111 research outputs found
Volume constrained minimizers of the fractional perimeter with a potential energy
We consider volume-constrained minimizers of the fractional perimeter with
the addition of a potential energy in the form of a volume inte- gral. Such
minimizers are solutions of the prescribed fractional curvature problem. We
prove existence and regularity of minimizers under suitable assumptions on the
potential energy, which cover the periodic case. In the small volume regime we
show that minimizers are close to balls, with a quantitative estimate
The isoperimetric problem for nonlocal perimeters
We consider a class of nonlocal generalized perimeters which includes
fractional perimeters and Riesz type potentials. We prove a general
isoperimetric inequality for such functionals, and we discuss some
applications. In particular we prove existence of an isoperimetric profile,
under suitable assumptions on the interaction kernel.Comment: 17 p
Symmetric self-shrinkers for the fractional mean curvature flow
We show existence of homothetically shrinking solutions of the fractional
mean curvature flow, whose boundary consists in a prescribed numbers of
concentric spheres. We prove that all these solutions, except from the ball,
are dynamically unstable.Comment: 12 page
Isoperimetric problems for a nonlocal perimeter of Minkowski type
We prove a quantitative version of the isoperimetric inequality for a non
local perimeter of Minkowski type. We also apply this result to study
isoperimetric problems with repulsive interaction terms, under convexity
constraints. We show existence of minimizers, and we describe the shape of
minimizers in certain parameter regimes
Minimizers for nonlocal perimeters of Minkowski type
We study a nonlocal perimeter functional inspired by the Minkowski content,
whose main feature is that it interpolates between the classical perimeter and
the volume functional. This problem is related by a generalized coarea formula
to a Dirichlet energy functional in which the energy density is the local
oscillation of a function.
These two nonlocal functionals arise in concrete applications, since the
nonlocal character of the problems and the different behaviors of the energy at
different scales allow the preservation of details and irregularities of the
image in the process of removing white noises, thus improving the quality of
the image without losing relevant features.
In this paper, we provide a series of results concerning existence, rigidity
and classification of minimizers, compactness results, isoperimetric
inequalities, Poincar\'e-Wirtinger inequalities and density estimates.
Furthermore, we provide the construction of planelike minimizers for this
generalized perimeter under a small and periodic volume perturbation.Comment: To appear in Calc. Var. Partial Differential Equation
Fractional mean curvature flow of Lipschitz graphs
We consider the fractional mean curvature flow of entire Lipschitz graphs. We
provide regularity results, and we study the long time asymptotics of the flow.
In particular we show that in a suitable rescaled framework, if the initial
graph is a sublinear perturbation of a cone, the evolution asymptotically
approaches an expanding self-similar solution. We also prove stability of
hyperplanes and of convex cones in the unrescaled setting
Minimal periodic foams with equal cells
We show existence of periodic foams with equal cells in
minimizing an anisotropic perimeter.Comment: 9 page
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