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Cauchy–Dirichlet Problem for First Order Nonlinear Systems

✍ Scribed by Bernard Dacorogna; Paolo Marcellini

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
582 KB
Volume
152
Category
Article
ISSN
0022-1236

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✦ Synopsis

Let 0/R n be open, u : 0 Ä R m and thus the gradient matrix Du # R m_n . We let E/R m_n be compact and denote by RcoE and PcoE the rank one convex and polyconvex hull of E, respectively. We show that if RcoE=PcoE (and two other hypotheses, named the segment property and the extreme points property) and if

x # 0 then there exists (a dense set of) u # W 1, (0; R m ) such that

a.e. in 0 on 0.

We apply this existence theorem to some relevant examples studied in the literature, as well as to problems with (x, u) dependence.

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