THE DIRICHLET PROBLEM FOR VARIABLE EXPONENT SMIRNOV CLASS HARMONIC FUNCTIONS IN DOUBLY-CONNECTED DOMAINS

In the present work we consider the Dirichlet problem in a doubly-connected domain D with an arbitrary piecewise smooth boundary F in a class of those harmonic functions which are real parts of analytic in D functions of Smirnov class E-P1(t),E-P2(t)(D) with variable exponents p(1) (t) and p(2) (t). It is shown that depending on the geometry of Gamma and the functions p(i), i = 1, 2, the problem may turn out to be uniquely and non-uniquely solvable or, generally speaking, unsolvable at all. In the latter case we have found additional (necessary and sufficient) conditions for the given on the boundary functions ensuring the existence of a solution. In all cases, where solutions do exist, they are constructed in quadratures.