R. Duduchava and B. Silbermann

Boundary Value Problems in Domains with Peaks

abstract:
We obtain criteria of solvability of the Dirichlet and the Neumann boundary value problems (BVPs) for the Laplacian in 2D domains with angular points and peaks on the boundary. We start with the correct formulation of BVPs and modify it for domains with outward peaks (classical conditions are incorrect). Boundary integral equations (BIEs), obtained by the indirect potential method, turn out to be equivalent to the corresponding BVPs only when inward peaks are absent. BIEs on boundary curve with angular points are investigated in different weighted function spaces. If boundary curve has a cusp, corresponding to an inward or an outward peak, equations are non--Fredholm in usual spaces and we should impose restrictions on the right--hand sides. The conditions are defined with the Cesaro--type integrals. We consider also equivalent reduction to boundary pseudo-differential equations (BPsDEs) of orders $\pm1$ by the direct potential method. Crucial role in our investigations of BVPs and of corresponding BIEs, PsDOs belongs to the equivalent reduction of BVPs to the Riemann--Hilbert problem for analytic functions on the unit disk. The latter problem can be investigated thoroughly, even when peaks are present and equations have non--closed image by invoking results on convolution equations with vanishing symbols.

Mathematics Subject Classification: 47A68, 35J25, 35J55.

Key words and phrases: Boundary integral equation, Convolution equation, Non-elliptic symbols, Logarithmic potentials, Boundary value problems