2010/10/21 by Norayr Matevosyan, Arshak Petrosyan · 3 citations
Mathematics · Engineering · #Differential Equations and Boundary Problems #Numerical methods in inverse problems #Numerical methods in engineering
paper · doi:10.1002/cpa.20349
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 in an infinite strip (global version) or a finite parabolic cylinder (localized version), where is a uniformly parabolic operator with double Dini continuous 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 C1,1-regularity in a fairly general class of quasi-linear obstacle-type free boundary problems. © 2010 Wiley Periodicals, Inc.