On the boundary values of harmonic functions (Q1081731)

Let D be a domain with smooth boundary \(\partial D\) in the Euclidean space of dimension \(n\geq 3\). Given a continuous function w on \(\partial D\) with a continuous extension to D having a finite Dirichlet integral, it is known that the Dirichlet solution in D of f is the orthogonal projection of w onto the Hilbert space H of harmonic functions in D with finite Dirichlet integrals. Under hypotheses prescribed above the author shows that this orthogonal projection tends to w at all points of \(\partial D\) irrespective of the dimension n.