The dirichlet problem for harmonic functions from variable exponent smirnov classes in domains with piecewise smooth boundary

Abstract

The Dirichlet problem is solved for harmonic functions from variable exponent Smirnov classes in domains with piecewise smooth boundaries. The solvability conditions are established. Depending on the boundary geometry and value of the space exponent at angular points, the Dirichlet problem may turn out to be unsolvable, solvable uniquely and non-uniquely. In the unsolvable case, for boundary functions the necessary and sufficient conditions are found, which govern the solvability. In all solvability cases, solutions are constructed in explicit form. Bibliography: 19 titles.