Uniform Equicontinuity for a family of Zero Order operators approaching the fractional Laplacian

Abstract

In this paper we consider a smooth bounded domain $\Omega \subset \R^N$ and a parametric family of radially symmetric kernels $K_\epsilon: \R^N \to \R_+$ such that, for each $\epsilon \in (0,1)$, its $L^1-$norm is finite but it blows up as $\epsilon \to 0$. Our aim is to establish an $\epsilon$ independent modulus of continuity in ${\Omega}$, for the solution $u_\epsilon$ of the homogeneous Dirichlet problem \begin{equation*} \left \{ \begin{array}{rcll} - \I_\epsilon [u] \&=\& f \& \mbox{in} \ \Omega. \\ u \&=\& 0 \& \mbox{in} \ \Omega^c, \end{array} \right . \end{equation*} where $f \in C(\bar{\Omega})$ and the operator $\I_\epsilon$ has the form \begin{equation*} \I_\epsilon[u](x) = \frac12\int \limits_{\R^N} [u(x + z) + u(x - z) - 2u(x)]K_\epsilon(z)dz \end{equation*} and it approaches the fractional Laplacian as $\epsilon\to 0$. The modulus of continuity is obtained combining the comparison principle with the translation invariance of $\I_\epsilon$, constructing suitable barriers that allow to manage the discontinuities that the solution $u_\epsilon$ may have on $\partial \Omega$. Extensions of this result to fully non-linear elliptic and parabolic operators are also discussed.