Minimal currents, geodesics, and relaxation of variational integrals on mappings of bounded variation (Q1196419)

The authors study the problem of extending the functional \(J(u)=\int_ \Omega f(x,u(x),\nabla u(x))dx\) defined on \(C^ 1\) to the space of mappings of bounded variation \(BV(\Omega;R^ m)\). Here \(f=f(x,y,z)\) is a nonnegative, continuous function which is convex in \(z\) and satisfies the growth condition \(c_ o| z|\leq f(x,y,z)\leq c| z|\), \(c,c_ o>0\). Moreover, some isotropic condition on \(f\) is assumed: \(f(x,y,z)\geq f(x,y,zq\otimes q)\) for \(q\in R^ n\), \(| q|=1\). The main result is a representation formula for the greatest lower semicontinuous function \(\overline J\) on \(BV(\Omega,R^ m)\) less than or equal \(J\) on \(C^ 1(\Omega,R^ m)\).