235 research outputs found
Brunn-Minkowski inequality for the 1-Riesz capacity and level set convexity for the 1/2-Laplacian
We prove that that the 1-Riesz capacity satisfi es a Brunn-Minkowski
inequality, and that the capacitary function of the 1/2-Laplacian is level set
convex.Comment: 9 page
Mean curvature flow with obstacles: existence, uniqueness and regularity of solutions
We show short time existence and uniqueness of \C^{1,1} solutions to the
mean curvature flow with obstacles, when the obstacles are of class \C^{1,1}.
If the initial interface is a periodic graph we show long time existence of the
evolution and convergence to a minimal constrained hypersurface
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
Volume-constrained minimizers for the prescribed curvature problem in periodic media
We establish existence of compact minimizers of the prescribed mean curvature
problem with volume constraint in periodic media. As a consequence, we
construct compact approximate solutions to the prescribed mean curvature
equation. We also show convergence after rescaling of the volume-constrained
minimizers towards a suitable Wulff Shape, when the volume tends to infinity.Comment: In this version the statement of Lemma 2.5 has been corrected with
respect to the published versio
The two obstacle problem for the parabolic biharmonic equation
We consider a two obstacle problem for the parabolic biharmonic equation in a
bounded domain. We prove long time existence of solutions via an implicit time
discretization scheme, and we investigate the regularity properties of
solutions.Comment: 20 page
A note on non lower semicontinuous perimeter functionals on partitions
We consider isotropic non lower semicontinuous weighted perimeter functionals
defined on partitions of domains in . Besides identifying a
condition on the structure of the domain which ensures the existence of
minimizing configurations, we describe the structure of such minima, as well as
their regularity
Crystalline Evolutions in Chessboard-like Microstructures
We describe the macroscopic behavior of evolutions by crystalline curvature
of planar sets in a chessboard--like medium, modeled by a periodic forcing
term. We show that the underlying microstructure may produce both pinning and
confinement effects on the geometric motion.Comment: 17 pages, 10 figures. arXiv admin note: text overlap with
arXiv:1707.0334
Existence and uniqueness for planar anisotropic and crystalline curvature flow
We prove short-time existence of \phi-regular solutions to the planar
anisotropic curvature flow, including the crystalline case, with an additional
forcing term possibly unbounded and discontinuous in time, such as for instance
a white noise. We also prove uniqueness of such solutions when the anisotropy
is smooth and elliptic. The main tools are the use of an implicit variational
scheme in order to define the evolution, and the approximation with flows
corresponding to regular anisotropies
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
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