Dimension reduction for functionals on solenoidal vector fields

TL;DR

This paper investigates the asymptotic behavior of integral functionals constrained to divergence-free vector fields in thin domains, showing that the Gamma-limit simplifies to a convexified energy density despite potential nonlocal effects.

Contribution

It establishes the Gamma-convergence of functionals on divergence-free fields in thin domains to a convexified energy functional, highlighting a surprising simplification.

Findings

Gamma-limit equals convexified energy density

Relaxation can lead to nonlocal functionals in general

Results hold under standard growth and coercivity assumptions

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.