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Structure of entropy solutions to the eikonal equation


De Lellis, C; Otto, F (2003). Structure of entropy solutions to the eikonal equation. Journal of the European Mathematical Society, 5(2):107-145.

Abstract

In this paper, we establish rectifiability of the jump set of an S1–valued conservation law in two space–dimensions. This conservation law is a reformulation of the eikonal equation and is motivated by the singular limit of a class of variational problems. The only assumption on the weak solutions is that the entropy productions are (signed) Radon measures, an assumption which is justified by the variational origin. The methods are a combination of Geometric Measure Theory and elementary geometric arguments used to classify blow–ups.¶The merit of our approach is that we obtain the structure as if the solutions were in BV, without using the BV–control, which is not available in these variationally motivated problems.

Abstract

In this paper, we establish rectifiability of the jump set of an S1–valued conservation law in two space–dimensions. This conservation law is a reformulation of the eikonal equation and is motivated by the singular limit of a class of variational problems. The only assumption on the weak solutions is that the entropy productions are (signed) Radon measures, an assumption which is justified by the variational origin. The methods are a combination of Geometric Measure Theory and elementary geometric arguments used to classify blow–ups.¶The merit of our approach is that we obtain the structure as if the solutions were in BV, without using the BV–control, which is not available in these variationally motivated problems.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:entropy solutions, partial regularity, singular perturbation, conservation laws, rectifiability
Language:English
Date:2003
Deposited On:17 Sep 2010 09:49
Last Modified:06 Dec 2017 20:49
Publisher:European Mathematical Society
ISSN:1435-9855
Additional Information:© 2003 EMS Publishing House. All rights reserved.
Publisher DOI:https://doi.org/10.1007/s10097-002-0048-7
Related URLs:http://www.ams.org/mathscinet-getitem?mr=1985613

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