Abstract
This chapter presents Dirichlet's problem and its applications. For the multiplicity and variety of its applications, Dirichlet's problem occupies an exceptional place in mathematics. The chapter discusses the two-dimensional problems that are of particular interest from the abundance of its applications and also for the high state of development and effectiveness of the methods for its solution. Dirichlet's problem consists of the following: to find the function that is harmonic in the interior of a region and that takes stipulated values on the boundary of the region. The Dirichlet problem is readily solved, if one knows the conformal transformation of the region onto a circle. Conversely, if for a certain simply-connected region one knows the solution of Dirichlet's problem, then it is possible to deduce the function conformally transforming this region onto a circle.
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