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Integral representation and \(\Gamma\)-convergence of variational integrals with \(p(x)\)-growth. (English) Zbl 1036.49022

Summary: We study the integral representation properties of limits of sequences of integral functionals like \(\int f(x,Du)\,dx\) under nonstandard growth conditions of \((p,q)\)-type: namely, we assume that \[ | z|^{p(x)}\leq f(x,z)\leq L(1+| z|^{p(x)})\,. \] Under weak assumptions on the continuous function \(p(x)\), we prove \(\Gamma \)-convergence to integral functionals of the same type. We also analyse the case of integrands \(f(x,u,Du)\) depending explicitly on \(u\); finally, we weaken the assumption allowing \(p(x)\) to be discontinuous on nice sets.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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