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

TL;DR

This paper establishes an epsilon-independent modulus of continuity for solutions to a family of operators approaching the fractional Laplacian, ensuring uniform regularity in bounded domains.

Contribution

It introduces a method to obtain uniform equicontinuity estimates for solutions of operators approaching the fractional Laplacian, extending to nonlinear elliptic and parabolic cases.

Findings

Uniform modulus of continuity independent of epsilon

Construction of barriers managing boundary discontinuities

Extension to nonlinear elliptic and parabolic operators

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.