On Compactness of Minimizing Sequences Subject to a Linear Differential Constraint

Abstract

For Ω ⊂ R N open (and possibly unbounded), we consider integral functionals of the formdened on the subspace of L p consisting of those vector elds u which satisfy Au = 0 on Ω in the sense of distributions. Here, A may be any linear dierential operator of rst order with constant coecients satisfying Murat's condition of constant rank. The main results provide sharp conditions for the compactness of minimizing sequences with respect to the strong topology in L p .

On the role of lower bounds in characterizations of weak lower semicontinuity of multiple integrals

“…The proof is based on an idea that already was used in [13] in a more general context. Let " > 0 and choose a function u 2 U such that…”

A weak-strong convergence property and symmetry of minimizers of constrained variational problems in $\mathbb{R}^N$

“…However, all of these articles yield strong convergence at most on bounded domains. Results on unbounded domains with the sequence constrained to solutions of a linear system of differential equations (which includes the gradient case by using the constraint curl u n = 0, but is not limited to it) were recently obtained by the second author in [16].…”

“…x ∈ Ω, every µ ∈ R and every ξ ∈ R N . Then we have the following theorem, essentially a variant of the results of Visintin [24], Evans & Gariepy [7], Zhang [25] and Sychev [22,23] for the scalar case on unbounded domains (for related results on unbounded domains also see [16]).…”

“…The results cited above are not applicable in the situation of Theorem 2.7, in particular because of the unbounded domain and the extremely weak lower bound. Nevertheless, it is not difficult to obtain a proof following the approach of [22,23] or [16] based on the theory of Young measures. In fact, since we only consider the scalar case (in particular, quasiconvexity reduces to convexity) and exploit the equiintegrability of the sequence considered, it can be simplified significantly.…”