Variational Problems with Two Phases and their Free Boundaries

Abstract

The problem of minimizing /[Vu|2 + q2(x)\2(v)] dx in an appropriate class of functions v is considered.Here q(x) ¥= 0 and A2(t>) = X2 if v < 0, = X22 if v > 0. Any minimizer u is harmonic in {u ¥= 0} and | Vu|2 has a jump a2(x)(\]-\22) across the free boundary [u ¥= 0).Regularity and various properties are established for the minimizer u and for the free boundary.Introduction.In this paper we consider the problem of minimizing and X2(u) is lower semicontinuous at v = 0; it is assumed that A2 > 0 and A = A2 -A22 =£ 0. The class K consists of all functions v in L'loc(S2), with Vu G L2(Í2) such that v -u° on a given open subset S of 3Í2, and ß is a domain in R".The analogous problem for functions in K+ = {v OE K, v > 0 a.e.} was studied in [1]; in that paper it was proved that any (local) minimizer of J(v) in K+ is Lipschitz continuous and, if n -2, the free boundary 8{w > 0} is analytic if q(x) is analytic.The present variational problem is motivated by applications to the now of two liquids in models of jets and cavities; these applications will be studied in other forthcoming papers [5,6].The present work is aimed at extending results of [1].In particular, we shall establish nondegeneracy theorems, the Lipschitz continuity of the solution, and some properties of the free boundary; for n = 2 the free boundary is proved to be continuously differentiable.A new and rather powerful tool introduced in this paper is the monotonicity formula (Lemma 5.1) asserting that, for a minimizer u, if u(x0) = 0 then r~4f P2~"\ Vu+\2dx ■ ( p2-"\Vu-\2dx/> if r/•.JBjtx0) JBr(x0)