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Double obstacle problems with obstacles given by non-C 2 Hamilton–Jacobi equations

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Andersson, John Erik, Shahgholian, Henrik and Weiss, Georg S. (2012) Double obstacle problems with obstacles given by non-C 2 Hamilton–Jacobi equations. Archive for Rational Mechanics and Analysis, Vol.206 (No.3). pp. 779-819. doi:10.1007/s00205-012-0541-4

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Official URL: http://dx.doi.org/10.1007/s00205-012-0541-4

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Abstract

We prove optimal regularity for double obstacle problems when obstacles are given by solutions to Hamilton-Jacobi equations that are not C (2). When the Hamilton-Jacobi equation is not C (2) then the standard Bernstein technique fails and we lose the usual semi-concavity estimates. Using a non-homogeneous scaling (different speeds in different directions) we develop a new pointwise regularity theory for Hamilton-Jacobi equations at points where the solution touches the obstacle. A consequence of our result is that C (1)-solutions to the Hamilton-Jacobi equation <Equation ID="Equa"> <MediaObject> </MediaObject> </Equation>, are, in fact, C (1,alpha/2), provided that . This result is optimal and, to the authors' best knowledge, new.

Item Type: Journal Article
Divisions: Faculty of Science > Mathematics
Journal or Publication Title: Archive for Rational Mechanics and Analysis
Publisher: Springer
ISSN: 0003-9527
Official Date: December 2012
Dates:
DateEvent
December 2012Published
Volume: Vol.206
Number: No.3
Page Range: pp. 779-819
DOI: 10.1007/s00205-012-0541-4
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access

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