scholarly journals Weak Lower Semicontinuity of Integral Functionals and Applications

SIAM Review ◽  
2017 ◽  
Vol 59 (4) ◽  
pp. 703-766 ◽  
Author(s):  
Barbora Benešová ◽  
Martin Kružík
Keyword(s):  
Lower Semicontinuity ◽  
Integral Functionals ◽  
Weak Lower Semicontinuity

scholarly journals On the relaxation of integral functionals depending on the symmetrized gradient

2020 ◽  
pp. 1-36
Author(s):  
Kamil Kosiba ◽  
Filip Rindler
Keyword(s):  
Lower Semicontinuity ◽  
Integral Functionals ◽  
Weak Lower Semicontinuity ◽  
Functions Of Bounded Deformation ◽  
Image Position

Abstract 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.

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

2017 ◽  
Vol 10 (1) ◽  
pp. 49-67 ◽  
Author(s):  
Jan Krämer ◽  
Stefan Krömer ◽  
Martin Kružík ◽  
Gabriel Pathó
Keyword(s):  
Lower Semicontinuity ◽  
Sufficient Conditions ◽  
Extension Property ◽  
Order Differential Operator ◽  
Equivalent Condition ◽  
Integral Functionals ◽  
Weak Lower Semicontinuity ◽  
Necessary And Sufficient ◽  
Quasiconvexity At The Boundary ◽  
Special Case

AbstractWe 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.

scholarly journals Nonlocal Variational Principles with Variable Growth

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Yongqiang Fu ◽  
Miaomiao Yang
Keyword(s):  
Sobolev Spaces ◽  
Lower Semicontinuity ◽  
Variable Exponent ◽  
Variational Principles ◽  
Young Measures ◽  
Bounded Set ◽  
Weak Lower Semicontinuity ◽  
Variable Exponent Sobolev Spaces ◽  
Regular Open

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.

Weak lower semicontinuity for polyconvex integrals in the limit case

2013 ◽  
Vol 51 (1-2) ◽  
pp. 171-193 ◽  
Author(s):  
M. Focardi ◽  
N. Fusco ◽  
C. Leone ◽  
P. Marcellini ◽  
E. Mascolo ◽  
...  
Keyword(s):  
Lower Semicontinuity ◽  
Limit Case ◽  
Weak Lower Semicontinuity
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