Which solutions of the third problem for the Poisson equation are bounded?

Abstract

This paper deals with the problem $\Delta u=g$ on $G$ and $\partial u/\partial n+uf=L$ on $\partial G$. Here, $G\subset {ℝ}^m$, $m>2$, is a bounded domain with Lyapunov boundary, $f$ is a bounded nonnegative function on the boundary of $G$, $L$ is a bounded linear functional on $W^{1,2}\left(G\right)$ representable by a real measure $\mu$ on the boundary of $G$, and $g\in L_2\left(G\right)\cap L_p\left(G\right)$, $p>m/2$. It is shown that a weak solution of this problem is bounded in $G$ if and only if the Newtonian potential corresponding to the boundary condition $\mu$ is bounded in $G$.