#### Document Type

Article

#### Date

11-26-2010

#### Embargo Period

11-14-2011

#### Disciplines

Mathematics

#### Description/Abstract

The central theme in this paper is the Hopf-Laplace equation, which represents stationary solutions with respect to the inner variation of the Dirichlet integral. Among such solutions are harmonic maps. Nevertheless, minimization of the Dirichlet energy among homeomorphisms often leads to nonharmonic solutions. We investigate the Hopf-Laplace equation for a certain class of topologically well behaved mappings which are almost homeomorphisms, called Hopf deformations. We establish Lipschitz continuity of Hopf deformations, the best possible regularity one can get. Thus in particular we show that the minimal-energy deformations are Lipschitz continuous, a result of considerable interest in the theory of minimal surfaces, calculus of variations, and PDEs, with potential applications to elastic plates.

#### Recommended Citation

Cristina, Jan; Iwaniec, Tadeusz; Kovalev, Leonid V.; and Onninen, Jani, "The Hopf-Laplace Equation" (2010). *Mathematics Faculty Scholarship.* Paper 54.

http://surface.syr.edu/mat/54

#### Source

Harvested from arXiv.org

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 3.0 License.

## Additional Information

This manuscript is from arXiv.org, for more information see http://arxiv.org/abs/1011.5934