Editing Dirichlet problem (section)

==Methods of solution==
For bounded domains, the Dirichlet problem can be solved using the [[Perron method]], which relies on the [[maximum principle]] for [[subharmonic function]]s. This approach is described in many text books.<ref>See for example:
* {{harvnb|John|1982}}
* {{harvnb|Bers|John|Schechter|1979}}
* {{harvnb|Greene|Krantz|2006}}
</ref> It is not well-suited to describing smoothness of solutions when the boundary is smooth. Another classical [[Hilbert space]] approach through [[Sobolev space]]s does yield such information.<ref>See for example:
* {{harvnb|Bers|John|Schechter|1979}}
* {{harvnb|Chazarain|Piriou|1982}}
* {{harvnb|Taylor|2011}}
</ref> 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]]. {{harvtxt|Bell|1992}} has outlined a different approach for establishing the smooth Riemann mapping theorem, based on the [[reproducing kernel]]s 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 operator]]s, for which the standard theory of [[compact operator|compact]] and [[Fredholm operator]]s is applicable. The same methods work equally for the [[Neumann problem]].<ref>See:
* {{harvnb|Folland|1995}}
* {{harvnb|Bers|John|Schechter|1979}}</ref>