The Lavrentiev Phenomenon for Free Discontinuity Problems

We study lower semicontinuity problems for a class of integral functionals depending on a bulk energy and a jump part energy. These functionals are naturally defined on functions of bounded variation, and can be extended by relaxation to a larger set of discontinuous functions. We give a representation formula for this extension showing the appearance of an additional "Lavrentiev" term which gives the energy necessary for the creation of a singularity with unbounded variation. In general the difference between the functional and its relaxation implies that the same minimization problem may have different solutions on BV functions and on larger spaces of discontinuous functions. In the special case of autonomous functionals the Lavrentiev term can be expressed as an inf-convolution between the jump-part energy density and a rescaled bulk energy density.