The Lavrentiev phenomenon for free discontinuity problems (Q1891775)

The paper deals with one dimensional lower semicontinuity and relaxation problems for functionals \(F(u)\) having an absolutely continuous energy density depending on \(u'\) and an atomic part depending on the discontinuity set \(S_u\) and the jumps of \(u\). The natural domain of these functionals, whose model is \[ F(u)= \int_I f(u'(t)) dt+ \sum_{t\in S_u\cap I} g(u(t_+)- u(t_-)),\qquad I\subset R\text{ open interval}, \] is the space \(\text{SBV}(I)\) of special functions with bounded variation in \(I\). In this context it is possible to analyze the ``Lavrentiev phenomenon'', trying to compute the energy which is necessary for the creation of point singularities such that \(u\not\in \text{SBV}\) near the singularity. It is shown in the paper that (even for nonautonomous functionals) this additional energy can be computed solving an auxiliary variational problem. In the special case of autonomous functionals, the Lavrentiev term can be expressed using the inf-convolution of the energy densities \(f\) and \(g\).