BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Leon Happ (TU Wien) DTSTART:20240311T144000Z DTEND:20240311T161000Z DTSTAMP:20260405T175107Z UID:NSCM/128 DESCRIPTION:Title: E xtension operators for homogenization that preserve target manifold constr aints\nby Leon Happ (TU Wien) as part of Nečas Seminar on Continuum M echanics\n\nLecture held in Room K3\, Faculty of Mathematics and Physics\ , Charles University\, Sokolovská 83 Prague 8..\n\nAbstract\nMotivated b y homogenization problems in micro-magnetics and plasticity\, we study the existence of extension operators for W^{1\,p} Sobolev maps on periodicall y perforated domains\, where the maps are only allowed to range in some pr escribed 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 li es 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 pre serving this feature. It turns out that this is a problem sitting right at the interface between analysis and algebraic topology.\n\nThe unconstrain ed 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 ans wer 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 differ entiation. It can be shown that\, due to Sobolev embeddings\, for p bigger or equal than d the question whether every Sobolev function on a perforat ed domain possesses an extension reduces to a similar question for continu ous functions. This is a problem well studied in algebraic topology and\, in theory\, there are tools available to determine its answer.\n\nThese ob servations lead us to focus on the case that p is smaller than d (e.g.\, i n the physically relevant instance that p=2\, d=3). Then\, for p>1\, we ca n indeed proof the existence of an extension operator as described above\, and under the additional assumption that the target manifold is [p-1] con nected (where [] is the floor function)\, meaning that the first [p-1] fun damental groups of N are trivial. The proof builds upon the already allude d result by E. Acerbi\, V. C. Piat\, G. Dal Maso\, and D. Percivale\, comb ining 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 wi ll not be linear. Already the target manifold constraint is not compatible with linearity. \n\nIn my talk\, I will put a special emphasis on illumin ating 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.\n LOCATION:https://researchseminars.org/talk/NSCM/128/ END:VEVENT END:VCALENDAR