Lower semicontinuity of a class of integral functionals on the space of functions of bounded deformation

2017 ◽  
Vol 10 (2) ◽  
pp. 183-207 ◽  
Author(s):  
Gianni Dal Maso ◽  
Gianluca Orlando ◽  
Rodica Toader
Keyword(s):  
Lower Semicontinuity ◽  
Linear Growth ◽  
Lower Semicontinuous ◽  
Euclidean Norm ◽  
Integral Functionals ◽  
Functions Of Bounded Deformation

AbstractWe study the lower semicontinuity of some free discontinuity functionals with linear growth defined on the space of functions with bounded deformation. The volume term is convex and depends only on the Euclidean norm of the symmetrized gradient. We introduce a suitable class of surface terms, which make the functional lower semicontinuous with respect to ${L^{1}}$ convergence.

scholarly journals Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints

2020 ◽  
Vol 13 (3) ◽  
pp. 219-255 ◽  
Author(s):  
Adolfo Arroyo-Rabasa ◽  
Guido De Philippis ◽  
Filip Rindler
Keyword(s):  
Lower Semicontinuity ◽  
Generalized Convexity ◽  
Linear Growth ◽  
Arbitrary Order ◽  
Integral Functionals ◽  
Vector Measures ◽  
First Order ◽  
Linear Pde ◽  
Side Constraints ◽  
Linear Pdes

AbstractWe show general lower semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known lower semicontinuity and relaxation theorems for BV, BD, and for more general first-order linear PDE side constrains. Our proofs are based on recent progress in the understanding of singularities of measure solutions to linear PDEs and of the generalized convexity notions corresponding to these PDE constraints.

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.

scholarly journals Equivalence of viscosity and weak solutions for a p-parabolic equation

2021 ◽  
Author(s):  
Jarkko Siltakoski
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Lower Semicontinuity ◽  
Lower Semicontinuous ◽  
Relationship Of ◽  
The Relationship

AbstractWe study the relationship of viscosity and weak solutions to the equation $$\begin{aligned} \smash {\partial _{t}u-\varDelta _{p}u=f(Du)}, \end{aligned}$$ ∂ t u - Δ p u = f ( D u ) , where $$p>1$$ p > 1 and $$f\in C({\mathbb {R}}^{N})$$ f ∈ C ( R N ) satisfies suitable assumptions. Our main result is that bounded viscosity supersolutions coincide with bounded lower semicontinuous weak supersolutions. Moreover, we prove the lower semicontinuity of weak supersolutions when $$p\ge 2$$ p ≥ 2 .

On the convergence of solutions of variational problems with pointwise functional constraints in variable domains

2020 ◽  
Vol 17 (4) ◽  
pp. 509-537
Author(s):  
Alexander Kovalevsky
Keyword(s):  
Bounded Domain ◽  
Convex Sets ◽  
Lower Semicontinuous ◽  
Variational Problems ◽  
Integral Functionals ◽  
Functional Constraints ◽  
Convergence Of Solutions ◽  
Minimum Value ◽  
Variable Domains

We consider a sequence of convex integral functionals $F_s:W^{1,p}(\Omega_s)\to\mathbb R$ and a sequence of weakly lower semicontinuous and, in general, nonintegral functionals $G_s:W^{1,p}(\Omega_s)\to\mathbb R$, where $\{\Omega_s\}$ is a sequence of domains in $\mathbb R^n$ contained in a bounded domain $\Omega\subset\mathbb R^n$ ($n\geqslant 2$) and $p>1$. Along with this, we consider a sequence of closed convex sets $V_s=\{v\in W^{1,p}(\Omega_s): M_s(v)\leqslant 0\,\,\text{a.e.\ in}\,\,\Omega_s\}$, where $M_s$ is a mapping from $W^{1,p}(\Omega_s)$ to the set of all functions defined on $\Omega_s$. We establish conditions under which minimizers and minimum values of the functionals $F_s+G_s$ on the sets $V_s$ converge to a minimizer and the minimum value of a functional on the set $V=\{v\in W^{1,p}(\Omega): M(v)\leqslant 0\,\,\text{a.e.\ in}\,\,\Omega\}$, where $M$ is a mapping from $W^{1,p}(\Omega)$ to the set of all functions defined on $\Omega$.

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