Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients
KAUST Grant NumberKUK-I1-007-43
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AbstractIn 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 = LA,b,cu := div(A(x, s)∇u) + b(x,s) · ∇u + c(x,s)u - δsu 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 C1,1-regularity in a fairly general class of quasi-linear obstacle-type free boundary problems. © 2010 Wiley Periodicals, Inc.
CitationMatevosyan N, Petrosyan A (2010) Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients. Comm Pure Appl Math 64: 271–311. Available: http://dx.doi.org/10.1002/cpa.20349.
SponsorsN. Matevosyan was partly supported by the WWTF (Wiener Wissenschafts, Forschungs und Technologiefonds) Project No. CI06 003 and by award no. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).A. Petrosyan was supported in part by National Science Foundation Grant DMS-0701015.