Abstract
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.
Highlights
Introduction and main resultsIn the study of the regularity of two-phase elliptic and parabolic problems, a key role is played by certain continuous perturbations of the solution, constructed as supremum of the solution itself over balls of variable radius
We show that if φ satisfies a suitable differential inequality, vφ(x) = sup Bφ(x)(x)u is a subsolution of Lu = 0 away from its zero set
The crucial fact is that if the radius satisfies a suitable differential inequality, modulus a small correcting term, the perturbations turn out to be subsolutions of the problem, suitable for comparison purposes. This kind of subsolutions have been introduced for the first time by Caffarelli in the classical paper [1] in order to prove that, in a general class of two-phase problems for the laplacian, Lipschitz free boundaries are C1,α
Summary
In the study of the regularity of two-phase elliptic and parabolic problems, a key role is played by certain continuous perturbations of the solution, constructed as supremum of the solution itself over balls of variable radius. The proof of Theorem 1.2 goes along well-known guidelines and consists in the following three steps: to improve the Lipschitz constant of the level sets of u far from F(u), to carry this interior gain to the free boundary, to rescale and iterate the first two steps. This procedure gives a geometric decay of the Lipschitz constant of F(u) in dyadic balls that corresponds to a C1,γ regularity of F(u) for a suitable γ.
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