Monotone Sobolev Mappings of Planar Domains and Surfaces

Abstract

Abstract. An approximation theorem of Youngs (1948) asserts that a continuous map between compact oriented topological 2-manifolds (surfaces) is monotone if and only if it is a uniform limit of homeomorphisms. Analogous approximation of Sobolev mappings is at the very heart of Geometric Function Theory (GFT) and Nonlinear Elasticity (NE). In both theories the mappings in question arise naturally as weak limits of energy-minimizing sequences of homeomorphisms. As a result of this, the energy-minimal mappings turn out to be monotone. In the present paper we show that, conversely, monotone mappings in the Sobolev space W 1,p , 1 < p < ∞ , are none other than W 1,p -weak (also strong) limits of homeomorphisms. In fact, these are limits of diffeomorphisms. By way of illustration, we establish the existence of energy-minimal deformations within the class of Sobolev monotone mappings for p -harmonic type energy integrals.

Limits of Sobolev homeomorphisms

“…Then every continuous monotone map h : X onto − − → Y of Sobolev class W 1,p (X, R 2 ), p > 1, can be approximated uniformly and strongly in W 1,p (X, R 2 ) by homeomorphisms h j : X onto − − → Y (and with diffeomorphisms as well). This result has been established by the present authors in [42]. The reader may wish to notice that for 1 < p < 2 the lack of uniform continuity estimates prevents a W 1,p -limit of homeomorphisms from being continuous and monotone.…”

Radial symmetry of minimizers to the weighted Dirichlet energy

“…It is quite easy to see that mappings in H 1,2 (A, A * ) extend as continuous monotone maps of A onto A * . As a converse Iwaniec and Onninen [27] proved a Sobolev variant of the classical Youngs approximation theorem. According to their result the class H 1,2 (A, A * ) equals the class of orientation-preserving monotone mappings from A onto A * in the Sobolev class W 1,2 (A, C) that also preserve the order of the boundary components of annuli.…”

The Radó–Kneser–Choquet theorem for $p$-harmonic mappings between Riemannian surfaces

“…As U t is C 1,β -smooth, this discussion justifies writing DŨ t L p (M ) in estimates below. Furthermore, the careful analysis of constants in the proof of Theorem 4.2 reveals that the above constant C is, in fact, independent of p t , again due to the uniform bound 2 ≤ p t ≤ p. In a consequence, (36) holds true for C independent of t.…”