BOUNDARY REGULARITY FOR A DEGENERATE ELLIPTIC EQUATION WITH MIXED BOUNDARY CONDITIONS

We consider a function U satisfying a degenerate elliptic equation on R-+(N+1) :-(0, +infinity) x R-N with mixed Dirichlet-Neumann boundary conditions. The Neumann condition is prescribed on a bounded domain Omega subset of R-N of class C-1,C-1, whereas the Dirichlet data is on the exterior of Omega. We prove Holder regularity estimates of U/d(Omega)(s), where d(Omega) is a distance function defined as d(Omega)(z) := dist(z, R-N \ Omega), for z is an element of<(R-+(N+1))over bar>. The degenerate elliptic equation arises from the Caffarelli-Silvestre extension of the Dirichlet problem for the fractional Laplacian. Our proof relies on compactness and blow-up analysis arguments.