On recent advances in boundary-value problems in the plane

Abstract

The survey is devoted to recent advances in nonclassical solutions of the main boundary-value problems such as the well-known Dirichlet, Hilbert, Neumann, Poincare, and Riemann problems in the plane. Such solutions are essentially different from the variational solutions of the classical mathematical physics and based on the nonstandard point of view of the geometric function theory with a clear visual sense. The traditional approach of the latter is the meaning of the boundary values of functions in the sense of the so-called angular limits or limits along certain classes of curves terminated at the boundary. This become necessary if we start to consider boundary data that are only measurable, and it is turned out to be useful under the study of problems in the field of mathematical physics as well. Thus, we essentially widen the notion of solutions and, furthermore, obtain spaces of solutions of the infinite dimension for all the given boundary-value problems. The latter concerns the Laplace equation, as well as its counterparts in the potential theory for inhomogeneous and anisotropic media.