1,721,070 research outputs found
Brunn-Minkowski inequality for the 1-Riesz capacity and level set convexity for the 1/2-Laplacian
No full textWe prove that that the 1-Riesz capacity satisfies a Brunn-Minkowski inequality, and that the capacitary function of the 1/2-Laplacian is level set convex
K-mean convex and K-outward minimizing sets
No full textWe consider the evolution of sets by nonlocal mean curvature and we discuss the preservation along the flow of two geometric properties, which are the mean convexity and the outward minimality. The main tools in our analysis are the level set formulation and the minimizing movement scheme for the nonlocal flow. When the initial set is outward minimizing, we also show the convergence of the (time integrated) nonlocal perimeters of the discrete evolutions to the nonlocal perimeter of the limit flow
Minimal Periodic Foams with Equal Cells
No full textWe show existence of periodic foams with equal cells in R-n minimizing an anisotropic perimeter
Nonlocal minimal clusters in the plane
Full text linkWe prove existence of partitions of an open set Ω with a given number of phases, which minimize the sum of the fractional perimeters of all the phases, with Dirichlet boundary conditions. In two dimensions we show that, if the fractional parameter s is sufficiently close to 1, the only singular minimal cone, that is, the only minimal partition invariant by dilations and with a singular point, is given by three half-lines meeting at 120 degrees. In the case of a weighted sum of fractional perimeters, we show that there exists a unique minimal cone with three phases
Minimal periodic foams with fixed inradius
Full text linkIn this note, we show existence and regularity of periodic tilings of the Euclidean space into equal cells containing a ball of fixed radius, which minimize either the classical or the fractional perimeter. We also discuss some qualitative properties of minimizers in dimensions 3 and 4
Minimisers of a general Riesz-type problem
Full text linkWe consider sets in RN which minimise, for fixed volume, the sum of the perimeter and a non-local term given by the double integral of a kernel g:RN∖{0}→R+. We establish some general existence and regularity results for minimisers. In the two-dimensional case we show that balls are the unique minimisers in the perimeter-dominated regime, for a wide class of functions g
Existence of minimizers for a generalized liquid drop model with fractional perimeter
No full textWe consider the minimization problem of the functional given by the sum of the fractional perimeter and a general Riesz potential, which is one generalization of Gamow's liquid drop model. We first show the existence of minimizers for any volumes if the kernel of the Riesz potential decays faster than that of the fractional perimeter. We also prove the existence of generalized minimizers for any volumes if the kernel of the Riesz potential just vanishes at infinity. Finally, we study the asymptotic behavior of minimizers when the volume goes to infinity and we prove that a sequence of minimizers converges to the Euclidean ball up to translations if the kernel of the Riesz potential decays sufficiently fast.(c) 2022 Elsevier Ltd. All rights reserved
Minimal Barriers for Geometric Evolutions
Full text linkWe study some properties of De Giorgi's minimal barriers and local minimal barriers for geometric flows of subsets of R-n. Concerning evolutions of the form partial derivative u/partial derivative t + F(del u, del(2)u) = 0, we prove a representation result for the minimal barrier M(E, F-F) when F is not degenerate elliptic; namely, we show that M(E, F-F) = M(E, FF+), where F+ is the smallest degenerate elliptic function above F. We also characterize the disjoint sets property and the joint sets property in terms of the Function F
Connected surfaces with boundary minimizing the Willmore energy
Full text linkFor a given family of smooth closed curves gamma(1),...,gamma(alpha) subset of R-3 we consider the problem of finding an elastic connected compact surface M with boundary gamma = gamma(1) boolean OR ... boolean OR gamma(alpha). This is realized by minimizing the Willmore energy W on a suitable class of competitors. While the direct minimization of the Area functional may lead to limits that are disconnected, we prove that, if the infimum of the problem is < 4 pi, there exists a connected compact minimizer of W in the class of integer rectifiable curvature varifolds with the assigned boundary conditions. This is done by proving that varifold convergence of bounded varifolds with boundary with uniformly bounded Willmore energy implies the convergence of their supports in Hausdorff distance. Hence, in the cases in which a small perturbation of the boundary conditions causes the non-existence of Area-minimizing connected surfaces, our minimization process models the existence of optimal elastic connected compact generalized surfaces with such boundary data. We also study the asymptotic regime in which the diameter of the optimal connected surfaces is arbitrarily large. Under suitable boundedness assumptions, we show that rescalings of such surfaces converge to round spheres. The study of both the perturbative and the asymptotic regime is motivated by the remarkable case of elastic surfaces connecting two parallel circles located at any possible distance one from the other. The main tool we use is the monotonicity formula for curvature varifolds ([15, 31]) that we extend to varifolds with boundary, together with its consequences on the structure of varifolds with bounded Willmore energy
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