Limits of Sobolev homeomorphisms

Abstract

Let $\mathbb{X},\mathbb{Y}\in {\mathbb{R}}^2$ be topologically equivalent bounded Lipschitz domains. We prove that weak and strong limits of homeomorphisms $h:\mathbb{X}\stackrel{\mathrm{onto}}{⟶}\mathbb{Y}$ in the Sobolev space ${\mathcal{W}}^{1,p}(\mathbb{X},{\mathbb{R}}^2),p\ge 2$, are the same. As an application, we establish the existence of 2D-traction free minimal deformations for fairly general energy integrals.