On the relaxation of integral functionals depending on the symmetrized gradient

We prove results on the relaxation and weak* lower semicontinuity of integral functionals of the form $${\cal F}[u]: = \int_\Omega f \left( {\displaystyle{1 \over 2}\left( {\nabla u(x) + \nabla u{(x)}^T} \right)} \right) \,{\rm d}x,\quad u:\Omega \subset {\mathbb R}^d\to {\mathbb R}^d,$$ over the space BD(Ω) of functions of bounded deformation or over the Temam–Strang space $${\rm U}(\Omega ): = \left\{ {u\in {\rm BD}(\Omega ):\;\,{\rm div}\,u\in {\rm L}^2(\Omega )} \right\},$$ depending on the growth and shape of the integrand f. Such functionals are interesting, for example, in the study of Hencky plasticity and related models.

Calculus of variations: A differential form approach

We study integrals of the form {\int_{\Omega}f(d\omega_{1},\dots,d\omega_{m})}, where {m\geq 1} is a given integer, {1\leq k_{i}\leq n} are integers, {\omega_{i}} is a {(k_{i}-1)}-form for all {1\leq i\leq m} and {f:\prod_{i=1}^{m}\Lambda^{k_{i}}(\mathbb{R}^{n})\rightarrow\mathbb{R}} is a continuous function. We introduce the appropriate notions of convexity, namely vectorial ext. one convexity, vectorial ext. quasiconvexity and vectorial ext. polyconvexity. We prove weak lower semicontinuity theorems and weak continuity theorems and conclude with applications to minimization problems. These results generalize the corresponding results in both classical vectorial calculus of variations and the calculus of variations for a single differential form.

𝒜$\mathcal{A}$-quasiconvexity at the boundary and weak lower semicontinuity of integral functionals

We state necessary and sufficient conditions for weak lower semicontinuity of integral functionals of the form ${u\mapsto\int_{\Omega}h(x,u(x))\,\mathrm{d}x}$, where h is continuous and possesses a positively p-homogeneous recession function, ${p>1}$, and ${u\in L^{p}(\Omega;\mathbb{R}^{m})}$ lives in the kernel of a constant-rank first-order differential operator ${\mathcal{A}}$ which admits an extension property. In the special case ${\mathcal{A}=\mathrm{curl}}$, apart from the quasiconvexity of the integrand, the recession function’s quasiconvexity at the boundary in the sense of Ball and Marsden is known to play a crucial role. Our newly defined notions of ${\mathcal{A}}$-quasiconvexity at the boundary, generalize this result. Moreover, we give an equivalent condition for the weak lower semicontinuity of the above functional along sequences weakly converging in ${L^{p}(\Omega;\mathbb{R}^{m})}$ and approaching the kernel of ${\mathcal{A}}$ even if ${\mathcal{A}}$ does not have the extension property.

Nonlocal Variational Principles with Variable Growth

This paper is concerned with the functionalJdefined byJ(u)=∫Ω×ΩW(x,y,∇u(x),∇u(y))dx dy, whereΩ⊂ℝNis a regular open bounded set andWis a real-valued function with variable growth. After discussing the theory of Young measures in variable exponent Sobolev spaces, we study the weak lower semicontinuity and relaxation ofJ.