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. 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. 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. The classical methods of potential theory allow the Dirichlet problem to be solved directly in terms of integral operators, for which the standard theory of compact and Fredholm operators is applicable. The same methods work equally for the Neumann problem.
Read more about this topic: Dirichlet Problem
Famous quotes containing the words methods of, methods and/or solution:
“All men are equally proud. The only difference is that not all take the same methods of showing it.”
—François, Duc De La Rochefoucauld (1613–1680)
“It would be some advantage to live a primitive and frontier life, though in the midst of an outward civilization, if only to learn what are the gross necessaries of life and what methods have been taken to obtain them.”
—Henry David Thoreau (1817–1862)
“There is a lot of talk now about metal detectors and gun control. Both are good things. But they are no more a solution than forks and spoons are a solution to world hunger.”
—Anna Quindlen (b. 1953)