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The interaction between bulk energy and surface energy in multiple integrals. (English) Zbl 0810.49015

This paper is concerned with a class of functionals that arise in a variety of applications and which are of the form \[ F(u)= \int_ \Omega f(\nabla u(x)) dx+ \int_{S_ u\cap \Omega} g((u^ +- u^ -)\otimes \nu_ u) d{\mathcal H}^{n-1}, \] in which \(\Omega\subset \mathbb{R}^ n\) is bounded and open, \(f\) has only the property that it is finite at least at one point, and \(g\) is positively homogeneous of degree 1. The natural setting for this class of problems is the space \(\text{BV}(\Omega; \mathbb{R}^ k)\) of functions of bounded variation, or its subspace \(\text{SBV}(\Omega; \mathbb{R}^ k)\) of special functions of bounded variation. With this setting \(S_ u\) denotes the set of jump points of \(u\), and the normal \(\nu_ u\) and traces \(u^ +\), \(u^ -\) on the two sides of \(S_ u\) are well-defined.
A major question is whether the direct method of the calculus of variations may be applied to the minimization problem associated with \(F\). Lower semicontinuity is an unresolved problem. With this in mind, the main purpose of the paper is to study the relaxed functional \(\overline{F}\) of \(F\), this being the greatest l.s.c. (in the \(L^ 1\)- topology) functional less than or equal to \(F\). The advantage of this approach is that the minimizing sequences of \(F\) may be described using \(\overline{F}\).
The main theorem of the paper gives an integral representation result for \(\overline{F}\), in the form \[ \overline{F}(u)= \int_ \Omega \phi(\nabla u(x)) dx+ \int_ \Omega \phi^ \infty\left( {D_ s u\over | D_ s u|}\right) | D_ s u|, \] in which \(\phi\) is quasiconvex with linear growth, \(\phi^ \infty\) is its recession function, \(D_ s u\) denotes the singular part of the measure \(Du\), and \(| \cdot|\) denotes its total variation. The bulk of the paper is taken up with the proof of this theorem, though many of the preliminaries are summarized in some detail.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49J40 Variational inequalities
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