Моногенні функції в комутативних алгебрах і еліптичні рівняння математичної фізики

Abstract

An algebraic-analytic approach to elliptic equations of mathe- matical physics is developed at the Department of Complex Analysis and Potential Theory of the Institute of Mathematics of the National Academy of Sciences of Ukraine. This approach means a finding of commutative Ba- nach algebra such that differentiable in the sense of Gâteaux functions with values in this algebra have components satisfying the given equation with par- tial derivatives. Such algebras are found for the biharmonic equation and the three-dimensional Laplace equation and elliptic equations degenerating on an axis that describe axial-symmetric potential fields. An use of differentiable in the sense of Gâteaux functions given in commutative Banach algebras com- bines the preservation of basic properties of analytic functions of a complex variable for the mentioned differentiable functions and the convenience and the simplicity of construction of solutions of PDEs.