Relaxation of quasiconvex functionals in \(BV(\Omega, \mathbb{R}^ N)\) for integrands \(f(x, u, \bigtriangledown u)\)
type: Item
instance of: scholarly article
title: Relaxation of quasiconvex functionals in \(BV(\Omega, \mathbb{R}^ N)\) for integrands \(f(x, u, \bigtriangledown u)\)
zbMATH Open document ID: 0788.49039
DOI: 10.1007/BF00386367
author: Irene Fonseca
author: Stefan Müller
published in: Archive for Rational Mechanics and Analysis
publication date: 1993-10-17T00:00:00Z
review text: 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 \(W\) vanishe…
Mathematics Subject Classification ID: 49Q15
zbMATH DE Number: 429198
zbMATH Keywords: quasiconvex functionals
zbMATH Keywords: bounded variation
zbMATH Keywords: quasiconvexity
zbMATH Keywords: relaxation
zbMATH Keywords: variational problems for phase transitions
zbMATH Keywords: minima of a bulk energy
zbMATH Keywords: linear growth condition
zbMATH Keywords: singular perturbation
zbMATH Keywords: energy densities
zbMATH Keywords: degenerate bounds
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00386367
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description: scientific article; zbMATH DE number 429198