Lipschitz regularity for inner-variational equations

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Abstract

We obtain Lipschitz regularity results for a fairly general class of nonlinear first-order partial differential equations. These equations arise from the inner variation of certain energy integrals. Even in the simplest model case of the Dirichlet energy the inner-stationary solutions need not be differentiable everywhere; the Lipschitz continuity is the best possible. But the proofs, even in the Dirichlet case, turn out to rely on topological arguments. The appeal to the inner-stationary solutions in this context is motivated by the classical problems of existence and regularity of the energy-minimal deformations in the theory of harmonic mappings and certain mathematical models of nonlinear elasticity, specifically, neo-Hookean-type problems.2013

Original languageEnglish (US)
Pages (from-to)643-672
Number of pages30
JournalDuke Mathematical Journal
Volume162
Issue number4
DOIs
StatePublished - Mar 2013

ASJC Scopus subject areas

  • Mathematics(all)

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