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The Grotzsch Problem in Higher Dimensions Tomasz Adamowicz

Summary: The Gršotzsch Problem in Higher Dimensions
Tomasz Adamowicz
Department of Mathematics, Syracuse University, 215 Carnegie Hall,
13244-1150 Syracuse, NY, USA (e-mail: tadamowi@syr.edu )
December 12, 2006
An extension of Gršotzsch Problem to higher dimensions is considered. The
problem is formulated and proven for a subclass of polyconvex energy integrals
and counterexamples in general case are given. A conjecture about the general-
ized distortion functions is stated.
Keywords: Gršotzsch Problem, distortion function, polyconvex, extremal map-
pings, calculus of variations.
Mathematics Subject Classification (2000): 30C35, 30C65, 30C70
1 Introduction
The purpose of this paper is to extend the Gršotzsch Problem in the plane to higher
dimensions (see below or cf. [2] for formulation of the problem). This elaboration
is obtained for a wide class of polyconvex energy integrals under certain conditions
imposed on them. The motivation for our work comes from recent developments in the
theory of mappings with integrable distortion [3] - a promissing, dynamically growing
branch of the calculus of variations.


Source: Adamowicz, Tomasz - Matematiska Institutionen, Linköpings Universitet


Collections: Mathematics