Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Subsolutions of Elliptic Operators in Divergence Form and Application to Two-Phase Free Boundary Problems | Hindawi Publishing Corporation Boundary Value Problems Volume 2007 Article ID 57049 21 pages doi 2007 57049 Research Article Subsolutions of Elliptic Operators in Divergence Form and Application to Two-Phase Free Boundary Problems Fausto Ferrari and Sandro Salsa Received 29 May 2006 Accepted 10 September 2006 Recommended by Jose Miguel Urbano Let L be a divergence form operator with Lipschitz continuous coefficients in a domain o and let u be a continuous weak solution of Lu 0 in u 0 . In this paper we show that if 0 satisfies a suitable differential inequality then V0 x supB x x U is a subsolution of Lu 0 away from its zero set. We apply this result to prove cu regularity of Lipschitz free boundaries in two-phase problems. Copyright 2007 F. Ferrari and S. Salsa. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction and main results 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 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 indeed c1 a. This result has been subsequently extended to more general operators Feldman 2 considers linear anisotropic operators with constant coefficients Wang 3 a class of concave fully nonlinear operators of the type F D2u and again Feldman 4 fully nonlinear operators not necessary concave of the type F D2u Du .
