Γ-convergence of nonconvex unbounded integrals in strongly connected sets

Abstract

We study Γ-convergence of nonconvex integrals of the calculus of variations in strongly connected sets when the integrands do not have polynomial growth and can take infinite values. Applications to homogenization of unbounded integrals in strongly perforated sets are also developed.

Keywords:

• Γ-Convergence
• strong connectedness
• perforated sets
• nonconvex unbounded integrals
• homogenization

MSC:

• 49J45

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The abbreviation ru-usc means radially uniformly upper semicontinuous.

2 The abbreviation lsc means lower semicontinuous.

3 Given a set X and a function $f:X\to [0,\mathrm{\infty }]$, by the effective domain of f we mean the set of $z\in X$ such that $f\left(z\right)<\mathrm{\infty }$.

4 By Alexandrov's theorem we mean the following result: ${\mu }_{ϵ}\rightharpoonup \mu$ weakly if and only if $\underset{ϵ\to 0}{lim}{\mu }_{ϵ}\left(B\right)=\mu \left(B\right)$ for all bounded Borel sets $B\subset O$ with $\mu \left(\mathrm{\partial B}\right)=0$.

5 For $a=(a_i{)}_{i\in \{1,\cdots ,m\}}\in {\mathbb{R}}^m$ and $b=(b_j{)}_{j\in \{1,\cdots ,N\}}\in {\mathbb{R}}^N$, $a\otimes b\in \mathbb{M}$ and is defined by $a\otimes b:=(a_i{b}_j{)}_{(i,j)\in \{1,\cdots ,m\}\times \{1,\cdots ,N\}}$.