On the relaxation of Dirichlet minimum problems for some classes of unbounded integral functionals
In the paper the relaxation in BV and Sobolev spaces of the integral functional \[ F(\Omega,u) = \int_{\Omega} f(\nabla u(x)) dx \] is considered. Above, \(\Omega\) denotes an open and bounded domain in \({\mathbb R}^n\), \(f\) is a Borel function with values in \([0,+\infty]\) which in applications take the form \(f = g + I_C\) where \(g\) is continuous and \(I_C\) is the indicator function of a convex set \(C \subset {\mathbb R}^n\). The constraint set \(\{x: \nabla u(x) \in C\) a.e. in \(\Omega\}\), in general, is not contained in \(\operatorname {dom} f = \{z: f(z) < + \infty\}\) for a function \(u \in W^{1,\infty}_{\text{loc}} ({\mathbb R}^n)\), so the functional \(F\) is unbounded. The authors obtain some integral representation formulae for the lower semicontinuous envelope (in \(L^1(\Omega)\) topology) of such functionals and then they give some applications to the relaxed minimum problems in BV spaces with nonhomogeneous Dirichlet or Neumann boundary conditions ( getting the existence of the min and the compactness of minimizing sequences). Such unbounded energy functionals have applications in elasto-plastic torsion theory, as well as in electrostatics.
scientific article; zbMATH DE number 1370135