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Bildhauer, Michael; Fuchs, Martin

Splitting-type variational problems with mixed linear-superlinear growth conditions. (English) Zbl 1466.49009

J. Math. Anal. Appl. 501, No. 1, Article ID 124452, 29 p. (2021).
Summary: Splitting-type variational problems with mixed linear-superlinear growth conditions are considered. In the two-dimensional case the minimization problem is given by \[ J[w]=\int_{\Omega}[f_1(\partial_1 w)+f_2(\partial_2 w)]\operatorname{d}x\to \min \] with respect to a suitable class of comparison functions. Here \(f_1\) is a convex energy density with linear growth and \(f_2\) is a convex energy density with superlinear growth, for instance given by an \(N\)-function or just bounded from below by an \(N\)-function. One motivation for this kind of problem located between the well-known splitting-type problems of superlinear growth and the splitting-type problems of linear growth (recently considered by M. Bildhauer and M. Fuchs [J. Math. Sci., New York 250, No. 2, 232–249 (2020; Zbl 1448.49011); translation from Probl. Mat. Anal. 105, 45–58 (2020)]) is the link to mathematical problems in plasticity (see [R. Temam, in: Computational methods in nonlinear mechanics, Proc. TICOM 2nd int. Conf. Austin/Texas 1979, 475–484 (1980; Zbl 0457.73017)]). Here we prove results in the appropriate way of relaxation, including approximation procedures, duality, and existence and uniqueness of solutions as well as some new higher-integrability results.

Cited in 5 Documents

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49Q20 Variational problems in a geometric measure-theoretic setting

Keywords:

nonstandard growth; splitting-type variational problems; relaxation; linear growth

Citations:

Zbl 1448.49011; Zbl 0457.73017
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Full Text: DOI

References:

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