Some new monotonicity theorems with applications to free boundary problems

The monotonicity theorem of Alt, Caffarelli, and Friedman [ACF, Lemma 5.1] plays a central role in the existence and regularity theory of two-phase free boundary problems. It has been extended, for example, to variable coefficient operators [C3, Lemma 1] and to eigenvalue problems [FL]. In this paper we reformulate the theorem so that it applies to inhomogeneous equations in which the right-hand side of the equation need not vanish at the free boundary. Of particular interest is the case in which Au takes two different constant values on the set where u > 0 and on the set where u < 0. The Prandtl-Batchelor