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A note on the smoothness of energy-minimizing incompressible deformations

Nirmalendu Chaudhuri
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Abstract

In this note we prove that any $W^{1,2}$ mapping $u$ in the plane that minimizes an appropriate quasiconvex energy functional subject to the Jacobian constraint ${\rm det} \na u=1$ a.e., are necessarily Lipschitz. Furthermore we show that the minimizers corresponding to uniformly convex energy are affine and give an example of non-affine minimizers subject to affine boundary data corresponding to a convex energy. We also discuss the regularity issues in dimension greater than or equal to 3.

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Nirmalendu Chaudhuri. 2005-10-16. A note on the smoothness of energy-minimizing incompressible deformations. https://arxiv.org/abs/math/0510324

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