Narrow Results

A family of the spread harmonic measures is naturally generated by partially reflected Brownian motion. Their relation to the mixed boundary value problem makes them important to characterize the transfer capacity of irregular interfaces in Laplacian transport processes. This family presents a continuous transition between the harmonic measure (Dirichlet condition) and the Hausdorff measure (Neumann condition). It is found that the scaling properties of the spread harmonic measures on prefractal boundaries are characterized by a set of multifractal exponent functions depending on the only scaling parameter. A conjectural extension of the spread harmonic measures to fractal boundaries is proposed. The developed concepts are applied to give a new explanation of the anomalous constant phase angle frequency behavior in electrochemistry.

Let f be a non-negative C¹-function on [0, ∞) such that f(u)/u is increasing and , where . Assume Ω ⊂ R^N is a smooth bounded domain, a is a real parameter and b ≥ 0 is a continuous function on , b≢0. We consider the problem Δu+au =b(x)f(u) in Ω and we prove a necessary and sufficient condition for the existence of positive solutions that blow-up at the boundary. We also deduce several existence and uniqueness results for a related problem, subject to homogeneous Dirichlet, Neumann or Robin boundary condition.