# Dirichlet Problem

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## Dirichlet problem

[‚dē·rē′klā ‚präb·ləm]
(mathematics)
To determine a solution to Laplace's equation which satisfies certain conditions in a region and on its boundary.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Dirichlet Problem

(named after P. G. L. Dirichlet), the problem of finding a harmonic function from its values given on the boundary of the region under consideration.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
It is known that the Poisson kernel function is an analogue of the Cauchy kernel for the analytic functions and the Poisson integral formula solves the Dirichlet problem for the inhomogeneous Laplace equation.
In this paper we study the following parametric nonlinear Dirichlet problem:
Consequently [v.sub.k] is the solution of Dirichlet problem
As a quasioptimal preconditioner, we suggest to use the spectrally adapted matrix of the corresponding Dirichlet problem with homogeneous isotropic coefficients in the same computational domain.
We recall that we are considering the shape optimization problem (7) where [u.sub.D] solves the pure Dirichlet problem (4) and [u.sub.N] solves the Neumann problem (6).
Zhang; Nonexistence of positive classical solutions of a singular nonlinear Dirichlet problem with a convection term, Nonlinear Analysis 8 (1996) 957-961
Grebennikov, The study of the approximation quality of GR-method for solution of the Dirichlet problem for Laplace equation.
From the uniqueness theorem for the exterior Dirichlet problem [5,6] we obtain div [u.sub.s] = 0 in [D.sub.e].
No date is cited for the first edition, but to this second has been added a final chapter on the existence of solutions, primarily the Dirichlet problem for various types of elliptic equations.
Many authors have investigated the Dirichlet problem in simply connected domains.
Zhang, Existence of solutionas for p(x)-Laplacian Dirichlet problem, Nonlinear Analysis: TMA, 52(2003), 1843-1852.
They cover the Cauchy problem, the Dirichlet problem, the Neumann problem, the Neumann problem for a nonlocal nonlinear diffusion equation, nonlocal p-Laplacian evolution problems, the nonlocal total variation flow, and nonlocal models for sandpiles.

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