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

Abstract

This paper deals with the problem Δu=g on G and ∂u/∂n+uf=L on ∂G. Here, G⊂ℝm, 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 W1,2(G) representable by a real measure μ on the boundary of G, and g∈L2(G)∩Lp(G), 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 μ is bounded in G.