Relaxation of singular functionals defined on Sobolev spaces

Hafedh Ben Belgacem

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 5, page 71-85
  • ISSN: 1292-8119

Abstract

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In this paper, we consider a Borel measurable function on the space of m × n matrices f : M m × n → ℝ ¯ taking the value + ∞ , such that its rank-one-convex envelope R f is finite and satisfies for some fixed p > 1 : - c 0 ≤ R f ( F ) ≤ c ( 1 + ∥ F ∥ p ) for all F ∈ M m × n , where c , c 0 > 0 . Let Ø be a given regular bounded open domain of ℝ n . We define on W 1 , p ( Ø ; ℝ m ) the functional I ( u ) = ∫ Ø f ( ∇ u ( x ) ) d x . Then, under some technical restrictions on f , we show that the relaxed functional I ¯ for the weak topology of W 1 , p ( Ø ; ℝ m ) has the integral representation: I ¯ ( u ) = ∫ Ø Q [ R f ] ( ∇ u ( x ) ) d x , where for a given function g , Q g denotes its quasiconvex envelope.

How to cite

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Ben Belgacem, Hafedh. "Relaxation of singular functionals defined on Sobolev spaces." ESAIM: Control, Optimisation and Calculus of Variations 5 (2010): 71-85. <http://eudml.org/doc/116558>.

@article{BenBelgacem2010,
abstract = { In this paper, we consider a Borel measurable function on the space of $\scriptstyle m\times n$ matrices $\scriptstyle f: M^\{m\times n\}\rightarrow \bar\{\mathbb\{R\}\}$ taking the value $ \scriptstyle +\infty$, such that its rank-one-convex envelope $\scriptstyle Rf$ is finite and satisfies for some fixed $\scriptstyle p>1$: $$\scriptstyle -c\_0\leq Rf(F)\leq c(1+\Vert F\Vert^p)\ \hbox\{for all\}\ F\in M^\{m\times n\},$$ where $\scriptstyle c,c_0>0$. Let $\scriptstyle\O$ be a given regular bounded open domain of $\scriptstyle \mathbb\{R\}^n$. We define on $\scriptstyle W^\{1,p\}(\O;\mathbb\{R\}^m)$ the functional $$\scriptstyle I(u)=\int\_\{\O\}f(\nabla u(x))\ dx.$$ Then, under some technical restrictions on $\scriptstyle f$, we show that the relaxed functional $\scriptstyle\bar I$ for the weak topology of $\scriptstyle W^\{1,p\}(\O;\mathbb\{R\}^m)$ has the integral representation: $$\scriptstyle \bar I(u)=\int\_\{\O\}Q[Rf](\nabla u(x))\ dx,$$ where for a given function $\scriptstyle g$, $\scriptstyle Qg$ denotes its quasiconvex envelope. },
author = {Ben Belgacem, Hafedh},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Rank-one convexity; quasiconvexity; weak lower semicontinuity.; rank-one convexity; weak lower semicontinuity; relaxation; integral functional},
language = {eng},
month = {3},
pages = {71-85},
publisher = {EDP Sciences},
title = {Relaxation of singular functionals defined on Sobolev spaces},
url = {http://eudml.org/doc/116558},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Ben Belgacem, Hafedh
TI - Relaxation of singular functionals defined on Sobolev spaces
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 5
SP - 71
EP - 85
AB - In this paper, we consider a Borel measurable function on the space of $\scriptstyle m\times n$ matrices $\scriptstyle f: M^{m\times n}\rightarrow \bar{\mathbb{R}}$ taking the value $ \scriptstyle +\infty$, such that its rank-one-convex envelope $\scriptstyle Rf$ is finite and satisfies for some fixed $\scriptstyle p>1$: $$\scriptstyle -c_0\leq Rf(F)\leq c(1+\Vert F\Vert^p)\ \hbox{for all}\ F\in M^{m\times n},$$ where $\scriptstyle c,c_0>0$. Let $\scriptstyle\O$ be a given regular bounded open domain of $\scriptstyle \mathbb{R}^n$. We define on $\scriptstyle W^{1,p}(\O;\mathbb{R}^m)$ the functional $$\scriptstyle I(u)=\int_{\O}f(\nabla u(x))\ dx.$$ Then, under some technical restrictions on $\scriptstyle f$, we show that the relaxed functional $\scriptstyle\bar I$ for the weak topology of $\scriptstyle W^{1,p}(\O;\mathbb{R}^m)$ has the integral representation: $$\scriptstyle \bar I(u)=\int_{\O}Q[Rf](\nabla u(x))\ dx,$$ where for a given function $\scriptstyle g$, $\scriptstyle Qg$ denotes its quasiconvex envelope.
LA - eng
KW - Rank-one convexity; quasiconvexity; weak lower semicontinuity.; rank-one convexity; weak lower semicontinuity; relaxation; integral functional
UR - http://eudml.org/doc/116558
ER -

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