Harmonic measures of perforated domains (Q1378849)

The author studies the harmonic measure of a problem with holes proving the weak* convergence to a measure \(H(x,.;\mu)\) defined by the relation \[ u(x)= \int_\Omega g(y) H(x,dy; \mu) \] for every \(g\) in \(H^1(\Omega) \cap C(\Omega)\), where \(\mu\) is a Borel measure not charging polar sets and \(u\) is the weak solution of the relaxed Dirichlet problem \[ Lu+(u-g) \mu=0, \quad u-g \in H^1_0 (\Omega) \cap L^2 (\Omega;\mu). \]