Relaxation of variational problems under trace constraints (Q1347442)

In this paper the authors consider the problem of the \(L^1\)-lower semicontinuous envelope for the functional \[ F(u,A)=\int _{A\cap \Omega}|\nabla u|dx +\int _{\Sigma \cap A}\beta_0(x,u^+,u^-)dH^{N-1}(x) \] defined on \(W^{1,1}(\Omega\setminus \Sigma,R^d)\), where \(\Sigma\subset\overline\Omega\) is a given surface (\(\Sigma\cap \overline\Omega\) not necessarily empty) and with the function \(\beta_0\) admitting the value \(+\infty\) (in order to include the study of problems with constraints). The authors give a complete integral representation of the lower semicontinuous envelope defined on the class of vector valued BV functions, even for the surface energy density on \(\Sigma \cap \partial \Omega\). In the last section the authors give also two examples of applications of these results; one with a trace condition (a Dirichlet problem with assigned value on \(\Sigma\)) and one with a trace energy.