Radó-Kneser-Choquet Theorem for simply connected domains ($p$-harmonic setting)

Abstract:

A remarkable result known as the Radó-Kneser-Choquet theorem asserts that the harmonic extension of a homeomorphism of the boundary of a Jordan domain $\Omega \subset \mathbb R^2$ onto the boundary of a convex domain $\mathcal Q\subset \mathbb R^2$ takes $\Omega$ diffeomorphically onto $\mathcal Q$. Numerous extensions of this result for linear and nonlinear elliptic PDEs are known, but only when $\Omega$ is a Jordan domain or, if not, under additional assumptions on the boundary map. On the other hand, the newly developed theory of Sobolev mappings between Euclidean domains and Riemannian manifolds demands extending this theorem to the setting of simply connected domains. This is the primary goal of our article. The class of the $p$-harmonic equations is wide enough to satisfy those demands. Thus we confine ourselves to considering the $p$-harmonic mappings.

The situation is quite different from that of Jordan domains. One must circumvent the inherent topological difficulties arising near the boundary.

Our main theorem is the key to establishing approximation of monotone Sobolev mappings with diffeomorphisms. This, in turn, leads to the existence of energy-minimal deformations in the theory of nonlinear elasticity.