Preprint 71/1999

Optimal existence theorems for nonhomogeneous differential inclusions

Stefan Müller and Mikhail A. Sytchev

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Submission date: 14. Dec. 1999
Pages: 37
published in: Journal of functional analysis, 181 (2001) 2, p. 447-475 
DOI number (of the published article): 10.1006/jfan.2000.3726
MSC-Numbers: 35F30, 35J55, 49K20, 73G05
Keywords and phrases: nonconvex variational problem, differential inclusions, hamilton-jacobi equations, quasiconvexity, solid-solid phase transitions
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In this paper we address the question of solvability of the differential inclusions
where tex2html_wrap_inline41 a.e. in tex2html_wrap_inline43, and where tex2html_wrap_inline45 is a multi-valued function.

Our approach to these problems is based on the idea to construct a sequence of approximate solutions which converges strongly and makes use of Gromov's idea (following earlier work of Nash and Kuiper) to control convergence of the gradients by appropriate selection of the elements of the sequence. In this paper we identify an optimal setting of this method.

We show that if for each tex2html_wrap_inline47, each tex2html_wrap_inline49, and each tex2html_wrap_inline51 we can find a piece-wise affine function tex2html_wrap_inline53 (here tex2html_wrap_inline55) with tex2html_wrap_inline57 and tex2html_wrap_inline59 a.e. for all (x',u') sufficiently close to (x,u), then we can resolve the differential inclusions. The result holds provided tex2html_wrap_inline65 is the zero set of a nonnegative upper semicontinuous function d such that for each (x,u) the set K(x,u) is compact and tex2html_wrap_inline73 if and only if tex2html_wrap_inline75. We also discuss some generalizations and applications of this result.

19.04.2013, 01:40