Abstract
We prove that planar homeomorphisms can be approximated by diffeomorphisms in the Sobolev space $W^{1,2}$ and in the Royden algebra. As an application, we show that every discrete and open planar mapping with a holomorphic Hopf differential is harmonic.
Highlights
It is a fundamental property of Sobolev spaces W 1,p, 1 p < ∞, that any element can be approximated strongly by C ∞ smooth functions, or by piecewise affine ones
In the context of vector-valued Sobolev functions, that is, mappings in W 1,p(Ω, Rn), invertibility comes into play
The studies of invertible Sobolev mappings are of great importance in nonlinear elasticity [2, 14, 26, 35]
Summary
It is a fundamental property of Sobolev spaces W 1,p, 1 p < ∞, that any element can be approximated strongly (i.e., in the norm) by C ∞ smooth functions, or by piecewise affine ones. Approximation, Sobolev homeomorphisms, Hopf differential, harmonic mappings. (In the converse direction, a piecewise-affine mapping can be smoothed in dimensions less than four [25], but not in general.) Partial results toward the Ball-Evans problem were obtained in [24] (for planar bi-Sobolev mappings that are smooth outside of a finite set) and in [6] (for planar bi-Holder mappings, with approximation in the Holder norm). Eells and Lemaire inquired about the possibility of a converse result, e.g., for mappings with finite energy and almost-everywhere positive Jacobian [11, (2.6)] In this setting a counterexample was provided by Jost [20], who proved the existence of W 1,2-solutions of (1.1) in every homotopy class of mappings between compact Riemann surfaces.
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