla u_{\varepsilon_j}(y^j)|\rho/y_1^j +\varepsilon_j^\gamma/(y_1^j)^2+v_j(0)$ for $\rho\varepsilon_j^\gamma/(y_1^j)^2\}$, \item $|\nabla v_j(0)|=|\nabla u_{\varepsilon_j}(y^j)|/|y^j|\leq 1$, \item $v_j=0$ on $x_1=-1$, \item $z_j=(x_j-y_j)/y_1^j\in B_{\sigma}\cap \{x_1\leq -\frac{\sigma}{2}\}\cap \partial\{v_{\varepsilon_j} >\varepsilon_j^\gamma/(y_1^j)^2\}$. \end{enumerate} All these properties together with Proposition \ref{prop:nondeg} allow us to pass to the limit for a subsequence and obtain a limit function $v_0$ and a limit point $z=\lim_j z_j$. We observe that the $C^{1,1}$-estimates give $$0=u_{\varepsilon_j}(y^j-y_1^je_1)\geq u_{\varepsilon_j}(y^j) -\nabla u_{\varepsilon_j}(y^j)\cdot e_1y_1^j-C(y_1^j)^2,$$ which implies $\varepsilon_j^\gamma0\}}$ for $x_1>-1$, \item $v_0\geq 0$, \item $\sup_{B_\rho\cap \{x_1>-1\}}v_0\leq C(M)\rho^2$ for $\rho>1$, \item $0\in \partial\{v_0>0\}$, \item $|\nabla v_0(0)|=0$, \item $v_0=0$ on $x_1=-1$, \item $z\in B_\sigma\cap \{x_1\leq - \sigma/2\}\cap \left(\{v_0>0\} \cup \partial\{v_0>0\}\right)$. \end{enumerate} Note that(1) follows from the fact that $\Delta v_j\in \mathcal{C}_{\varepsilon_j/(y_1^j)^2}$ where $\varepsilon_j/(y_1^j)^2\to 0$. Moreover, (5) is a simple consequence of (4). Theorem B in \cite{SU03} now implies that $v_0=(x_1^+)^2/2$. This contradicts (7) whenever $1>\sigma>0$. \noindent\textbf{Case 2:} $\varepsilon_j^\gamma/(y_1^j)^2\to t>0$. This is similar to case 1. The only difference is that the origin will be in the $t$-level set instead of in the zero level set which also means that we do not know that the gradient vanishes at the origin. Instead of using the result from \cite{SU03} we use Proposition \ref{prop:class} to say that $v_0$ must be one-dimensional and monotone. This yields a contradiction for $1>\sigma >0$. \end{proof} \begin{corollary}\label{cor:c1} Let $u_\varepsilon\in P_1(M,\varepsilon)$. Then for any $0<\gamma<1$ there is an $\varepsilon_0(M,\gamma,n)$ such that $y\in\partial\{u_\varepsilon>\varepsilon^\gamma\}$ and $y_1<\varepsilon_0$ imply that $\partial \{u_\varepsilon>\varepsilon^\gamma\}\cap B_{ry_1}(y)$ is a uniform (with respect to $\varepsilon$) $C^1$-graph for some $r>0$. \end{corollary} \begin{proof} If $|\nabla u(y)|\geq y_1$ then, by the implicit function theorem, $\partial \{u_\varepsilon>\varepsilon^\gamma\}\cap B_{ry_1}(y)$ is a $C^{1,\alpha}$-graph for some $r>0$. In the other case, $|\nabla u_\varepsilon(y)|\varepsilon^\gamma/y_1^2\}\cap B_{r}(y)$ is a $C^{1,\alpha}$-graph for some $r>0$. By rescaling and combining the two cases, we obtain the desired result. \end{proof} \begin{remark} \label{rmk2} \rm Noteworthy is that in the proof of Corollary \ref{cor:c1} we apply the result in \cite{Red93a,Red93b} for the rescaled function $v$. For this we need $\varepsilon/y_1^2$ to be small. This is assured by the assumption $\gamma<1$. To extend this result to $\gamma=1$ we need to come up with something different. In addition we note that in Corollary \ref{cor:c1} we obtain that the level sets are $C^1$ but that the $C^1$-norm might blow up as $y_1\to 0$. \end{remark} \section{Tangential touching property of the $\varepsilon^\gamma$-level sets} Here we prove that the $\varepsilon^\gamma$-level set can be approximated with planes, provided that $\varepsilon$ is small enough and that the $\varepsilon^\gamma$-level set is close enough to the fixed boundary, see Figure \ref{fig:cones}. In what follows, we will use the notation $$K_\delta(z)=\{|x_1-z_1|<\delta\sqrt{(x_2-z_2)^2+\cdots+(x_n-z_n)^2}\}.$$ To avoid cumbersome notation we will also write $K_\delta(0)=K_\delta$. \begin{figure}[ht] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(0,0) \includegraphics[width=0.8\textwidth]{fig2} %\input{cones.pdf_t} \end{picture} \begin{picture}(100,50)(0,0) \put(50,44){$\partial\{u_\varepsilon>\varepsilon^\gamma\}$} \put(90,35){$B_1^+$} \put(37,35){$K_\delta(y)$} \put(39,20){$y$} \put(63,16){Angle$=\arctan\delta$} \put(3,9){$B_\rho(y)$} \end{picture} \end{center} \caption{The level set $\partial\{u_\varepsilon>\varepsilon^\gamma\}$ is inside $K_\delta(y)$ when $y$ is close to the fixed boundary.} \label{fig:cones} \end{figure} \begin{proposition}\label{prop:dirmono} Let $u\in P_1(\varepsilon,M)$. Then for any $\delta>0$ there are $\varepsilon_0(M,\gamma,n,\delta)$ and $\rho(M,\gamma,n,\delta)$ such that $y \in \partial\{u_\varepsilon>\varepsilon^\gamma\}$ and $\max(\varepsilon,y_1,|\nabla u_\varepsilon(y)|)<\varepsilon_0$ imply $$\label{eq:tang} \partial\{u_\varepsilon>\varepsilon^\gamma\}\cap B_\rho(y)\cap B_1^+ \subset K_\delta(y)\cap B_\rho(y)\cap B_1^+.$$ \end{proposition} \begin{proof} If this is not true then for $\delta>0$ there are $\varepsilon_j\to 0$, $y^j\in \partial\{u_{\varepsilon_j}>\varepsilon_j^\gamma\}$ with $\max(|y^j_1|,|\nabla u_{\varepsilon_j}(y^j)|)|\to 0$ and $x^j$ with $r_j=|x^j-y^j|\to 0$, such that $$\label{eq:notcone} x^j\in K_\delta(y^j)^c\cap \partial\{u_{\varepsilon_j}>\varepsilon_j^\gamma\}.$$ We want to prove that this cannot be true. The proof is divided into two different cases with some sub-cases depending on the rate of convergence of $y_1^j$, $\varepsilon_j$, $|\nabla u_{\varepsilon_j}(y^j)|$ and $r_j$. \noindent\textbf{Case A:} $|\nabla u_{\varepsilon_j}(y^j)|-y_1^j/r_j$, \item $v_j\geq 0$, \item $\sup_{B_\rho\cap \{x_1>-y_1^j/r_j\}}v_j \leq C(M)\rho^2+C|\nabla u_{\varepsilon_j}(y^j)|\rho/r_j +\varepsilon_j^\gamma/r_j^2+v_j(0)$ for $\rho0$, \item $0\in \partial\{v_j>\varepsilon_j^\gamma/r_j^2\}$ with $\varepsilon_j^\gamma/r_j^2\leq C(M)|y^j|^2/r_j^2\to 0$, \item $|\nabla v_j(0)|=|\nabla u_{\varepsilon_j}(y^j)/r_j|\leq |y^j_1|/r_j\to 0$, \item $v_j=0$ on $x_1=- y_1^j/r_j$, \item $z_j=(x^j-y^j)/r_j\in \partial B_1 \cap K^c_\delta\cap\partial \{v_j>\varepsilon_j^\gamma/r_j^2\}$. \end{enumerate} We can, by standard compactness arguments, pass to the limit for a subsequence $v_{j_k}\to v_0$. Using Proposition \ref{prop:nondeg}, we see that $v_0$ satisfies \begin{enumerate} \item $\Delta v_0=\chi_{\{v_0>0\}}$ in $x_1>0$, \item $v_0\geq 0$, \item $\sup_{B_\rho^+}v_0\leq C(M)\rho^2$ for all $\rho>1$, \item $0\in \partial\{v_0>0\}$, \item $|\nabla v_0(0)|=0$, \item $v_0=0$ for $x_1=0$, \item $z_0\in \partial B_1\cap K^c_\delta\cap \partial \{v_0>0\}$. \end{enumerate} \cite[Theorem B]{SU03} implies together with (1)-(6) that $$v_0=\frac{(x_1^+)^2}{2},$$ which contradicts (7). \noindent\textbf{Case A2:} $0<1/A<|y_1^j|/r_j0$ for $x_1>-1$, \item $v_j\geq 0$, \item $\sup_{B_\rho\cap \{x_1>-1\}}v_j \leq C(M)\rho^2+C|\nabla u_{\varepsilon_j}(y^j)|\rho/y_1^j +\varepsilon_j^\gamma/(y_1^j)^2+v_j(0)$ for $\rho\varepsilon_j^\gamma/(y_1^j)^2\}$, \item $|\nabla v_j(0)|=\nabla u_{\varepsilon_j}(y^j)/|y^j|\leq 1$, \item $v_j=0$ on $x_1=-1$, \item $z_j=(x^j-y^j)/y_1^j\in K^c_\delta\cap \{v_j>\varepsilon_j^\gamma/(y_1^j)^2\}$. \end{enumerate} When $y_1^j$ is small enough, Corollary \ref{cor:c1} assures that $\partial\{v_j>\varepsilon_j^\gamma/(y_1^j)^2\}\cap B_{c_0}$ is a uniform $C^1$-graph for $c_0$ small. Since $|z_j|\to 0$, (7) implies that $\partial\{v_j>\varepsilon_j^\gamma/(y_1^j)^2\}$ has a tangential direction in $K_\delta^c$. All these properties allow us, together with Proposition \ref{prop:nondeg}, to pass to the limit for a subsequence and obtain a function $v_0$ satisfying \begin{enumerate} \item $\Delta v_0=\chi_{\{v_0>0\}}$ for $x_1>-1$, \item $v_0\geq 0$, \item $\sup_{B_\rho\cap \{x_1>-1\}}v_0\leq C(M)\rho^2+t$ for all $\rho >1$, \item $0\in \partial\{v_0>t\}$ for some bounded $t$, \item $|\nabla v_0(0)|\leq 1$, \item $v_0=0$ on $x_1=-1$, \item The set $\partial\{v_0>t\}\cap B_{c_0}$ has a tangential direction in $K_\delta^c$, for some small $c_0$. \end{enumerate} As in cases 1 and 2 in the proof of Proposition \ref{prop:md}, (1)-(7) imply that $v_0$ is one-dimensional. Therefore it can only depend on $x_1$, which contradicts (7), since (7) says that the level set of $v_0$ has a tangent lying in $K_\delta^c$. \noindent\textbf{Case B:} $|\nabla u_{\varepsilon_j} (y^j)|>y_1^j$. This implies by the $C^{1,1}$-estimates that $$0=u(y^j-y^j_1e_1)\geq u(y^j)-|\nabla u(y^j)\cdot e_1y^j_1|-C(y^j_1)^2 \geq \varepsilon_j^\gamma-C|\nabla u_{\varepsilon_j}(y^j)|^2,$$ which gives $\varepsilon_j^\gamma\leq C|\nabla u(y^j)|^2$. Let $s_j=|\nabla u(y^j)|$. For the first two cases B1 and B2, let us define $$v_j(x)=\frac{u_{\varepsilon_j}(s^jx+y^j)}{s_j^2}.$$ Then \begin{enumerate} \item $\Delta v_j=\beta_{\varepsilon_j}(v_js_j^2)\in \mathcal{C}_{\varepsilon_j/s_j^2}$ on $v_j>0$ for $x_1>-y_1^j/s_j$, \item $v_j\geq 0$, \item $\sup_{B_\rho\cap \{x_1>-y_1^j/s_j\}}|v_j|\leq C(M)\rho^2 +\varepsilon_j^\gamma/(s_j)^2+v_j(0)$ for $\rho\varepsilon_j^\gamma/(s_j)^2\}$, \item $|\nabla v_j(0)|=1$, \item $v_j=0$ on $x_1=-y_1/s_j$, \item With $z_j=(x^j-y^j)/s_j\in K^c_\delta\cap \partial \{v_j>\varepsilon_j^\gamma/(s_j)^2\}$. \end{enumerate} For this, we consider two different cases. \textbf{Case B1:} $|z_j|=r_j/s_j\to 0$. This yields a situation similar to case A3. Via Corollary \ref{cor:c1} and Proposition \ref{prop:class} we obtain a contradiction. \textbf{Case B2:} $A<|z_j|<1/A$. Then the situation is similar to the one in case A2. Thus, we obtain, either by Proposition \ref{prop:class} or Theorem B in \cite{SU03}, that the limit is one dimensional. This leads to a contradiction. \textbf{Case B3:} $|z_j|\to \infty$, i.e. $s_j=o(r_j)$. Then $\varepsilon_j^\gamma\leq Cs_j^2=o(r_j^2)$. By the assumptions in this case, $y_1^j=o(r_j)$. Let $$v_j(x)=\frac{u_{\varepsilon_j}(r_jx+y^j)}{r_j^2}.$$ Then similar arguments as in case A1 leads to contradiction. Since all cases lead to a contradiction, the theorem is proved. \end{proof} \subsection{The $C^1$ regularity of the $\varepsilon^\gamma$-level set} Now we are ready to prove the uniform $C^1$ regularity of the $\varepsilon^\gamma$-level sets. \begin{proof}[Proof of Theorem \ref{thm:C1}] We apply Corollary \ref{cor:c1} and obtain that for $y_1<\varepsilon_0$, the level set $\partial\{u_\varepsilon>\varepsilon^\gamma\}\cap B_{ry_1}(y)$ is differentiable. In addition, we know that the normal at a point $y$, denoted by $n_y$, is continuous with modulus of continuity $\sigma(\cdot/y_1)$, where $\sigma$ is a modulus of continuity. Hence, it might blow up when $y_1\to 0$. We need to prove that $n_y$ is uniformly continuous. Take two points $y,z\in \partial\{u_\varepsilon>\varepsilon^\gamma\}\cap \{x_1<\varepsilon_0\}$. From Proposition \ref{prop:dirmono} it follows that for any $\tau>0$ there is $\delta_\tau$ such that $y_1<\delta_\tau$ implies $\|n_y-e_1\|<\tau/2$. We split the proof into three cases: \noindent\textbf{Case 1:} $y_1,z_1<\delta_\tau/2$. Then obviously $\|n_y-n_z\|\leq \tau$. \noindent\textbf{Case 2:} $y_1<\delta_\tau/2$ and $z_1>\delta_\tau/2$. Then $\|z-y\|<\delta_\tau/2$ implies $\|n_y-n_z\|\leq \tau$. \noindent\textbf{Case 3:} $y_1,z_1>\delta_\tau/2$. From the arguments above, $$\|n_y-n_z\|\leq \sigma(2\|y-z\|/\delta_\tau),$$ which implies that $\|n_y-n_z\|\leq \tau$ if $\|y-z\|$ is small enough. \end{proof} Finally we give the proof of Corollary \ref{cor:limit}. \begin{proof}[Proof of Corollary \ref{cor:limit}] By Theorem \ref{thm:C11} we have uniform $C^{1,1}$-estimates and by Theorem \ref{thm:C1} we have uniform $C^1$-estimates of the sets $\partial \{u_\varepsilon>\varepsilon^\gamma\}$ when $y_1$ and $\varepsilon$ are smaller than $\varepsilon_0$. Letting $\varepsilon\to 0$ we have $u_\varepsilon\to u_0$. Observing that Proposition \ref{prop:nondeg} implies that the limiting set is indeed the free boundary, $\partial\{u_0>0\}$, the result follows. \end{proof} \subsection*{Acknowledgements} I would like to take the opportunity to thank my supervisor Henrik Shahgholian for introducing me to the problem and for all his pertinent comments and suggestions. Furthermore, I am grateful to R\'egis Monneau for carefully reading the paper and coming with many valuable comments. This research project is part of the ESF program GLOBAL. The author has been supported by grant KAW 2005.0098 from the Knut and Alice Wallenberg Foundation. Part of this work was carried out during a one-month stay at the Petroleum Institute, Abu Dhabi, partially supported by STINT foundation. Moreover, I thank the Petroleum Institute for the hospitality during my visit. \begin{thebibliography}{1} \bibitem{ACF84} Hans~Wilhelm Alt, Luis~A. Caffarelli, and Avner Friedman. \newblock Variational problems with two phases and their free boundaries. \newblock {\em Trans. Amer. Math. Soc.}, 282(2):431--461, 1984. \bibitem{BK73} Ha\"\i m Br\'ezis and David Kinderlehrer. \newblock The smoothness of solutions to nonlinear variational inequalities. \newblock {\em Indiana Univ. Math. J.}, 23:831--844, 1973/74. \bibitem{Caf98} L.~A. 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