Boundary trace of positive solutions of semilinear elliptic equations in Lipschitz domains

TL;DR

This paper investigates the boundary behavior of positive solutions to semilinear elliptic equations in Lipschitz domains, defining a boundary trace and analyzing singularities, removability, and existence of solutions with prescribed boundary data.

Contribution

It introduces a boundary trace concept for solutions in Lipschitz domains and characterizes singularities, removability, and existence results, especially for power nonlinearities and polyhedral domains.

Findings

Sharp criteria for removability of boundary singularities.

Existence and uniqueness of solutions with arbitrary boundary trace.

Characterization of boundary measures and removable sets using Besov spaces.

Abstract

We study the generalized boundary value problem for nonnegative solutions of $-\Delta u+g(u)=0$ in a bounded Lipschitz domain $\Gw$, when $g$ is continuous and nondecreasing. Using the harmonic measure of $\Gw$, we define a trace in the class of outer regular Borel measures. We amphasize the case where $g(u)=|u|^{q-1}u$, $q>1$. When $\Gw$ is (locally) a cone with vertex $y$, we prove sharp results of removability and characterization of singular behavior. In the general case, assuming that $\Gw$ possesses a tangent cone at every boundary point and $q$ is subcritical, we prove an existence and uniqueness result for positive solutions with arbitrary boundary trace. We obtain sharp results involving Besov spaces with negative index on k-dimensional edges and apply our results to the characterization of removable sets and good measures on the boundary of a polyhedron.