Optimal existence theorems for nonhomogeneous differential inclusions
Stefan Müller and Mikhail A. Sytchev
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Submission date: 14. Dec. 1999
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 a.e. in , and where 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 , each , and each we can find a piece-wise affine function (here ) with and a.e. for all (x',u') sufficiently close to (x,u), then we can resolve the differential inclusions. The result holds provided 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 if and only if . We also discuss some generalizations and applications of this result.