arXiv subjects

Giles Shaw

Publications and source records attributed to Giles Shaw.

4 recordsLinked to original sources

arXiv2020

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 $(u_j)_j \in \mathrm{BV}(Ω;\mathbb{R}^m)$ under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functional \[ \mathcal{F}\colon u\to\int_Ωf(x,u(x),\nabla u(x)) \;\mathrm{dx},\quad u\in\mathrm{W}^{1,1}(Ω;\mathbb{R}^m),\quad Ω\in\mathbb{R}^d\text{ open}, \] to the space $\mathrm{BV}(Ω; \mathbb{R}^m)$. Lower semicontinuity results of this type were first obtained by Fonseca and Müller [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 Carathéodory 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.

math.AP↗

arXiv2020

Relaxation for partially coercive integral functionals with linear growth

We prove an integral representation theorem for the $\mathrm{L}^1(Ω;\mathbb{R}^m)$-relaxation of the functional \[ \mathcal{F}\colon u\mapsto\int_Ωf(x,u(x),\nabla u(x))\;\mathrm{dd } x,\quad u\in\mathrm{W}^{1,1}(Ω;\mathbb{R}^m),\quadΩ\subset\mathbb{R}^d\text{ open,} \] to the space $\mathrm{BV}(Ω;\mathbb{R}^m)$ under very general assumptions, requiring principally that $f$ be Carathéodory, partially coercive, and quasiconvex in the final variable. Our result is the first of its kind which applies to integrands which are unbounded in the $u$-variable and thus allows to treat many problems from applications. Such functionals are out of reach of the classical blow-up approach introduced by Fonseca & Müller [Arch. Ration. Mech. Anal. 123 (1993), 1--49]. Our proof relies on an intricate truncation construction (in the $x$ and $u$ arguments simultaneously) made possible by the theory of liftings as introduced in the companion paper arXiv:1708.04165, and features techniques which could be of use for other problems featuring $u$-dependent integrands.

math.AP↗

arXiv2018

Counterexamples in Calculus of Variations in $L^\infty$ through the vectorial Eikonal equation

We show that for any regular bounded domain $Ω\subseteq \mathbb R^n$, $n=2,3$, there exist infinitely many global diffeomorphisms equal to the identity on $\partial Ω$ which solve the Eikonal equation. We also provide explicit examples of such maps on annular domains. This implies that the $\infty$-Laplace system arising in vectorial Calculus of Variations in $L^\infty$ does not suffice to characterise either limits of $p$-Harmonic maps as $p\to \infty$, or absolute minimisers in the sense of Aronsson.

math.AP↗

arXiv2014

Strictly continuous extension of functionals with linear growth to the space BV

The main result of this paper is a proof of the continuity of a family of integral functionals defined on the space of functions of bounded variation with respect to a topology under which smooth functions are dense. These functionals occur often in the Calculus of Variations as the extension of integral problems defined over weakly differentiable functions with linear growth, and the result in this paper sheds light on the question of what the 'correct' extension is in this context. The result is proved via a combination of Reshetnyak's Continuity Theorem and a map assigning a lifting $μ[u]\in\mathbf{M}(Ω\times\mathbb{R}^m;\mathbb{R}^{m\times d})$ to each $u\in BV(Ω;\mathbb{R}^{m})$ and is valid for a large class of integrands satisfying $|f(x,y,A)|\leq C(1+|y|^{d/(d-1)}+|A|)$. In the case where $f$ exhibits $d/(d-1)$ growth in the $y$ variable, an embedding result from the theory of concentration-compactness is needed.

math.AP↗