1,720,992 research outputs found
The relaxed energy of fractional Sobolev maps with values into the circle
Get PDFWe deal with the weak sequential density of smooth maps in the fractional Sobolev classes of W-s,W-p maps in high dimension domains and with values into the circle. When s is lower than one, using interpolation theory we introduce a natural energy in terms of optimal extensions on suitable weighted Sobolev spaces. The relaxation problem is then discussed in terms of Cartesian currents. When sp=1, the energy gap of the relaxed functional is always finite and is given by the minimal connection of the singularities times an energy weight, obtained through a minimum problem for one dimensional W-1/p,W-p maps with degree one. When sp>1, instead, concentration on codimension one sets needs unbounded energy. We finally treat the case where s is greater than one, obtaining an almost complete picture
The relaxed p-energy of manifold constrained mappings
No full textThe p-energy of Sobolev mappings between Riemannian manifolds is studied, for each integer p greater than two. We analyse the lower semicontinuous extension of the energy to currents. We then restrict to mappings with values into the p-sphere, by giving an explicit relaxed p-energy formula, whose proof depends on a strong density result. Finally, a related coarea formula is obtained
Strict convergence with equibounded area and minimal completely vertical liftings
No full textMinimal lifting measures of vector-valued functions of bounded variation were introduced by Jerrard-Jung. They satisfy strong continuity properties with respect to the strict convergence in BV. Moreover, they can be described in terms of the action of the optimal Cartesian currents enclosing the graph of u. We deal with a good notion of completely vertical lifting for maps with values into the two dimensional Euclidean space. We then prove lack of uniqueness in the high codimension case. Relationship with the relaxed area functional in the strict convergence is also discussed. (C) 2022 Elsevier Ltd. All rights reserved
On the Curvature Energy of Cartesian Surfaces
Get PDFWe analyze the lower semicontinuous envelope of the curvature functional of Cartesian surfaces in codimension one. To this aim, following the approach by Anzellotti–Serapioni–Tamanini, we study the class of currents that naturally arise as weak limits of Gauss graphs of smooth functions. The curvature measures are then studied in the non-parametric case. Concerning homogeneous functions, some model examples are studied in detail. Finally, a new gap phenomenon is observed
On Generalized Nonparametric Minimal Hyperfurfaces in High Dimension
No full textNonparametric g-surfaces in Euclidean space have recently been characterized by Bildhauer-Fuchs in terms of closure of a 1-form associated to the so called asymptotic normal. This 1-form can be written by means of the pull-back of a canonical vector-valued 1-form through a suitable map depending on the asymptotic normal, that in the minimal surfaces case agrees with the Gauss graph map. We show that a similar characterization holds true for g-hypersurfaces of any high dimension N, but this time in terms of a canonical vector valued form of degree N - 1. In the minimal hypersurfaces case, we finally discuss the lack of a relationship between the previous result and existence of good parameterizations, when N is greater than two
Relaxation results for a class of functionals with linear growth defined on manifold constrained mappings
No full textIn this paper we study the lower semicontinuous envelope of a class of functionals with linear growth defined
on mappings from the n-dimensional ball into RN that are constrained to take values into a smooth submanifold Y
of R^N
Weak elastic energy of irregular curves
No full textA weak notion of elastic energy for (not necessarily regular) rectifiable curves in any space dimension is proposed. Our p-energy is defined through a relaxation process, where a suitable p-rotation of inscribed polygons is adopted. The discrete p-rotation we choose has a geometric flavour: a polygon is viewed as an approximation to a smooth curve, and hence its discrete curvature is spread out into a smooth density. For any exponent p greater than 1, the p-energy is finite if and only if the arc-length parametrization of the curve has a second-order summability with the same growth exponent. In that case, moreover, the energy agrees with the natural extension of the integral of the pth power of the scalar curvature. Finally, a comparison with other definitions of discrete curvature is provided. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'
The BV-energy of maps into a manifold: relaxation and density results
No full textLet Y be a smooth compact oriented Riemannian manifold without boundary, and assume
that its 1-homology group has no torsion. Weak limits of graphs of smooth maps uk:Bn !
Y with equibounded total variation give rise to equivalence classes of Cartesian currents in
cart1,1(Bn × Y) for which we introduce a natural BV-energy. Assume moreover that the first
homotopy group of Y is commutative. In any dimension n we prove that every element T in
cart1,1(Bn ×Y) can be approximated weakly in the sense of currents by a sequence of graphs
of smooth maps uk:Bn !Y with total variation converging to the BV-energy of T. As a consequence,
we characterize the lower semicontinuous envelope of functions of bounded variations
from Bn into Y
Maps of bounded variation with values into a manifold: total variation and relaxed energy
No full textGraphs of W1,1 maps with values into S1: relaxed energies, minimal connections and lifting
No full textThe aim of this paper is to link the analytic results of [6] [7] [19] relative to W^1;1-mappings from B^n into
S1 to the measure theoretical geometric results in [12] [15]. The paper also contains a few remarks about
mappings in W^1;p, p>=2, with values into S2
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