Uniform Equicontinuity for a Family of Zero Order Operators Approaching the Fractional Laplacian

Abstract

In this paper we consider a smooth bounded domain < subset of>(N) and a parametric family of radially symmetric kernels K-epsilon: (N)(+) such that, for each epsilon (0, 1), its L-1-norm is finite but it blows up as epsilon 0. Our aim is to establish an epsilon independent modulus of continuity in , for the solution u(epsilon) of the homogeneous Dirichlet problem {-J(epsilon)[u] = f in Omega. u = 0 in Omega(c,) where f epsilon c((Omega) over bar and the operator J(epsilon) has the form J(epsilon)[u](x) = integral N-R[u(x+z) - u(x)K-epsilon(z)dz

and it approaches the fractional Laplacian as epsilon 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 . Extensions of this result to fully non-linear elliptic and parabolic operators are also discussed.