Boundary singularities of solutions to elliptic viscous Hamilton-Jacobi equations

TL;DR

This paper investigates boundary singularities of solutions to elliptic viscous Hamilton-Jacobi equations, establishing existence, boundary trace characterization, and singularity behavior under various growth conditions of the nonlinear term.

Contribution

It provides new existence results for measure boundary data, characterizes boundary traces for power-type nonlinearities, and analyzes boundary singularities of solutions.

Findings

Existence of solutions with measure boundary data under integrability conditions.

Boundary traces can be outer regular Borel measures, possibly unbounded.

Solutions exhibit boundary singularities depending on the growth of the nonlinear term.

Abstract

We study the boundary value problem with measures for (E1) $-\Gd u+g(|\nabla u|)=0$ in a bounded domain $\Gw$ in $\BBR^N$, satisfying (E2) $ u=\gm$ on $\prt\Gw$ and prove that if $g\in L^1(1,\infty;t^{-(2N+1)/N}dt)$ is nondecreasing (E1)-(E2) can be solved with any positive bounded measure. When $g(r)\geq r^q$ with $q>1$ we prove that any positive function satisfying (E1) admits a boundary trace which is an outer regular Borel measure, not necessarily bounded. When $g(r)=r^q$ with $1