On non-local reflection for elliptic equations of the second order in R^2 (the Dirichlet condition)

TL;DR

This paper develops a non-local reflection formula for solutions of analytic elliptic PDEs across real-analytic curves, extending classical point-to-point reflection principles known for harmonic functions.

Contribution

It introduces a novel non-local reflection formula applicable to a broad class of elliptic equations with Dirichlet conditions, surpassing traditional harmonic function reflection methods.

Findings

Derived a non-local reflection formula for elliptic PDEs

Identified conditions under which the formula simplifies to point-to-point reflection

Extended reflection principles beyond harmonic functions to general elliptic equations

Abstract

Point-to-point reflection holding for harmonic functions subject to the Dirichlet or Neumann conditions on an analytic curve in the plane almost always fails for solutions to more general elliptic equations. We develop a non-local, point-to-compact set, formula for reflecting a solution of an analytic elliptic partial differential equation across a real-analytic curve on which it satisfies the Dirichlet conditions. We also discuss the special cases when the formula reduces to the point-to-point forms.