Relaxation of quasiconvex functionals in \(BV(\Omega, \mathbb{R}^ N)\) for integrands \(f(x, u, \bigtriangledown u)\) (Q687181): Difference between revisions

The relaxation \(\mathcal F\) in \(BV(\Omega,\mathbb{R}^ N)\) of the functional \(E(u):=\int_ \Omega f(x,u(x),\nabla u(x))dx\) is obtained, where \(f(x,u,\cdot)\) is quasiconvex, grows at most linearly with possibly degenerate bounds and satisfies some technical continuity conditions. The relaxation is defined by \[ {\mathcal F}(u):= \inf_{u_ n}\left\{\liminf_{n\to +\infty} f(x,u_ n(x), \nabla u_ n(x))dx\mid u_ n\in W^{1,1}\text{ and } u_ n\to u\text{ in }L^ 1\right\} \] and the following integral representation is obtained, \[ \begin{aligned} {\mathcal F}(u) &= \int_ \Omega f(x,u(x), \nabla u(x))dx+ \int_{S(u)} K(x,u^ -(x),u^ +(x),\nu(x))dH^{N-1}(x)\\ & +\int_ \Omega f^ \infty(x,u(x), dC(u)),\end{aligned} \] where the distributional derivative \(Du\) is written as \(Du= \nabla ud{\mathcal L}^ N+ (u^ +- u^ -)\otimes \nu dH^{N-1}\lfloor S(u)+C(u)\). Here \(\nabla u\) is the density of the absolutely continuous part of \(Du\) with respect to the Lebesgue measure \({\mathcal L}^ N\), \(H^{N-1}\) is the \(N-1\)-dimensional Hausdorff measure, \((u^ +- u^ -)\) is the jump of \(u\) across the interface \(S(u)\), and \(C(u)\) is the Cantor part of \(Du\), i.e. the part singular with respect to \({\mathcal L}^ N\) and \(H^{N-1}\lfloor S(u)\). Finally, \(f^ \infty\) denotes the recession function \[ f^ \infty(x,u,A):= \limsup_{t\to +\infty} {f(x,u,tA)\over t}. \] This problem was motivated by the analysis of variational problems for phase transitions and the study of the development of cracks. Equilibrium of such materials are often associated to minima of a bulk energy \(E(u)\) where \(f(x,u,\cdot)\) is non-convex and where the function spaces involved should allow discontinuous vector- valued \(u\). This, together with a linear growth condition on \(f(x,u,\cdot)\), suggests the need to relax \(E(\cdot)\) in \(BV\). In addition, singular perturbation problems derived from phase transitions lead to energy densities of the type \(f(x,u,A)=\sqrt{W(u)}h(A)\), where \(W\) vanishes at more than one point, thus preventing the coerciveness of \(f(x,u,\cdot)\) and imposing the need to consider degenerate bounds for \(f\).