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MiS Preprint
71/1999

Optimal existence theorems for nonhomogeneous differential inclusions

Stefan Müller and Mikhail A. Sytchev

Abstract

In this paper we address the question of solvability of the differential inclusions Du()K(,u()),u|Ω=f,uW1,(Ω;Rm), where Df()F(,f()) a.e. in Ω, and where F:Ω×Rm2Rm×n 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 (x,u)Ω×Rm, each ε>0, and each vF(x,u) we can find a piece-wise affine function ϕlv+W01,(Ω;Rm) (here lv(y)=vy)) with ||dist(Dϕ,K(x,u))||L1ε and DϕF(x,u) a.e. for all (x',u') sufficiently close to (x,u), then we can resolve the differential inclusions. The result holds provided {(x,u,v,):vK(x,u)} 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 d(x,u,vj)0 if and only if dist(vj,K(x,u))0. We also discuss some generalizations and applications of this result.

Received:
14.12.99
Published:
14.12.99
MSC Codes:
35F30, 35J55, 49K20, 73G05
Keywords:
nonconvex variational problem, differential inclusions, hamilton-jacobi equations, quasiconvexity, solid-solid phase transitions

Related publications

inJournal
2001 Repository Open Access
Stefan Müller and Mikhail A. Sychev

Optimal existence theorems for nonhomogeneous differential inclusions

In: Journal of functional analysis, 181 (2001) 2, pp. 447-475