Un résultat d'existence pour les ensembles minimaux par optimisation sur des grilles polyédrales

Abstract : Let us recall that a subset of Rn is said to be minimal if its d-dimensional Hausdorff measure cannot be decreased by a deformation taken in a suitable class. One can give as an example the standard Plateau problem, which can be rewritten in terms of finding a minimal set under deformations that only move a relative compact subset of points of a given domain. In that case the boundary of the domain acts as a topological constrainst. An Almgren quasiminimal set can have its measure decreased by deformations, but only in a controlled manner in regard of the measure of the points being affected. For instance, graphs of lipschitz functions from Rd into Rn-d are quasiminimal, and it is known (see [A]) that quasiminimal sets are rectifiable. Furthermore (see [DS]), cores of quasiminimal sets — which are obtained by removing the points whose contribution to the measure of the set is null, operation denoted as making the set reduced — contain big pieces of lipschitz graphs and are uniformly rectifiable. Another interesting property shows up for Hausdorff limits of reduced quasiminimal sets. In fact, not only is the limit quasiminimal and reduced, but the Hausdorff measure is lower semicontinuous (see for instance [D1]), which is usually not true. This property makes limits of minimizing sequences of quasiminimal sets the ideal candidates for solving existence problems with topological constrainst stable under deformations. We introduce a theorem of existence, in the case of an open domain of Rn and in arbitrary dimension and codimension, using an automatic method to build a minimizing sequence of quasiminimal sets, by minimizing over the d-dimensional faces of polyhedral grids. Such grids must be carefully designed, since we want to approximate a given d-rectifiable set by following the directions of some of its approximate tangent planes to keep control on the measure increase introduced by the approximation, while keeping uniform bounds on the shape of the polyhedrons to avoid making them too flat. These uniform bounds are used when approximating other competitors using successive radial projections on decreasing dimensional polyhedrons of the grid till dimension d, to obtain quasiminimality constants depending only on d and n. The quasiminimal sequence we finally obtain converges on every compact subset of the domain to a minimal set — or almost-minimal when minimizing the functional defined as Jd h(E) = R hdHd where h is continuous on the domain, with values in [1,M]. If we can find lipschitz retractions onto the limit (we can obtain them for instance when n = 3 and d = 2 using Jean Taylor's regularity theorem in [T]) we can claim that the limit is still in the topological class we used. Our result could also be generalized in some cases of boundaryless manifolds, or on closed domains when there exists a lipschitz retraction of a neighborhood onto the boundary.
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https://tel.archives-ouvertes.fr/tel-00348735
Contributor : Vincent Feuvrier <>
Submitted on : Sunday, December 21, 2008 - 3:27:37 PM
Last modification on : Thursday, January 11, 2018 - 6:12:18 AM
Long-term archiving on : Tuesday, June 8, 2010 - 6:01:11 PM

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Vincent Feuvrier. Un résultat d'existence pour les ensembles minimaux par optimisation sur des grilles polyédrales. Mathématiques [math]. Université Paris Sud - Paris XI, 2008. Français. ⟨tel-00348735⟩

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