Subsolutions of Elliptic Operators in Divergence Form and Application to Two-Phase Free Boundary Problems
1 Dipartimento di Matematica, Università di Bologna, Piazza di Porta S.~Donato 5, Bologna 40126, Italy
2 C.I.R.A.M., Via Saragozza 8, Bologna 40123, Italy
3 Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 7, Milano 20133, Italy
Boundary Value Problems 2007, 2007:057049 doi:10.1155/2007/57049Published: 5 December 2006
Let be a divergence form operator with Lipschitz continuous coefficients in a domain , and let be a continuous weak solution of in . In this paper, we show that if satisfies a suitable differential inequality, then is a subsolution of away from its zero set. We apply this result to prove regularity of Lipschitz free boundaries in two-phase problems.