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85(1): 2024/11/15(金)23:23 ID:ImmeQPY/(2/4) AAS
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Methods of solution
For bounded domains, the Dirichlet problem can be solved using the Perron method, which relies on the maximum principle for subharmonic functions. This approach is described in many text books.[2] It is not well-suited to describing smoothness of solutions when the boundary is smooth. Another classical Hilbert space approach through Sobolev spaces does yield such information.[3] The solution of the Dirichlet problem using Sobolev spaces for planar domains can be used to prove the smooth version of the Riemann mapping theorem. Bell (1992) has outlined a different approach for establishing the smooth Riemann mapping theorem, based on the reproducing kernels of Szegő and Bergman, and in turn used it to solve the Dirichlet problem.

外部ﾘﾝｸ:en.wikipedia.org
Perron method
In the mathematical study of harmonic functions, the Perron method, also known as the method of subharmonic functions, is a technique introduced by Oskar Perron for the solution of the Dirichlet problem for Laplace's equation. The Perron method works by finding the largest subharmonic function with boundary values below the desired values; the "Perron solution" coincides with the actual solution of the Dirichlet problem if the problem is soluble.

The characterization of regular points on surfaces is part of potential theory. Regular points on the boundary of a domain
Ω are those points that satisfy the Wiener criterion:
略
The Wiener criterion was first devised by Norbert Wiener; it was extended by Werner Püschel to uniformly elliptic divergence-form equations with smooth coefficients, and thence to uniformly elliptic divergence form equations with bounded measureable coefficients by Walter Littman, Guido Stampacchia, and Hans Weinberger.
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