Preprint 59/2001

Partial regularity of strong local minimizers in the multi-dimensional calculus of variations

Jan Kristensen and Ali Taheri

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Submission date: 29. Aug. 2001
Pages: 27
published in: Archive for rational mechanics and analysis, 170 (2003) 1, p. 63-89 
DOI number (of the published article): 10.1007/s00205-003-0275-4

Let tex2html_wrap_inline14 be a bounded domain and tex2html_wrap_inline16 a given strongly quasiconvex integrand of class tex2html_wrap_inline18 satisfying the growth condition
for some c>0 and tex2html_wrap_inline24. Consider the multiple integral
where tex2html_wrap_inline28. The main result of the paper is that any strong local minimizer of tex2html_wrap_inline30 is of class tex2html_wrap_inline32 for any tex2html_wrap_inline34 on an open set of full n-dimensional measure. In the case of weak local minimizers we establish the same result under the extra assumption that the oscillations in the gradient of the minimizer are not too large. Without such an assumption weak local minimizers need not be partially regular. This is shown by a class of examples that are obtained by suitably modifying the arguments of S. Müller and V. Sverak.

24.11.2021, 02:11