Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients

Abstract

In this paper we extend the results of Caffarelli, Jerison, and Kenig [Ann. of Math. (2) 155 (2002)] and Caffarelli and Kenig [Amer. J. Math. 120 (1998)] by establishing an almost monotonicity estimate for pairs of continuous functions satisfying u ± ≥ 0, Lu ± ≥ -1, u + · u - = 0, in an infinite strip (global version) or a finite parabolic cylinder (localized version), where L is a uniformly parabolic operator Lu = L A,b,c u := div(A(x,s)∇u) + b(x,s) · ∇u + c(x,s)u - ∂ s u with double Dini continuous A and uniformly bounded b and c. We also prove the elliptic counterpart of this estimate. This closes the gap between the known conditions in the literature (both in the elliptic and parabolic case) imposed on u± in order to obtain an almost monotonicity estimate. At the end of the paper, we demonstrate how to use this new almost monotonicity formula to prove the optimal C 1,1 -regularity in a fairly general class of quasi-linear obstacle-type free boundary problems.