Relaxation of Non-Convex Integral Functionals in the Multidimensional Scalar Case
Abstract
We study integral functionals defined on scalar Sobolev spaces of the form E[f]:u ∫ f(x,u(x),∇ u(x)) d x, with an emphasis on the non-convex case, and the difficulties it involves to prevent the Lavrentiev phenomenon. We determine a formulation of the lower semicontinuous envelope of E[f] with respect to various topologies and with fixed Lipschitz Dirichlet boundary conditions.
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