Stability of free boundaries

Abstract

IN THIS paper we consider a unilateral problem relative to a nonlinear elliptic operator A-G of the second order. A is a linear operator whose principal part is not in divergence form and G is a Nemytsky operator relative to a continuous function with quadratic growth in the gradient. We introduce unilateral problems that are “regularizing”, relative to a sequence of operators A” G” which converge to A G. The An’s have regular coefficients; the G”‘s have linear growth in the gradient. The operators A” can obviously be written in divergence form. In this paper we show that the free boundaries of the “regular” problems converge to the free boundary of the “nonregular” problem; the rate of convergence can be expressed in terms of the convergence of the operators. The convergence in the uniform norm of the solutions was proved in a more general framework by the present authors in [6]. In this paper an exact estimate of the rate of convergence of the solutions in terms of the convergence of the coefficients of the operators is also obtained, (for the linear case see also Troianiello [S]). This kind of estimate will be used in the present paper. Caffarelli [3] and Friedman [5] introduce an assumption of “sign” and of regularity on the data in order to study the “regularity” of the free boundary of a variational inequality relative to the Laplacian and with zero obstacle. Under this assumption they prove a maximum principle in the noncoincidence set, together with a property of “nondegeneracy” (i.e. the solution leaves the obstacle with a certain minimum speed). Brezzi and Caffarelli consider the dam problem and give an estimate of the rate of convergence of the “discrete” free boundariesrelative to the approximation with affine finite elements-to the “continuous” free boundary of the dam problem [2]. They make use of the regularity properties of the solution Wi and of the free boundary, of the property of nondegeneracy proved in [3] and [5] and of the estimate of the convergence of the discrete solutions in the uniform norm. The question we deal with in the present paper is, in a sense, opposite to that considered by Brezzi and Caffarelli; what we want to do is to approximate a problem having a “nonregular” solution by means of “regularizing” problems. We take up the hypothesis of the sign on the data again, and prove a maximum principle for the solutions, uniform in n, together