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Journal of Convex Analysis 10 (2003), No. 2, 445--464
Copyright Heldermann Verlag 2003

Examples of the Lavrentiev Phenomenon with Continuous Sobolev Exponent Dependence

M. Foss
Kansas State University, Dept. of Mathematics, Manhattan, KS 66506-2602, U.S.A.,

We construct variational problems with infima that have non-trivial continuous dependence upon the Sobolev space from which the competing functions are taken. It is shown, for each m in a particular class of continuous functions, that there is a variational integral and boundary conditions such that, for every p from [1, infinity], the infimum is equal to m(p) if the admissible class is a subset of W1, p. Thus, the manner in which the infimum depends upon the Sobolev exponent may be prescribed.

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