Nonlinear boundary value problems relative to harmonic functions

Abstract

We study the problem of finding a function u verifying --$\Delta$u = 0 in $\Omega$ under the boundary condition $\partial$u $\partial$n + g(u) = $\mu$ on $\partial$$\Omega$ where $\Omega$ $\subset$ R N is a smooth domain, n the normal unit outward vector to $\Omega$, $\mu$ is a measure on $\partial$$\Omega$ and g a continuous nondecreasing function. We give sufficient condition on g for this problem to be solvable for any measure. When g(r) = |r| p--1 r, p > 1, we give conditions in order an isolated singularity on $\partial$$\Omega$ be removable. We also give capacitary conditions on a measure $\mu$ in order the problem with g(r) = |r| p--1 r to be solvable for some $\mu$. We also study the isolated singularities of functions satisfying --$\Delta$u = 0 in $\Omega$ and $\partial$u $\partial$n + g(u) = 0 on $\partial$$\Omega$ \ {0}.