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Relaxation and attainment results for an integral functional with unbounded energy well

Published online by Cambridge University Press:  12 July 2007

Baisheng Yan
Baisheng Yan
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA (yan@math.msu.edu)
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Abstract

Consider the functional I(u) = ∫Ω ‖Du|n − L det Du| dx, whose energy well consists of matrices satisfying |ξ|n = L det ξ. We show that the relaxations of this functional in various Sobolev spaces are significantly different. We also make several remarks concerning various p-growth semiconvex hulls of the energy-well set and prove an attainment result for a special Hamilton-Jacobi equation, |Du|n = L det Du, in the so-called grand Sobolev space W1,q)(Ω; Rn), with q = nL/(L + 1).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 132 , Issue 6 , December 2002 , pp. 1513 - 1523
Copyright
Copyright © Royal Society of Edinburgh 2002

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