Perron's solutions for two-phase free boundary problems with distributed sources

TL;DR

This paper extends Perron method techniques to construct weak solutions for two-phase free boundary problems with sources, generalizing Caffarelli's earlier work on homogeneous cases.

Contribution

It introduces a new approach to handle distributed sources in two-phase free boundary problems using Perron solutions.

Findings

Successfully constructs weak solutions with sources.

Extends classical results to more general source terms.

Provides a framework for future research in free boundary problems.

Abstract

We use Perron method to construct a weak solution to a two-phase free boundary problem with right-hand-side. We thus extend the results of the pioneer work of Caffarelli for the homogeneous case.