Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients

Handle URI:
http://hdl.handle.net/10754/597493
Title:
Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients
Authors:
Matevosyan, Norayr; Petrosyan, Arshak
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 = 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.
Citation:
Matevosyan 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.
Publisher:
Wiley-Blackwell
Journal:
Communications on Pure and Applied Mathematics
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
21-Oct-2010
DOI:
10.1002/cpa.20349
Type:
Article
ISSN:
0010-3640
Sponsors:
N. 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.
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Full metadata record

DC FieldValue Language
dc.contributor.authorMatevosyan, Norayren
dc.contributor.authorPetrosyan, Arshaken
dc.date.accessioned2016-02-25T12:40:49Zen
dc.date.available2016-02-25T12:40:49Zen
dc.date.issued2010-10-21en
dc.identifier.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.en
dc.identifier.issn0010-3640en
dc.identifier.doi10.1002/cpa.20349en
dc.identifier.urihttp://hdl.handle.net/10754/597493en
dc.description.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.en
dc.description.sponsorshipN. 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.en
dc.publisherWiley-Blackwellen
dc.titleAlmost monotonicity formulas for elliptic and parabolic operators with variable coefficientsen
dc.typeArticleen
dc.identifier.journalCommunications on Pure and Applied Mathematicsen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
dc.contributor.institutionPurdue University, West Lafayette, United Statesen
kaust.grant.numberKUK-I1-007-43en
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