Dimension reduction for functionals on solenoidal vector fields

Abstract

We study integral functionals constrained to divergence-free vector fields in $L^p$ on a thin domain, under standard $p$-growth and coercivity assumptions, $1<p<\infty$. We prove that as the thickness of the domain goes to zero, the Gamma-limit with respect to weak convergence in $L^p$ is always given by the associated functional with convexified energy density wherever it is finite. Remarkably, this happens despite the fact that relaxation of nonconvex functionals subject to the limiting constraint can give rise to a nonlocal functional as illustrated in an example.

25 pages

References in corpus (3)

• The von Karman equations for plates with residual strain
• Compensated Compactness, Separately convex Functions and interpolatory Estimates between Riesz Transforms and Haar Projections
• On the relationship between rank-$(n-1)$ convexity and $\mathcal{S}$-quasiconvexity

Cited by in corpus (1)

• Another approach to the thin-film Gamma-limit of the micromagnetic free energy in the regime of small samples