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Proceedings of the London Mathematical Society Advance Access published online on May 14, 2009
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdp016
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© 2009 London Mathematical Society

Deformations of finite conformal energy: existence and removability of singularities

Tadeusz Iwaniec

Department of Mathematics
Syracuse University
215 Carnegie Building
Syracuse, NY 13244-1150
USA
tiwaniec@syr.edu

Jani Onninen

Department of Mathematics
Syracuse University
215 Carnegie Building
Syracuse, NY 13244-1150
USA

Received 27 July 2008. Revision received 10 February 2009.

This paper features a class of mappings Formula between bounded domains X, Y sub Rn, having finite n-harmonic energy, such that we have


Formula

The fundamental question is whether or not the domains X, Y sub Rn of the same topological type admit a homeomorphism Formula in a given homotopy class having finite energy. The examples of non-existence, somewhat testing our theory, arise when we remove from bounded smooth domains X and Y thin subsets Formula and Formula , referred to as cracks or fractures. We are looking for homeomorphisms Formula of finite energy for which Formula is the cluster set of h over Formula . In general, infinite energy is required in order to increase the dimension of a crack Formula that is, when Formula . Suppose now that a bounded deformation Formula of finite energy is given. Does h extend continuously to X and, if so, is the extension injective on X? We give affirmative answers to these questions.

2000 Mathematics Subject Classification 30C60, 35J15, 35J70 (Primary).

Iwaniec was supported by the NSF grant DMS-0800416, and Onninen by the NSF grant DMS-0701059.

{dagger} When p<n, the topology can inhibit the space Formula from being dense in Formula see [6, 21, 22].


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