Liftings, Young measures, and lower semicontinuity

This work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs $(u_j,Du_j)j$ for $(u_j)_j \in \mathrm{BV}(\Omega;\mathbb{R}^m)$ under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functional $$\mathcal{F}\colon u\to\int_\Omega f(x,u(x),\nabla u(x)) \;\mathrm{dx},\quad u\in\mathrm{W}^{1,1}({\Omega};\mathbb{R}^m),\quad {\Omega}\in\mathbb{R}^d\text{ open},$$ to the space $\mathrm{BV}(\Omega; \mathbb{R}^m)$. Lower semicontinuity results of this type were first obtained by Fonseca and M\"uller [Arch. Ration. Mech. Anal. 123 (1993), 1-49] and later improved by a number of authors, but our theorem is valid under more natural, essentially optimal, hypotheses than those currently present in the literature, requiring principally that $f$ be Carath\'eodory and quasiconvex in the final variable. The key idea is that liftings provide the right way of localising $\mathcal{F}$ in the $x$ and $u$ variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow-up procedure.Comment: 75 pages. Updated to correct a series of minor typos/ inaccuracies. The statement and proof of Theorem have also been amended- subsequent steps relying upon the Theorem did not require updatin

Piecewise affine approximations for functions of bounded variation

BV functions cannot be approximated well by piecewise constant functions, but this work will show that a good approximation is still possible with (countably) piecewise affine functions. In particular, this approximation is area-strictly close to the original function and the $\mathrm{L}^1$-difference between the traces of the original and approximating functions on a substantial part of the mesh can be made arbitrarily small. Necessarily, the mesh needs to be adapted to the singularities of the BV function to be approximated, and consequently, the proof is based on a blow-up argument together with explicit constructions of the mesh. In the case of $\mathrm{W}^{1,1}$-Sobolev functions we establish an optimal $\mathrm{W}^{1,1}$-error estimate for approximation by piecewise affine functions on uniform regular triangulations. The piecewise affine functions are standard quasi-interpolants obtained by mollification and Lagrange interpolation on the nodes of triangulations, and the main new contribution here compared to for instance Cl\'{e}ment (RAIRO Analyse Num\'{e}rique 9 (1975), no.~R-2, 77--84) and Verf\"{u}rth (M2AN Math.~Model.~Numer.~Anal. 33 (1999), no. 4, 695-713) is that our error estimates are in the $\mathrm{W}^{1,1}$-norm rather than merely the $\mathrm{L}^1$-norm.Comment: 14 pages, 1 figur

Relaxation for partially coercive integral functionals with linear growth

We prove an integral representation theorem for the $\mathrm{L}^1$-relaxation of the functional $\mathcal{F}\colon u\mapsto\int_\Omega f(x,u(x),\nabla u(x))\;\mathrm{d} x,\quad u\in\mathrm{W}^{1,1}(\Omega;\mathbb{R}^m)$, where $\Omega\subset\mathbb{R}^d$ ($d \geq 2$) is a bounded Lipschitz domain, to the space $\mathrm{BV}(\Omega;\mathbb{R}^m)$ under very general assumptions: we require principally that $f$ is Carathéodory, that the partial coercivity and linear growth bound $g(x,y)|A|\leq f(x,y,A)\leq Cg(x,y)(1+|A|)$, hold, where $g\colon\overline{\Omega}\times\mathbb{R}^m\to[0,\infty)$ is a continuous function satisfying a weak monotonicity condition, and that $f$ is quasi-convex in the final variable. Our result is the first that applies to integrands which are unbounded in the $u$-variable and, therefore, allows for the treatment of many problems from applications. Such functionals are out of reach of the classical blowup approach introduced by Fonseca and Müller [Arch. Ration. Mech. Anal., 123 (1993), pp. 1--49]. Our proof relies on an intricate truncation construction (in the $x$- and $u$-arguments simultaneously) made possible by the theory of liftings developed in a previous paper by the authors [Arch. Ration. Mech. Anal., 232 (2019), pp. 1227--1328], and features techniques which could be of use for other problems involving $u$-dependent integrands