Scale independent extension operators for manifold valued Sobolev maps on perforated domains

Abstract

Motivated by homogenisation problems in micro-magnetics and plasticity, my talk is concerned with the existence of extension operators for Sobolev maps on periodically perforated domains, where the maps are only allowed to range in some prescribed manifold.For applications, one demands that separate Lp bounds for the extension and its gradient, independent of the scale of the perforations, hold. I want to stress that the novelty is really that all functions are subject to a constrained in the target space. The challenge is then to find an extension operator preserving this target manifold constraint. Such extension operators prove to be of immense benefit in the field of homogenisation, e.g., for deducing compactness results. Mainly due to Sobolev embeddings, the problem calls for a case differentiation in regard to the relation between the Sobolev exponent p and the dimension n of the domain. The new result I present focuses on the case that p is smaller than n (e.g., in the physically relevant instance that p=2, n=3). Then we can prove the existence of a bounded extension operator for W1,p maps on perforated domains that preserves the described manifold constrained in the target space, under the additional assumption that the target manifold is [p-1] connected. (where [] is the floor function). The unconstrained case was already treated in an influential paper by E. Acerbi, V. C. Piat, G. Dal Maso, and D. Percivale from 1992, providing a positive answer for all p except infinity. Our work builds upon their result, combining it with an appropriate retraction, firstly suggested by R. Hardt and F.-H. Lin in 1987, and originating from the field of obstruction theory in homology. I will present and discuss both ingredients separately; and then, how, by a careful construction, one can derive an extension operator (in the above sense with scale-independent Lp bounds) from them. I will complete my talk by pointing out how a relaxed version of the extension problem is closely entangled – and even equivalent in some cases – with the question about the surjectivity of the trace operator for Sobolev maps between a manifold with boundary and another target manifold. By studying this relation, it is possible to deduce necessary conditions for the existence of our desired extension operator – also in cases exceeding the assumption of a [p-1] connected target manifold as above.