Perron’s Method

Abstract

In 1923 O. Perron published a method for solving the Dirichlet boundary value problem

$$ {\left\{ \begin{array}{ll} \Delta h=0 \text { in }\Omega ,\\ h=g \text { on }\partial \Omega \end{array}\right. } $$

and it is of interest, especially if $\partial \Omega$ or g are irregular. The same method works with virtually no essential modifications for many other partial differential equations obeying a comparison principle. We will treat it for the p-Laplace equation. The p-superharmonic and p-subharmonic functions are the building blocks. This chapter is based on [GLM].