The weak and strong closures of Sobolev homeomorphisms are the same

TL;DR

This paper proves that for Sobolev homeomorphisms between certain planar domains, the weak and strong closures in the Sobolev space coincide, which has significant implications for calculus of variations and nonlinear elasticity.

Contribution

It establishes the equality of weak and strong closures of Sobolev homeomorphisms in multiply connected planar domains for p ≥ 2, a result previously unconfirmed.

Findings

Weak and strong closures of Sobolev homeomorphisms are identical for p ≥ 2.

The result supports existence theories in calculus of variations.

Applications to nonlinear elasticity are facilitated by this closure equality.

Abstract

Let X and Y be bounded multiply connected Lipschitz domains in \R^2. We consider the class H_p (X, Y) of homeomorphisms h : X -> Y in the Sobolev space W^{1,p} (X, \R^2). We prove that the weak and strong closures of H_p (X, Y), 2 \le p< \infty, are equal. The importance of this result to the existence theory in the calculus of variations and anticipated applications to nonlinear elasticity are captured by Theorem 1.5.