TY - JOUR

T1 - Monotone Sobolev Mappings of Planar Domains and Surfaces

AU - Iwaniec, Tadeusz

AU - Onninen, Jani

N1 - Funding Information:
T. Iwaniec was supported by the NSF grant DMS-1301558 and the Academy of Finland project 1128331. J. Onninen was supported by the NSF grant DMS-1301570.
Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - An approximation theorem of Youngs (Duke Math J 15, 87–94, 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 (Formula presented.), are none other than (Formula presented.),p-weak (also strong) limits of homeomorphisms. In fact, these are limits of diffeomorphisms. By way of illustration, we establish the existence of traction free energy-minimal deformations for p -harmonic type energy integrals.

AB - An approximation theorem of Youngs (Duke Math J 15, 87–94, 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 (Formula presented.), are none other than (Formula presented.),p-weak (also strong) limits of homeomorphisms. In fact, these are limits of diffeomorphisms. By way of illustration, we establish the existence of traction free energy-minimal deformations for p -harmonic type energy integrals.

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U2 - 10.1007/s00205-015-0894-6

DO - 10.1007/s00205-015-0894-6

M3 - Article

AN - SCOPUS:84952629746

SN - 0003-9527

VL - 219

SP - 159

EP - 181

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

IS - 1

ER -