1,721,067 research outputs found
Curvature of closed subsets of Euclidean space and minimal submanifolds of arbitrary
Full text linkA second fundamental form is introduced for arbitrary closed subsets of Euclidean space, extending the same notion introduced by J. Fu for sets of positive reach. We extend well known integral-geometric formulas to this general setting and we provide a structural result in terms of second fundamental forms of submanifolds of class that is new even for sets of positive reach. In the case of a large class of minimal submanifolds, which include viscosity solutions of the minimal surface system and rectifiable stationary varifolds of arbitrary codimension and higher multiplicities, we prove the area formula for the generalized Gauss map in terms of the discriminant of the second fundamental form and, adapting techniques from the theory of viscosity solutions of elliptic equations to our geometric setting, we conclude a natural second-order-differentiability property almost everywhere. Moreover the trace of the second fundamental form is proved to be zero for stationary integral varifolds
Rectifiability and approximate differentiability of higher order for sets
Full text linkThe main goal of this paper is to develop a concept of approximate differentiability of higher order for subsets of the Euclidean space that allows to characterize higher order rectifiable sets, extending somehow well known facts for functions. We emphasize that for every subset of the Euclidean space and for every integer we introduce the approximate differential of order of and we prove it is a Borel map whose domain is a (possibly empty) Borel set. This concept could be helpful to deal with higher order rectifiable sets in applications
Distance functions with dense singular sets
No full textWe characterize the denseness of the singular set of the distance function from a (Formula presented.) -hypersurface in terms of an inner ball condition and we address the problem of the existence of viscosity solutions of the Eikonal equation whose singular set (i.e. set of non-differentiability points) is not no-where dense
Normal bundle and Almgren's geometric inequality for singular varieties of bounded mean curvature
Full text linkIn this paper we deal with a class of varieties of bounded mean curvature in the viscosity sense that has the remarkable property to contain the blow up sets of all sequences of varifolds whose mean curvatures are uniformly bounded and whose boundaries are uniformly bounded on compact sets. We investigate the second-order properties of these varieties, obtaining results that are new also in the varifold's setting. In particular we prove that the generalized normal bundle of these varieties satisfies a natural Lusin (N) condition, a property that allows to prove a Coarea-type formula for their generalized Gauss map. Then we use this formula to extend a sharp geometric inequality of Almgren and the associated soap bubble theorem. As a consequence of the geometric inequality we obtain sufficient conditions to conclude that the area-blow-up set is empty for sequences of varifolds whose first variation is controlled
Second order rectifiability of varifolds of bounded mean curvature
Full text linkWe prove that the support of an m dimensional rectifiable varifold with a uniform lower bound on the density and bounded generalized mean curvature can be covered Hm almost everywhere by a countable union of m dimensional submanifolds of class C2. The C2-regularity of the submanifolds is optimal
Solid Mechanics. – Two-phase models for elastic membranes with soft inclusions, by Mario Santilli and Bernd Schmidt, communicated on 16 December 2022
Full text linkA procedure for magnetic design and optimisation of permanent magnet brushless dc motors
No full textA geometric second-order-rectifiable stratification for closed subsets of Euclidean space
No full textDefining the m-th stratum of a closed subset of an n dimensional Euclidean space to consist of those points, where it can be touched by a ball from at least n - m linearly independent directions, we establish that the m-th stratum is second-order rectifiable of dimension m and a Borel set. This was known for convex sets, but is new even for sets of positive reach. The result is based on a sufficient condition of parametric type for second-order rectifiability
Regularity of the distance function from arbitrary closed sets
Full text linkWe investigate the distance function delta(phi)(K) from an arbitrary closed subset K of a finite-dimensional Banach space (R-n, phi), equipped with a uniformly convex l(2)-norm phi. These spaces are known as Minkowski spaces and they are one of the fundamental spaces of Finslerian geometry (see Martini et al. in Expo Math 19:97-142, 2001, https://doi.org/10.1016/S0723-0869(01)80025-6) . We prove that the gradient of delta(phi)(K) satisfies a Lipschitz property on the complement of the phi-cut-locus of K (a.k.a. the medial axis of R-n similar to K) and we prove a structural result for the set of points outside K where delta(phi)(K) is pointwise twice differentiable, providing an answer to a question raised by Hiriart-Urruty (Am Math Mon 89:456-458, 1982, hups://doi.org/10.2307/2321379). Our results give sharp generalisations of some classical results in the theory of distance functions and they are motivated by critical low-regularity examples for which the available results gives no meaningful or very restricted informations. The results of this paper find natural applications in the theory of partial differential equations and in convex geometry
Il gruppo dei pari come risorsa nell’orientamento
No full textRassegna critica sugli interventi con l'utilizzo della peer education nei casi di dispersione scolastic
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