G. Khuskivadze, V. Kokilashvili, and V. Paatashvili
abstract:
In the present work we consider the Dirichlet problem in a doubly-connected
domain $D$ with an arbitrary piecewise smooth boundary $\Gamma$ in a class of
those harmonic functions which are real parts of analytic in $D$ functions of
Smirnov class $E^{p_1(t),p_2(t)}(D)$ with variable exponents $p_1(t)$ and
$p_2(t)$. It is shown that depending on the geometry of $\Gm$ 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.
Mathematics Subject Classification: 30E20, 30E25, 30D55, 47B38, 42B20
Key words and phrases: Hardy and Smirnov classes, variable exponent, Cauchy type integral, harmonic functions, Dirichlet problem, doubly-connected domain, piecewise smooth boundary