\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small Two nonlinear days in Urbino 2017,\newline \emph{Electronic Journal of Differential Equations}, Conference 25 (2018), pp. 55--63.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \setcounter{page}{55} \title[\hfilneg EJDE-2018/conf/25\hfil Regularity in transmission problems] {Regularity of transmission problems for uniformly elliptic fully nonlinear equations} \author[D. De Silva, F. Ferrari, S. Salsa \hfil EJDE-2018/conf/25\hfilneg] {Daniela De Silva, Fausto Ferrari, Sandro Salsa} \address{Daniela De Silva \newline Department of Mathematics, Barnard College, Columbia University, New York, NY 10027, USA} \email{desilva@math.columbia.edu} \address{Fausto Ferrari \newline Dipartimento di Matematica dell'Universit\`a di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna, Italy} \email{fausto.ferrari@unibo.it} \address{Sandro Salsa \newline Dipartimento di Matematica del Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy} \email{sandro.salsa@polimi.it} \dedicatory{Dedicated to our loved Anna Aloe} \thanks{Published September 15, 2018} \subjclass[2010]{35J60, 35B65} \keywords{Transmission problems; fully nonlinear equations; \hfill\break\indent regularity of solutions} \begin{abstract} We investigate the regularity of transmission problems for a general class of uniformly elliptic fully non linear equations. We prove that, if the forcing term is Lipschitz, then viscosity solution are $C^{1,\gamma}$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction and main results} In this article we consider viscosity solutions to the transmission problem \begin{equation} \begin{gathered} \mathcal{F}^{+}(D^2u)=f^{+} \quad \text{in }B_1^{+}:=B_1(0) \cap \{x_n>0\} \\ \mathcal{F}^{-}(D^2u)=f^{-} \quad \text{in }B_1^{-}:=B_1(0) \cap \{x_n<0\}\\ a( u_n) ^{+}-b( u_n) ^{-}=0 \quad \text{on }B_1':=B_1( 0) \cap \{ x_n=0\}, \end{gathered} \label{tp} \end{equation} where $( u_n) ^{+}$ and $( u_n) ^{-}$ denote the derivatives in the $e_n$ direction of $u$ restricted to the upper and lower half ball, respectively. Here $a>0,b\geq 0$ and $\mathcal{F}^{\pm }$ are fully nonlinear uniformly elliptic operators, with ellipticity constants $\Lambda \geq \lambda >0$ and $\mathcal{F}^{\pm }(0)=0$. That is, for every $M,N\in \mathcal{S}^{n}$, \[ \mathcal{M}_{\lambda,\Lambda }^{-}(N)\leq \mathcal{F}^\pm(M+N)-\mathcal{F}^\pm (M)\leq \mathcal{M}_{\lambda,\Lambda }^{+}(N), \] where $\mathcal{S}^{n}$ denotes the set of square symmetric $n\times n$ matrices and $\mathcal{M}_{\lambda ,\Lambda }^{-}$, $\mathcal{M}_{\lambda,\Lambda }^{+}$ are the extremal Pucci operators defined by \begin{gather*} \mathcal{M}_{\lambda ,\Lambda }^{-}(N) =\inf_{A\in \mathcal{A}_{\lambda ,\Lambda }}\operatorname{Tr}(AN),\quad \mathcal{M}_{\lambda ,\Lambda }^{+}(N)=\sup_{A\in \mathcal{A}_{\lambda ,\Lambda }} \operatorname{Tr}(AN), \\ \mathcal{A}_{\lambda ,\Lambda }=\{A\in \mathcal{S}^{n}: \lambda I\leq A\leq \Lambda I\}. \end{gather*} In the sequel we write $\mathcal{F}^{\pm }(D^2u)=f^{\pm }$, in $B_1^{\pm }$ to denote both interior equations in \eqref{tp}. Now we define viscosity solution for problem \eqref{tp}. \begin{definition} \label{viscosity} \rm We say that $u\in C( B_1) $ is a viscosity subsolution (supersolution) to \eqref{tp} if \begin{itemize} \item[(i)] $\mathcal{F}^{\pm }(D^2u)\geq f^{\pm }$ $(\leq f^{\pm })$ in $B_1^{\pm }$ in the viscosity sense; \item[(ii)] Let $x_0\in \{ x_n=0\} $, $\delta >0$ small, and $\varphi \in C^2(\overline{B}_{\delta }^{+ }( x_0) )\cap C( \overline{B}_{\delta }^-( x_0))$. If $\varphi $ touches $u$ from above (below) at $x_0$, then \[ a( \varphi _n) ^{+}( x_0) -b( \varphi_n) ^{-}( x_0) \geq 0 \quad ( \leq 0) . \] \end{itemize} \end{definition} We say that $u$ is a viscosity solution if it is both a viscosity subsolution and supersolution. It is easily seen that condition (ii) in Definition \ref{viscosity} can be replaced by the following condition \begin{itemize} \item[(ii')] Let \[ \psi ( x) =P( x') +px_n^{+}-qx_n^{-}, \] where $P$ is a quadratic polynomial. If $\psi $ touches from above (below) $u $ at $x_0\in \{ x_n=0\} $, then \[ ap-bq\geq 0 \quad (\leq 0) . \] \end{itemize} Transmission problems as \eqref{tp} play a key role for example in the regularity theory for two-phase free boundary problems developed by the authors in \cite{DFS,DFS2,DFS3}. Our purpose here is to review and extend to the case of distributed sources the regularity results about transmission problems provided in \cite{DFS3} for the homogeneous case. Our main result is the following one. \begin{theorem} \label{main} Let $u$ be a viscosity solution to \eqref{tp} in $B_1$. Assume that $f^{\pm }\in C^{0,1}(B_1^{\pm })\cap L^{\infty }( B_1) $. Then, for any $\rho <1$, $u\in C^{1,\alpha }(\bar{B}_{\rho}^{\pm })$ with norm bounded by a constant depending on $n$, $\lambda,\Lambda $, $\rho$, $\| u\| _{\infty }$, $\|f\|_\infty$ and $\|f^{\pm }\| _{C^{0,1}}$. In particular the transmission condition is satisfied in the classical sense. \end{theorem} \section{H\"{o}lder continuity} In this section we prove the H\"{o}lder continuity of a solution $u$ to problem \eqref{tp}. Here we only need $f^\pm\in C( B^\pm_1) \cap L^{\infty }( B_1) $. We introduce a special class of functions, based on the extremal Pucci operators, in the spirit of \cite{CC}. Since $a>0$ and $b\geq 0$ are defined up to a multiplicative constant and the problem is invariant under reflection with respect to $\{x_n=0\}$, we can assume that $a=1$, $0\leq b\leq 1$. We denote by $\underline{\mathcal{S}}_{\lambda ,\Lambda }( f^\pm) $ the class of continuous functions $u$ in $B_1$ such that \begin{gather*} \mathcal{M}_{\lambda,\Lambda }^{+}( D^2u) \geq f^{\pm }\quad \text{in }B_1^{\pm}, \quad\text{and}\\ ( u_n) ^{+}-b( u_n) ^{-}\geq 0\quad \text{on } B_1' \end{gather*} in the sense of Definition \ref{viscosity} with comparison functions touching $u$ from above. Analogously, we denote by $\overline{\mathcal{S}}_{\lambda ,\Lambda }( f^{\pm }) $ the class of continuous functions $u$ in $B_1$ such that \begin{gather*} \mathcal{M}_{\lambda,\Lambda }^{-}( D^2u) \leq f^{\pm } \quad\text{in }B_1^{\pm }, \quad\text{and}\\ ( u_n) ^{+}-b( u_n) ^{-}\leq 0\quad \text{on }B_1' \end{gather*} in the sense of Definition \ref{viscosity} with comparison functions touching $u$ from below. Finally we set \[ \mathcal{S}_{\lambda ,\Lambda }^{\ast }( f^{\pm }) =\underline{\mathcal{S}}_{\lambda ,\Lambda }( -| f^{\pm }|) \cap \overline{\mathcal{S}}_{\lambda ,\Lambda }(| f^{\pm }| ) . \] \begin{theorem} \label{thm2.1} Let $u\in \mathcal{S}_{\lambda ,\Lambda }^{\ast }( f^{\pm }) $ with $f^{\pm }\in C( B^\pm_1) \cap L^{\infty }( B_1) $. Then $u\in C^{\alpha }( B_{1/2}) $ with $\alpha $ and $\|u\|_{C^{\alpha}(B_{1/2})}$ depending on $n,\lambda ,\Lambda , \|f^{\pm }\| _{\infty }$ and $\| u\| _{\infty }$. \end{theorem} It is sufficient to show that $u\in C^{\alpha }( B_{\rho _0}) $ with $\rho _0$ small depending only on $\| f^{\pm }\| _{\infty }$. Then after the rescaling $u( x) \to u(rx) $, the theorem follows easily from the iteration of the following lemma. Here constants depending on $n, \lambda, \Lambda$ are called universal. \begin{lemma} Let $u\in \mathcal{S}^*_{\lambda ,\Lambda }( f^{\pm }) $ with $\| u\| _{\infty }\leq 1$ and $\| f^{\pm}\| _{\infty }\leq \varepsilon _0$ in $B_1$, $\epsilon_0$ small universal. Assume that at $\bar{x}=\frac{1}{5}e_n$ \begin{equation} u( \bar{x}) >0. \label{pos} \end{equation} Then $u\geq -1+c$ in $B_{1/3}$ with $0\max \{0,\frac{\Lambda}{\lambda}(n-1)-1\}$ and $\eta$ to be chosen later. Since $\Gamma ^{\gamma }$ is a radial function, in an appropriate system of coordinates \[ D^2w=\eta \gamma r^{-\gamma -2}\text{diag}\{( \gamma +1) ,-1,\dots,-1\}. \] Hence, in $D$, \begin{equation} \mathcal{M}_{\lambda,\Lambda }^{-}( D^2w) \geq \eta (\gamma r^{-\gamma -2} (\lambda ( \gamma +1) -\Lambda ( n-1))) +\varepsilon _0 \geq \| f^{\pm}\| _{\infty }. \label{ps} \end{equation} Since $\partial _n\Gamma ^{\gamma }>0$ on $\{x_n=0\}$, the transmission condition \begin{equation} ( w_n) ^{+}-b( w_n) ^{-}>0\quad \text{on } \{x_n=0\}, \label{ps1} \end{equation} is satisfied. Now choose $\eta, \epsilon_0$ small universal so that \[ w\leq c_0\quad \text{on }\partial B_{1/20}( \bar{x}) . \] Then, by choosing $\epsilon_0$ possibly smaller, we also have $w\leq 0$ on $\partial B_{3/4}( \bar{x}) $. Thus, in view of \eqref{cc} we obtain that \[ w\leq u+1\quad \text{on }\partial D. \] From \eqref{ps}, \eqref{ps1} $w$ is a strict classical subsolution of the transmission problem for the operator $\mathcal{M}_{\lambda ,\Lambda }^{-}$ with right hand sides $|f^\pm|$. By the definition of viscosity solution, we conclude that it must be \[ w\leq u+1\quad \text{in }D. \] Since $w\geq c$ in $B_{1/3}$ the proof is complete. \end{proof} \section{Upper and lower envelopes} Given a continuous function $u$ in $B_1( 0) $ and $\overline{B}_{\rho }\subset B_1( 0) $ we define for $\varepsilon >0$ the upper $\varepsilon$-envelope of $u$ in the $x'$-direction, \[ u^{\varepsilon }( y',y_n) =\sup_{x\in \overline{B} _{\rho }\cap \{ x_n=y_n\} }\{ u( x',y_n) -\frac{1}{\varepsilon }| x'-y'| ^2\} \quad y=( y',y_n) \in B_{\rho }. \] Note that there is $y_{\varepsilon }\in \overline{B}_{\rho }\cap \{ x_n=y_n\} $ such that \[ u^{\varepsilon }( y) =u( y_{\varepsilon }) -\frac{1}{ \varepsilon }| y_{\varepsilon }'-y'| ^2 \] with $| y'-y_{\varepsilon }'| \leq \sqrt{2\varepsilon \| u\|_{\infty }}$, since $u^{\varepsilon }( y) \geq u( y) $ and \[ \frac{1}{\varepsilon }| y_{\varepsilon }'-y'| ^2=u( y_{\varepsilon }) -u^{\varepsilon }(y) \leq u( y_{\varepsilon }) -u( y) . \] \begin{lemma} \label{conv} The following properties hold: \begin{itemize} \item[(1)] $u^{\varepsilon }\in C( B_{\rho }) $ and $u^{\varepsilon }\downarrow u$ uniformly in $B_{\rho }$ as $ \varepsilon \to 0$. \item[(2)] $u^{\varepsilon }$ is $C^{1,1}$ in the $ x'$-direction by below in $B_{\rho }$. Thus $ u^{\varepsilon }$ is pointwise second order differentiable in the $x'$-direction al almost every point of $B_{\rho}$. \item[(3)] If $u$ is a viscosity solution to \eqref{tp}, then, in a smaller ball $B_{r}$, $r\leq \rho -2\sqrt{\varepsilon \| u\| _{\infty }}$, $u^{\varepsilon }$ is a viscosity subsolution to \begin{equation} \begin{gathered} \mathcal{F}^{\pm }( D^2u^\epsilon) =f^{\pm }-\omega _{f^{\pm }}( \sqrt{2\varepsilon \| u\| _{\infty }}) \quad \text{in }B_{r}^{\pm } \\ ( u_n^{\varepsilon }) ^{+}-b( u_n^{\varepsilon })^{-}=0 \quad \text{on }B_{r}' \end{gathered} \label{pl1} \end{equation} where $\omega _{f^{\pm }}$ denotes the modulus of continuity of $f^{\pm }$. \end{itemize} \end{lemma} \begin{proof} (1) follows from \[ | u^{\varepsilon }( y_0) -u^{\varepsilon }(y_1) | \leq \frac{6\rho }{\varepsilon }|y_0'-y_1'| \] and \[ 0\leq u^{\varepsilon }( y) -u( y) =u(y_{\varepsilon }) -\frac{1}{\varepsilon }| y_{\varepsilon }'-y'| ^2-u( y) \leq \omega_u( \sqrt{2\varepsilon \| u\|_{\infty }}) \] where $\omega _u$ is the modulus of continuity of $u$. (2) follows from Alexandrov Theorem on concave/convex functions, since \[ u^{\varepsilon }( y',y_n) +\frac{1}{\varepsilon }| y'| ^2 =\sup_{x\in \overline{B}_{\rho }\cap \{ x_n=y_n\} } \big\{ u( x',y_n) -\frac{1}{\varepsilon }| x'| ^2+\frac{2}{\varepsilon } \langle x',y'\rangle \big\} \] is convex, being the supremum of a family of affine functions of $y'$. (3) Let $\varphi \in C^2( B_{r}) $ touch from above $u^{\varepsilon }$ at a point $\bar{x}\in B_{r}^{+}$. Then ($\varepsilon $ small) \[ u^{\varepsilon }( \bar{x}) =u( \bar{x}_{\varepsilon }) -\frac{1}{\varepsilon }| \bar{x}_{\varepsilon }'-\bar{x}'| ^2 \] with $| \bar{x}_{\varepsilon }'-\bar{x}'| ^2\leq 2\varepsilon \| u\| _{\infty }$. Consider the function \[ \Phi ( y) =\varphi ( y+\bar{x}-\bar{x}_{\varepsilon }) +\frac{1}{\varepsilon }| \bar{x}_{\varepsilon }'-\bar{x} _{\varepsilon }'| ^2. \] With our choice of $r$ and $y \in B_\rho^+$ close enough to $\bar x_\epsilon$, the point $y+\bar{x}-\bar{x}_{\varepsilon}\in B_{\rho}^{+}$. Thus, by the definition of $u^{\varepsilon }$, \[ u( y) \leq u^{\varepsilon }( y+\bar{x}-\bar{x}_{\varepsilon }) +\frac{1}{\varepsilon }| \bar{x}'-\bar{x} _{\varepsilon }'| ^2 \] and therefore \[ u( y) \leq \varphi ( y+\bar{x}-\bar{x}_{\varepsilon }) +\frac{1}{\varepsilon }| \bar{x}'-\bar{x}_{\varepsilon}'| ^2. \] with equality at $y=\bar{x}_{\varepsilon }$, since $\varphi ( \bar{x}) =u^{\varepsilon }( \bar{x}) $. Thus the function $\Phi $ touches from above $u$ at $\bar{x}_{\varepsilon }$. Therefore \[ \mathcal{F}^{+}( D^2\varphi ( \bar{x}) ) =\mathcal{F} ^{+}( D^2\Phi ( \bar{x}_{\varepsilon }) ) \geq f^{+}( x_{\varepsilon }) \geq f^{+}( \bar{x}) -\omega _{f^{+}}\\big(\sqrt{2\varepsilon \| u\| _{\infty }}\big). \] Similarly, we can check the transmission condition \[ ( u_n^{\varepsilon }) ^{+}-b( u_n^{\varepsilon }) ^{-}\geq 0 \quad \text{on }B_{r}'. \] Let \[ \varphi ( x) =P( x') +px_n^{+}-qx_n^{-} \] touch from above $u^{\varepsilon }$ at a point $\bar{x}=( \bar{x}',0) \in B'_{r}$, with $P$ quadratic polynomial. Then \[ P( x') +px_n^{+}-qx_n^{-}\geq u^{\varepsilon }(x) \] near $\bar{x}$ and \[ P( \bar{x}') =u^{\varepsilon }( \bar{x}) =u( \bar{x}_{\varepsilon }) -\frac{1}{\varepsilon }| \bar{x}'-\bar{x}_{\varepsilon }'| ^2. \] Then, for $r$ small and $y$ close enough to $\bar x_\epsilon$ \begin{align*} u( y) &\leq u^{\varepsilon }( y+\bar{x}-\bar{x} _{\varepsilon }) +\frac{1}{\varepsilon }| \bar{x}'- \bar{x}_{\varepsilon }'| ^2 \\ &\leq \varphi ( y+\bar{x}-\bar{x}_{\varepsilon }) +\frac{1}{ \varepsilon }| \bar{x}_{\varepsilon }'-\bar{x} _{\varepsilon }'| ^2 \end{align*} with equality at $y=\bar{x}_{\varepsilon }$ since $P( \bar{x}') =u^{\varepsilon }( \bar{x}) $. Hence $\varphi ( y+ \bar{x}-\bar{x}_{\varepsilon }) +\frac{1}{\varepsilon }| \bar{ x}_{\varepsilon }'-\bar{x}_{\varepsilon }'| ^2$ touches from above $u$ at $y=\bar{x}_{\varepsilon }$ and therefore $p-bq\geq 0$, as desired. \end{proof} Analogously we can define \[ u_{\varepsilon }( y',y_n) =\inf_{x\in \overline{B} _{\rho }\cap \{ x_n=y_n\} } \big\{ u( x',y_n) +\frac{1}{\varepsilon }| x'-y'| ^2\big\} \quad y=( y',y_n) \in B_{\rho }. \] the lower $\varepsilon$-envelope of $u$ in the $x'$-direction. Properties (1)--(3) hold with obvious changes: \begin{itemize} \item[(1')] $u_{\varepsilon }\in C( B_{\rho }) $ and $u_\epsilon \uparrow u$ uniformly in $B_{\rho }$ as $\varepsilon \to 0$. \item[(2')] $u_{\varepsilon }$ is $C^{1,1}$ in the $ x'$-direction by above in $B_{\rho }$. Thus $ u_{\varepsilon }$ is pointwise second order differentiable in the $x'$-direction al almost every point of $B_{\rho}$. \item[(3')] If $u$ is a viscosity solution to \eqref{tp}, then, in a smaller ball $B_{r}$, $r\leq \rho -2 \sqrt{\varepsilon \| u\| _{\infty }}$, $u_{\varepsilon }$ is a viscosity supersolution to \begin{equation} \begin{gathered} \mathcal{F}^{\pm }( D^2u_\epsilon) =f^{\pm }+\omega _{f^{\pm }}( \sqrt{2\varepsilon \| u\| _{\infty }}) \quad \text{in } B_{r}^{\pm } \\ ( ({u_{\epsilon}})_n) ^{+}-b( ({u_{\epsilon}})_n) ^{-}=0 \quad \text{on }B_{r}'. \end{gathered} \label{sp} \end{equation} \end{itemize} \section{Proof of Theorem \ref{main}} For the proof of Theorem \ref{main} we use the following pointwise regularity result (see \cite{MW}). \begin{theorem} \label{reg} Let $\mathcal{F}$ be a uniformly elliptic operator and $u$ be a viscosity solution to the Dirichlet problem \[ \mathcal{F}( D^2u) =f\text{ in }B_1^{+}, \quad u( x',0) =g( x') \text{ on } B_1'. \] Assume that $g$ is pointwise $C^{1,\alpha }$ at $0$ and $f\in L^{\infty}( B_1^{+}) $. Then $u$ is poinwise $C^{1,\alpha }$ at 0, that is there exists a linear function $L_u$ such that for all $r$ small \[ | u-L_u| \leq Cr^{1+\alpha }\quad \text{in }\bar{B}_{r}^{+} \] with C depending only on $n,\lambda ,\Lambda ,\| f\|_{\infty }$ and the pointwise $C^{1,\alpha }$ bound of $g$ at $0$. \end{theorem} Our main lemma reads as follows. \begin{lemma} Let $f^\pm\in C^{0,1}( B_1^{\pm }) $ and $u$ be a viscosity solution to \eqref{tp} in $B_1$. Then, for any $\sigma >0$ small, and any unit vector $e'$ in the $x'$ direction, \[ u( x+\sigma e') -u( x) \in S_{\lambda,\Lambda }^{\ast }( \omega _{f^{\pm }}( \sigma ) )\,. \] \end{lemma} \begin{proof} Let $v=u( x+\sigma e') $, $w=v-u$. Following the proof of \cite[Theorem 5.3]{CC}, with minor modification, it follows that \[ \mathcal{M}_{\lambda,\Lambda }^{+}( D^2w) \geq -\omega _{f^{\pm }}( \sigma ) \quad\text{and}\quad \mathcal{M}_{\lambda ,\Lambda }^{-}( D^2w) \leq \omega _{f^{\pm }}( \sigma ) \] in $B_1^{\pm }$. To show that the free boundary condition is satisfied in the viscosity sense, we slightly modify the technique in \cite{DFS3} for the homogeneous case. \[ \varphi ( x) =P( x')+px_n^{+}-qx_n^{-}-Cx_n^2 \] touch $w$ from above at $x_0=( x_0',0) $, where $P$ is a quadratic polynomial. Choosing suitably $C$ and possibly restricting the neighborhood around $x_0$, we can assume that \begin{equation} \label{M} \mathcal{M}_{{\lambda},\Lambda }^{+}( D^2\varphi ) \leq -2\omega _{f^{\pm }}( \sigma ) -2\omega_{f^\pm}(\sqrt{2 \| u\| _{\infty }}). \end{equation} Assume by contradiction that \[ p-bq<0. \] Let $\delta >0$ small, and consider the ring $A_{\delta}( x_0) =\bar{B}_{2\delta }( x_0) \backslash B_{\delta }( x_0) $. Without loss of generality we can assume that $\varphi $ touches $w$ strictly from above and therefore that \thinspace $\varphi -w\geq \eta >0$ on $A_{\delta }( x_0) $. Since $w_{\varepsilon }=u^{\varepsilon }-v_{\varepsilon }$ converges uniformly to $u-v$, for $\varepsilon $ small enough (up to adding a small constant), we have that $\varphi $ touches $w_{\varepsilon }$ from above at some point $x_{\varepsilon }$ and $\varphi -w_{\varepsilon }\geq \eta /2$ on $\partial B_{\delta }( x_{\varepsilon }) $. In view of \eqref{M}, we have \[ \mathcal{M}_{\lambda ,\Lambda }^{+}( D^2w_{\varepsilon }) \geq -\omega _{f^{\pm }}( \sigma ) -2\omega_{f^\pm}(\sqrt{2\varepsilon \| u\| _{\infty }})>\mathcal{M}_{\lambda ,\Lambda }^{+}( D^2\varphi )\] and therefore $x_{\varepsilon }\in \{ x_n=0\} $. Call \[ \psi =\varphi -w_{\varepsilon }-\eta /2. \] Since $\psi \geq 0$ on $\partial B_{\delta }( x_{\varepsilon }) $, $\psi ( x_{\varepsilon }) <0$ and $\varphi $ touches $w_{\varepsilon }$ from above at $x_{\varepsilon }$, from ABP estimates (\cite[Lemma 3.5]{CC}) it follows that the set of points in $B_{\delta }(x_{\varepsilon }) \cap \{ x_n=0\} $ where $\psi $ admits a touching plane $l( x') $ from below, in the $x'$ direction, of slope less than some arbitrary small number has positive measure. We choose the slope of $l$ small enough to have that $\bar{\varphi} =\varphi -l-\eta /2\geq w_{\varepsilon }$ on $\partial B_{\delta }(x_{\varepsilon }) $ and hence in the interior. By property (2) in Lemma \ref{conv} we can deduce that $\bar{\varphi}(y_{\varepsilon }) =w_{\varepsilon }( y_{\varepsilon }) $ for some $y_{\varepsilon }=( y_{\varepsilon }',0) $ where both $v^{\varepsilon }$ and $u_{\varepsilon }$ are twice pointwise differentiable in the $x'$ direction. Now we call $\bar{u}^{\varepsilon }$ the solution to $$ \mathcal{F}^{\pm }( D^2\bar u^\epsilon(x)) =f^\pm(x) -\omega_{f^\pm}(\sqrt{2\varepsilon \| u\| _{\infty }}) \quad \text{in $B_{\delta }^{\pm }( x_{\varepsilon }) $} $$ with $w=u^{\varepsilon }$ on $\partial B_{\delta}( x_{\varepsilon })$, and similarly let $\bar v_\epsilon$ the solution to $$ \mathcal{F}^{\pm }( D^2\bar v_\epsilon(x)) =f^\pm(x+\sigma e') +\omega_{f^\pm}(\sqrt{2\varepsilon \| u\| _{\infty }}) \quad \text{in $B_{\delta }^{\pm }( x_{\varepsilon }) $} $$ with $w=v_{\varepsilon }$ on $\partial B_{\delta}( x_{\varepsilon })$, Set $$ \bar{w}_{\varepsilon }=\bar{u}^{\varepsilon }-\bar{v}_{\varepsilon }. $$ Then $\mathcal{M}_{\lambda,\Lambda }^{+}( D^2\bar{w}_{\varepsilon }) \geq -\omega _{f^{\pm }}( \sigma ) -2\omega_{f^\pm}(\sqrt{2\varepsilon \| u\| _{\infty }})> \mathcal{M}_{\lambda,\Lambda }^{+}( D^2\bar{\varphi})$ in $B_{\delta }( x_{\varepsilon }) $ with $\bar{\varphi}\geq w_{\varepsilon }$ on the boundary. Thus, by comparison, we deduce that $\bar{\varphi}\geq \bar{w}_{\varepsilon }\geq w_{\varepsilon }$ also in the interior with contact from above at $y_{\varepsilon }$. Also note that, by comparison, the replacements $\bar{u}^{\varepsilon }$ and $\bar{v}_{\varepsilon }$ are sub and super solution of the transmission condition on $\{ x_n=0\} $, respectively. By the pointwise $C^{1,\alpha }$ differentiability at $y_{\varepsilon }$ of the boundary data, we conclude the $C^{1,\alpha }$ differentiability of $\bar{v}^{\varepsilon }$ and $\bar{u}_{\varepsilon }$ up to $y_{\varepsilon }$. Thus there are linear functions $L_{v},L_u$ such that, for all small $r$, \begin{gather*} | \bar{v}_{\varepsilon }-L_{v}| \leq Cr^{1+\alpha }, \\ | \bar{u}^{\varepsilon }-L_u| \leq Cr^{1+\alpha } \end{gather*} in $B_{r}^{+}( y_{\varepsilon }) $. Since $\bar{\varphi}$ touches $\bar{w}_{\varepsilon }$ from above, we get $p\geq p_u^{+}-p_{v}^{+}$, where \[ p_u^{+}=( \bar{u}^{\varepsilon }) _n^{+}( y_{\varepsilon}) , \quad p_{v}^{+}=( \bar{v}_{\varepsilon })_n^{+}( y_{\varepsilon }) . \] Arguing similarly in $B_{r}^{-}( y_{\varepsilon }) $ we infer $q\leq q_u^{-}-q_{v}^{-}$, where \[ q_u^{-}=( \bar{u}^{\varepsilon }) _n^{-}( y_{\varepsilon}) , \quad q_{v}^{-}=( \bar{v}_{\varepsilon })_n^{-}( y_{\varepsilon }) . \] In the next lemma we show that \[ p_u^{+}-p_{v}^{+}-b( q_u^{-}-q_{v}^{-}) \geq 0 \] thus reaching a contradiction. \end{proof} \begin{lemma} Let $g^{\pm }\in C( B_1^{\pm })$ and $u$ be a viscosity solution to $(0\leq b\leq 1)$ \begin{equation} \begin{gathered} \mathcal{F}^{\pm }( D^2u) =g^{\pm } \quad \text{in }B_1^{\pm } \\ ( u_n) ^{+}-b( u_n) ^{-}\geq 0\quad (\leq 0) \quad \text{on }B_1'. \end{gathered} \label{llp} \end{equation} Assume that $u$ is twice differentiable at $x=0$ in the $x'$-direction. Then $u$ is differentiable at $0$ and $$ u_n^{+}( 0) -bu_n^{-}( 0) \geq 0 \quad \text{($\leq 0$)}. $$ \end{lemma} \begin{proof} From Theorem \ref{reg} there exists a linear function $L_u$ such that, for all small $r$, \[ | u-L_u| \leq Cr^{1+\alpha }\quad \text{in }\overline{B}_{r}^{+}. \] Without loss of generality, by subtracting a linear function, we may assume that \[ L_u=( u_n) ^{+}( 0) x_n:=d^{+}x_n. \] Let $w$ be the solution to \[ \mathcal{F}^{+}( D^2w) =g^{+}\text{ in }B_{r}^{+}, \quad w=\varphi _{r} \text{ on }\partial B_{r}^{+} \] where \[ \varphi _{r}=\begin{cases} 2C| x| ^{1+\alpha } & \text{on }\partial B_{r}^{+}\cap \{ x_n>0\} \\ 2Cr^{\alpha -1}| x'| ^2 & \text{on } B_{r}'. \end{cases} \] Since $u$ is twice pointwise differentiable at $0$, we get, for $r$ small enough, \[ u-d^{+}x_n\leq \varphi _{r}\quad \text{on }\partial B_{r}^{+} \] and, by comparison, \begin{equation} u-d^{+}x_n\leq w\quad \text{in }B_{r}^{+}. \label{comp} \end{equation} Now, the rescaling $W_{r}( x) =r^{-1-\alpha }w( rx) $ solves \[ \mathcal{G}( D^2W_{r}) =r^{1-\alpha }g^+( rx)\quad \text{in }B_1^{+},\quad W_{r}( x') =2C| x'| ^2\text{ on }B_1' \] where $\mathcal{G}( M) =r^{1-\alpha }\mathcal{F}^{+}(r^{\alpha -1}M) $ has the same ellipticity constants of $\mathcal{F}^{+}$. By boundary $C^{1,\alpha }$ estimates we obtain that \[ \| W_{r}\| _{C^{1,\alpha }(B_{1/2}^{+})}\leq \bar{C} \] for a universal $\bar{C}$. In particular \[ W_{r}( x) \leq 2C| x'| ^2+\bar C x_n \quad \text{in }\bar{B}_{1/2}^{+}. \] Rescaling back we get \[ w( x) \leq 2Cr^{\alpha -1}| x'| ^2+\bar C r^{\alpha }x_n\quad \text{in }\bar{B}_{r/2}^{+}. \] From \eqref{comp}, \[ u\leq 2Cr^{\alpha -1}| x'| ^2+\bar C r^{\alpha }x_n+\ d^{+}x_n\quad \text{in }B_{r/2}^{+}. \] Arguing similarly in $B_{r}^{-}$ we find \[ u\leq 2Cr^{\alpha -1}| x'| ^2+\bar C r^{\alpha }x_n+\ d^{-}x_n\quad \text{in }B_{r/2}^{-} \] with $d^{-}=( u_n) ^{+}( 0) $. Thus \[ \varphi ( x) =2Cr^{\alpha -1}| x'| ^2+( \bar C r^{\alpha }+\ d^{+}) x_n^{+}-( -\bar C r^{\alpha }+\ d^{-}) x_n^{-} \] touches $u$ by above at zero. Therefore \[ ( \bar C r^{\alpha }+\ d^{+}) -b( -\bar C r^{\alpha }+d^{-}) \geq 0 \] for all small $r$ so that $d^{+}-bd^{-}\geq 0$. \end{proof} From Theorem \ref{reg} and the arguments in Chapter 5 in \cite{CC} we deduce the following result. \begin{corollary}\label{coro} Let $u$ be a viscosity solution to \eqref{tp}. Then $u\in C^{1,\alpha }$ in the $x'$-direction in $B_{3/4}$ with norm bounded by a constant depending on $n$, $\lambda ,\Lambda $, $\| u\| _{\infty }$ and $\| f^{\pm }\| _{C^{0,1}}$. \end{corollary} We are now ready to give the proof of our main Theorem \ref{main}. \begin{proof}[Proof of Theorem \ref{main}] Let, say $\rho =1/2$. The $C^{1,\alpha }$ regularity and the bounds on $\| u\| _{C^{1,\alpha }(B_{1/2}) }$ follow from Corollary \ref{coro} and the regularity theory for fully nonlinear uniformly elliptic equations in \cite{MW} or \cite{MiSi}. It remains to show that the transmission condition is satisfied in the classical sense. Let us prove that at $x=0$, $ap-bq\leq 0$ where $p=(u_n) ^{+}( 0) ,q=( u_n) ^{-}( 0)$. By the $C^{1,\alpha }$ regularity of $u$, after possibly subtracting the linear function $u(0)+\nabla _{x'}u( 0) \cdot x'$, we can write \begin{equation} | u( x) -( px_n^{+}-qx_n^{-}) | \leq Cr^{1+\alpha }\quad | x| \leq r. \label{est} \end{equation} For $r$ small, define \[ w_{r}( x) =Cr^{\alpha -1}(-| x|^2+Kx_n^2)-2r^{\alpha }CK| x_n| +px_n^{+}-qx_n^{-}. \] Choose $K$ large to have \begin{equation} \mathcal{M}_{\lambda,\Lambda }^{-}( D^2 w_r ) \geq C(2 \lambda (K-1) -2\Lambda n)>\| f^{\pm }\| _{\infty }. \label{super} \end{equation} Using \eqref{est}, we get $w_{r}