Calculus of Variations and Geometric Measure Theory

E. Acerbi - G. Bouchitté - I. Fonseca

Relaxation of convex functionals: the gap problem

created on 30 Nov 2001
modified on 10 Dec 2003


Published Paper

Inserted: 30 nov 2001
Last Updated: 10 dec 2003

Journal: Ann. Inst. H. Poincaré Anal. Non Linéaire
Volume: 20
Number: 3
Pages: 359-390
Year: 2003


It is shown that the relaxed energy $$ \mathcal F(u,A) :=\inf\Bigl\{\liminf{n\to+\infty} \intA f(x, \nabla un) dx:\ \{un\} \subset \W(A)\, ,\ un \to u\ \text{in}\ {L}1(A) \Bigr\}, $$ admits the representation $$\mathcal F(u,A) = \intA f(x,\nabla u) dx+ \mus(A)\ ,$$ where $f$ is a convex, Carathéodory integrand satisfying a non standard ``$\alpha$-$\beta$'' growth hypothesis, $ \beta\in [\alpha, N\alpha/(N-1))$. Sufficient conditions guaranteeing that $\mu_s=0$ are discussed. An example asserting that this representation may fail in the quasiconvex case is provided.

Keywords: convexity, Besicovitch covering theorem, Radon-Nikodym derivative, Lavrentiev phenomenon