## Convexity properties of solutions to some classical variational problems

**Abstract**

COMM. IN PARTIAL DIFFERENTTAL EQUATIONS, 7(ll), 1337-1379 (1982) CONVEXITY PROPERTIES OF SOLUTIONS TO SOME CLASSICAL VARIATIONAL PROBLEMS Luis A. ~affarelli New York University 251 Mercer St. New York, N.Y. Joel Spruck Brooklyn College CUNY Brooklyn, N.Y. 1. Introduction In this work, we study several classical varia- tional problems with the view of developing methods to show the existence of solutions that inherit convexity properties of their domain of definition. Our main effort centers on a free boundary problem arising in plasma physics [TI which may be formulated as follows: given R c R", a bounded simply connected domain, we seek a function u in $2 satisfying t Supported in part by NSF Grants MCS-8100802 and BK 780-5485. Copyright 0 1982 by Marcel Dekker, Inc. 1338 CAFFARELLI AND SPRUCX -Au = Au+ in R, u+=max(ulO) I constant (free) (1.1) ula~ Xu+ dx = I (prescribed) is given. where A > A1(R) The plasma region is the set D = Eu > 0) and the free boundary is the set r= aD. We ask for a solution that satisfies the implicit free boundary condition: u is c1 across r. Our main result (Theorem 4.1 and Corollary 4.2) asserts that if Q