Mappings of least Dirichlet energy and their Hopf differentials

TL;DR

This paper studies energy-minimizing mappings between planar domains, characterizing their Hopf differentials and crack formation, with implications for nonlinear elasticity and complex analysis.

Contribution

It establishes existence, uniqueness, and characterization of least Dirichlet energy mappings via Hopf differentials, including crack formation mechanisms.

Findings

Energy-minimal mappings are characterized by analytic and boundary-real Hopf differentials.

Cracks occur only at boundary points where the target domain is non-convex.

Cracks propagate along vertical trajectories of the Hopf differential from boundary inward.

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 $\bar{\mathscr H}_2(X, Y)$ 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 $h_z \bar{h_{\bar{z}}} dz \otimes dz$ and its trajectories. For a pair of doubly connected domains, in which $X$ has finite conformal modulus, we establish the following principle: A mapping $h \in \bar{\mathscr H}_2(X, Y)$ 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.