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The distance function and defect energy

Published online by Cambridge University Press:  14 November 2011

Patricio Aviles  and
Yoshikazu Giga
Patricio Aviles
Affiliation:
Department of Mathematics and Physics, University of Oxford, Oxford, U.K.; ETH. CH-8592, Zurich, Switzerland
Yoshikazu Giga
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060, Japan
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Abstract

Several energies measuring jump discontinuities of a unit length gradient field are considered and are called defect energies. The main example is a total variation I(φ) of the hessian of a function φ in a domain. It is shown that the distance function is the unique minimiser of I(φ) among all non-negative Lipschitz solutions of the eikonal equation |grad φ| = 1 with zero boundary data, provided that the domain is a two-dimensional convex domain. An example shows that the distance function is not a minimiser of I if the domain is noncovex. This suggests that the selection mechanism by I is different from that in the theory of viscosity solutions in general. It is often conjectured that the minimiser of a defect energy is a distance function if the energy is formally obtained as a singular limit of some variational problem. Our result suggests that this conjecture is very subtle even if it is true.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 5 , 1996 , pp. 923 - 938
Copyright
Copyright © Royal Society of Edinburgh 1996

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