## Duke Mathematical Journal

### Lipschitz regularity for inner-variational equations

#### 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.

#### Article information

Source
Duke Math. J., Volume 162, Number 4 (2013), 643-672.

Dates
First available in Project Euclid: 15 March 2013

https://projecteuclid.org/euclid.dmj/1363355690

Digital Object Identifier
doi:10.1215/00127094-2079791

Mathematical Reviews number (MathSciNet)
MR3039677

Zentralblatt MATH identifier
1319.49055

Subjects
Primary: 49N60: Regularity of solutions
Secondary: 35B65: Smoothness and regularity of solutions 73C50

#### Citation

Iwaniec, Tadeusz; Kovalev, Leonid V.; Onninen, Jani. Lipschitz regularity for inner-variational equations. Duke Math. J. 162 (2013), no. 4, 643--672. doi:10.1215/00127094-2079791. https://projecteuclid.org/euclid.dmj/1363355690

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