Related papers: Lower semicontinuous envelopes in $w^{1,1}\times l…

Verifying lower-semicontinuity of integral functionals in the weak topology of Sobolev spaces is a central theme in the calculus of variations. For integral functionals with $p$-growth, quasiconvexity is a necessary condition for weak…

In this paper, we present a new extension of the famous Serrin's lower semicontinuity theorem for the variational functional $\int_{\Omega}f(x,u,u')dx$,we prove its lower semicontinuity in $W_{loc}^{1,1}(\Omega)$ with respect to the strong…

We provide relaxation for not lower semicontinuous supremal functionals of the type $W^{1,\infty}(\Omega;\mathbb R^d) \ni u \mapsto\supess_{ x \in \Omega}f(\nabla u(x))$ in the vectorial case, where $\Omega\subset \mathbb R^N$ is a…

We consider the functional $$F_\infty(u)=\int_{\Omega}f(x,u(x),\nabla u(x)) dx \quad\quad u\in \varphi+ W_0^{1,\infty}(\Omega,\mathbb{R})$$ where $\Omega$ is an open bounded Lipschitz subset of $\mathbb{R}^N$ and $\varphi\in…

The weak lower semicontinuity of the functional $$ F(u)=\int_{\Omega}f(x,u,\nabla u)\, dx$$ is a classical topic that was studied thoroughly. It was shown that if the function $f$ is continuous and convex in the last variable, the…

We prove results on the relaxation and weak* lower semicontinuity of integral functionals of the form \[ \mathcal{F}[u] := \int_{\Omega} f \bigg( \frac{1}{2} \bigl( \nabla u(x) + \nabla u(x)^T \bigr) \bigg)\,\mathrm{d} x, \qquad u : \Omega…

We provide necessary and sufficient conditions on the density $W:\mathbb R^d\times\mathbb R ^d\to\mathbb R$ in order to ensure the sequential weak* lower semicontinuity of the functional $J: W^{1,\infty}(I;\mathbb R^d)\to \mathbb R$,…

The purpose of this paper is to study the lower semicontinuity with respect to the strong $L^1$-convergence, of some integral functionals defined in the space SBD of special functions with bounded deformation. Precisely, let $U$ be a…

We study variational problems involving nonlocal supremal functionals $L^\infty(\Omega;\mathbb{R}^m) \ni u\mapsto {\rm ess sup}_{(x,y)\in \Omega\times \Omega} W(u(x), u(y)),$ where $\Omega\subset \mathbb{R}^n$ is a bounded, open set and…

In this note we present a sufficient condition ensuring lower semicontinuity for nonlocal supremal functionals of the type $$W^{1,\infty}(\Omega;\mathbb R^d)\ni u \mapsto \sup{\rm ess}_{(x,y)\in \Omega} W(x,y, \nabla u(x),\nabla u(y)),$$…

We prove the lower semicontinuity of functionals of the form \[ \int \limits_\Omega \! V(\alpha) \, \mathrm{d} |\mathrm{E} u| \, , \] with respect to the weak converge of $\alpha$ in $W^{1,\gamma}(\Omega)$, $\gamma > 1$, and the weak*…

We study integral functionals defined on scalar Sobolev spaces of the form $$E[f]:u\mapsto \int_\Omega f(x,u(x),\nabla u(x)) d x,$$ with an emphasis on the non-convex case, and the difficulties it involves to prevent the Lavrentiev…

We isolate a general condition, that we call "localization principle", on the integrand L:\MM\to[0,\infty], assumed to be continuous, under which W^{1,q}-quasiconvexity with q\in[1,\infty] is a sufficient condition for I(u)=\int_\Omega…

Let $\Omega$, $\Omega'\subset\mathbb{R}^n$ be bounded domains and let $f_m\colon\Omega\to\Omega'$ be a sequence of homeomorphisms with positive Jacobians $J_{f_m} >0$ a.e. and prescribed Dirichlet boundary data. Let all $f_m$ satisfy the…

We introduce a notion of quasiconvexity for continuous functions $f$ defined on the vector bundle of linear maps between the tangent spaces of a smooth Riemannian manifold $(M,g)$ and $\mathbb{R}^m$, naturally generalizing the classical…

This article deals with the lower compactness property of a sequence of integrands and the use of this key notion in various domains: convergence theory, optimal control, non-smooth analysis. First about the interchange of the weak…

A uniformly continuously integrable sequence of real-valued measurable functions, defined on some probability space, is relatively compact in the $\sigma(L^1,L^\infty)$ topology. In this paper, we link such a result to weak convergence…

We state necessary and sufficient conditions for weak lower semicontinuity of $u\mapsto\int_\Omega h(x,u(x))\,d x$ where $|h(x,s)|\le C(1+|s|^p)$ is continuous and possesses a recession function, and $u\in L^p(\Omega;\mathbb{R}^m)$, $p>1$,…

We consider solutions $u\in W^{1,p}\big(\Omega;\mathbb{R}^{N}\big)$ of the $p$-Laplacian PDE \begin{equation} \nabla\cdot\big(a(x)|Du|^{p-2}Du\big)=0,\notag \end{equation} for $x\in\Omega\subseteq\mathbb{R}^{n}$, where $\Omega$ is open and…

We provide explicit examples to show that the relaxation of functionals $$ L^p(\Omega;\mathbb{R}^m) \ni u\mapsto \int_\Omega\int_\Omega W(u(x), u(y))\, dx\, dy, $$ where $\Omega\subset\mathbb{R}^n$ is an open and bounded set, $1<p<\infty$…