Mappings of least Dirichlet energy and their Hopf differentials

Abstract

The paper is concerned with mappings between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in the domain) of the energy-minimal mappings is established within the class ${\bar{\mathcal{H}}}_2(X,Y)$ of strong limits of homeomorphisms in the Sobolev space $W^{1,2}(X,Y)$, a result of considerable interest in the mathematical models of Nonlinear Elasticity. The inner variation leads to the Hopf differential $h_z\overline{h_{\stackrel{ˉ}{z}}}dz\otimes dz$ and its trajectories. For a pair of doubly connected domains, in which $X$ has finite conformal modulus, we establish the following principle: A mapping $h\in {\bar{\mathcal{H}}}_2(X,Y)$ is energy-minimal if and only if its Hopf-differential is analytic in $X$ and real along the boundary of $X$. In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of cracks in $X$. Nevertheless, cracks are triggered only by the points in the boundary of $Y$ where $Y$ fails to be convex. The general law of formation of cracks reads as follows: Cracks propagate along vertical trajectories of the Hopf differential from the boundary of $X$ toward the interior of $X$ where they eventually terminate before making a crosscut.

arXiv comment51 pages, 4 figures

References in corpus (3)

• Diffeomorphic approximation of Sobolev homeomorphisms
• Existence of energy-minimal diffeomorphisms between doubly connected domains
• Approximation up to the boundary of homeomorphisms of finite Dirichlet energy

Cited by in corpus (1)

• Harmonic maps with prescribed degrees on the boundary of an annulus and bifurcation of catenoids