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 Dirichlet problem is to find a harmonic function in a domain, with boundary conditions given by a continuous function ${\displaystyle \varphi (x)}$. The Perron solution is defined by taking the pointwise supremum over a family of functions ${\displaystyle S_{\varphi }}$,

where ${\displaystyle S_{\varphi }}$ is the set of all subharmonic functions such that ${\displaystyle v(x)\leq \varphi (x)}$ on the boundary of the domain.

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