Conditions for equality of hulls in the calculus of variations

Georg Dolzmann, Bernd Kirchheim, and Jan Kristensen

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Submission date: 06. Jan. 2000
Pages: 8
published in: Archive for rational mechanics and analysis, 154 (2000) 2, p. 93-100 
DOI number (of the published article): 10.1007/s002050000098
Bibtex
MSC-Numbers: 26B15, 74N15
Keywords and phrases: gradient young measures, quaiconvex hull, distance function
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Abstract:
We simplify and sharpen several results by K. Zhang concerning properties of quasiconvex hulls of sets and quasiconvex envelopes of their distance functions. The approach emphasizes the underlying geometry and in particular we show that ¡b¿K¡sup¿pc¡/sup¿=K¡sup¿c¡/sup¿¡/b¿ implies ¡b¿K¡sup¿rc¡/sup¿=K¡sup¿c¡/sup¿¡/b¿ if and only if ¡b¿minm,n ¡= 2¡/b¿ thus answering a question raised in
¡i¿K.W. Zhang, On various semiconvex hulls in the calculus of variations, Calc. Var. PDE 6 (1998), 143 - 160.¡/i¿