Elliptic equations with diffusion coefficient vanishing at the boundary: Theoretical and computational aspects

Abstract

Abstract.A class of degenerate elliptic PDEs is considered. Specifically, it is assumed that the diffusion coefficient vanishes on the boundary of the domain. It is shown that if the diffusion coefficient vanishes fast enough, then the problem has a unique solution in the class of smooth functions even if no boundary conditions are supplied. A numerical method is derived to compute solutions for such degenerate equations. The problem is motivated by a certain approach to the recovery of the phase of a wave from intensity measurements.1. Introduction and motivation. Our goal in this paper is to study elliptic partial differential equations, where the diffusion coefficient vanishes at the boundary. We shall present a number of theoretical results on the well-posedness of such problems and a numerical algorithm for solving them.One motivation to study such problems comes from a basic question in optics: Can the phase of a wave be reconstructed from its intensity? This is an important problem that appears in many applications. In general, measuring the phase is a difficult and expensive process. In contrast to direct phase measurement, measuring the wave's intensity is relatively easy. It is therefore tempting to seek methods for finding the phase from intensity measurements. Indeed, Teague [8] proposed such a phase sensor. His method was further developed by Roddier and others.To explain the idea behind intensity-based sensors (sometimes called "curvature sensors"), consider a complex-valued wave function W in the Fresnel (paraxial) regime,