Dimension reduction for functionals on solenoidal vector fields
Universität zu Köln, Köln, Germany
Received: 21 April 2010
We study integral functionals constrained to divergence-free vector fields in Lp on a thin domain, under standard p-growth and coercivity assumptions, 1 < p < ∞. We prove that as the thickness of the domain goes to zero, the Gamma-limit with respect to weak convergence in Lp 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.
Mathematics Subject Classification: 49J45 / 35E99
Key words: Divergence-free fields / Gamma-convergence / dimension reduction
© EDP Sciences, SMAI, 2010