Extension operators for homogenization that preserve target manifold constraints

Abstract: Motivated by homogenization problems in micro-magnetics and plasticity, we study the existence of extension operators for W^{1,p} Sobolev maps on periodically perforated domains, where the maps are only allowed to range in some prescribed Riemannian manifold. For applications, one demands that separate L^p bounds for the extension and its gradient, independent of the scale of the perforations, hold. We stress that the main novelty of our work lies in the fact that all functions are subject to a manifold constrained in the target space. The challenge is then to find an extension operator preserving this feature. It turns out that this is a problem sitting right at the interface between analysis and algebraic topology.

The unconstrained case was famously 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. However, as soon as one starts to include manifold constraints, the existence problem of extensions becomes highly sensitive towards the relation between p and d and calls for a case differentiation. It can be shown that, due to Sobolev embeddings, for p bigger or equal than d the question whether every Sobolev function on a perforated domain possesses an extension reduces to a similar question for continuous functions. This is a problem well studied in algebraic topology and, in theory, there are tools available to determine its answer.

These observations lead us to focus on the case that p is smaller than d (e.g., in the physically relevant instance that p=2, d=3). Then, for p>1, we can indeed proof the existence of an extension operator as described above, and under the additional assumption that the target manifold is [p-1] connected (where [] is the floor function), meaning that the first [p-1] fundamental groups of N are trivial. The proof builds upon the already alluded result by E. Acerbi, V. C. Piat, G. Dal Maso, and D. Percivale, combining it with an appropriate retraction onto the target manifold, firstly suggested by R. Hardt and F.-H. Lin in 1987, and originating again from algebraic topology. It is important to note that the extension operator will not be linear. Already the target manifold constraint is not compatible with linearity.

In my talk, I will put a special emphasis on illuminating the [p-1]-connectedness condition that arises from the construction of the retraction onto N. The same assumption is already well-known in the context of the Sobolev functions between manifolds, where, for instance, it determines the surjectivity of the trace operator. This is only one example of a deeper connection that I will unveil.

MathematicsPhysics

Audience: researchers in the topic

( video )

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Nečas Seminar on Continuum Mechanics

Series comments: This seminar was founded on December 14, 1966.

Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 8. If not written otherwise, we will meet on Mondays at 15:40 in lecture hall K3 (2nd floor)

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