145 research outputs found
Global calibrations for the non-homogeneous Mumford-Shah functional
Using a calibration method we prove that, if is a
closed regular hypersurface and if the function is discontinuous along
and regular outside, then the function which solves is in turn discontinuous along
and it is the unique absolute minimizer of the non-homogeneous
Mumford-Shah functional over ,
for large enough. Applications of the result to the study of the
gradient flow by the method of minimizing movements are shown.Comment: 33 page
Cascade of minimizers for a nonlocal isoperimetric problem in thin domains
For \Omega_\e=(0,\e)\times (0,1) a thin rectangle, we consider minimization
of the two-dimensional nonlocal isoperimetric problem given by \inf_u
E^{\gamma}_{\Omega_\e}(u) where E^{\gamma}_{\Omega_\e}(u):=
P_{\Omega_\e}(\{u(x)=1\})+\gamma\int_{\Omega_\e}\abs{\nabla{v}}^2\,dx and
the minimization is taken over competitors u\in BV(\Omega_\e;\{\pm 1\})
satisfying a mass constraint \fint_{\Omega_\e}u=m for some .
Here P_{\Omega_\e}(\{u(x)=1\}) denotes the perimeter of the set
in \Omega_\e, \fint denotes the integral average and denotes the
solution to the Poisson problem -\Delta
v=u-m\;\mbox{in}\;\Omega_\e,\quad\nabla v\cdot
n_{\partial\Omega_\e}=0\;\mbox{on}\;\partial\Omega_\e,\quad\int_{\Omega_\e}v=0.
We show that a striped pattern is the minimizer for \e\ll 1 with the number
of stripes growing like as We then present
generalizations of this result to higher dimensions.Comment: 20 pages, 2 figure
Functionals depending on curvatures with constraints
We deal with a family of functionals depending on curvatures and we prove for
them compactness and semicontinuity properties in the class of closed and
bounded sets which satisfy a uniform exterior and interior sphere condition. We
apply the results to state an existence theorem for the Nitzberg and Mumford
problem under this additional constraint.Comment: 20 pages. To appear on Rendiconti del Seminario Matematico
dell'Universita' di Padov
Local calibrations for minimizers of the Mumford-Shah functional with a regular discontinuity set
Using a calibration method, we prove that, if w is a function which satisfies
all Euler conditions for the Mumford-Shah functional on a two-dimensional
domain, and the discontinuity set S of w is a regular curve connecting two
boundary points, then there exists a uniform neighbourhood U of S such that w
is a minimizer of the Mumford-Shah functional on U with respect to its own
boundary conditions. We show that Euler conditions do not guarantee in general
the minimality of w in the class of functions with the same boundary value of w
and whose extended graph is contained in a neighbourhood of the extended graph
of w, and we give a sufficient condition in terms of the geometrical properties
of the domain and the discontinuity set under which this kind of minimality
holds.Comment: 31 pages, 2 figure
Nonlocal curvature flows
This paper aims at building a unified framework to deal with a wide class of
local and nonlocal translation-invariant geometric flows. First, we introduce a
class of generalized curvatures, and prove the existence and uniqueness for the
level set formulation of the corresponding geometric flows.
We then introduce a class of generalized perimeters, whose first variation is
an admissible generalized curvature. Within this class, we implement a
minimizing movements scheme and we prove that it approximates the viscosity
solution of the corresponding level set PDE.
We also describe several examples and applications. Besides recovering and
presenting in a unified way existence, uniqueness, and approximation results
for several geometric motions already studied and scattered in the literature,
the theory developed in this paper allows us to establish also new results
Minimality via second variation for a nonlocal isoperimetric problem
We discuss the local minimality of certain configurations for a nonlocal
isoperimetric problem used to model microphase separation in diblock copolymer
melts. We show that critical configurations with positive second variation are
local minimizers of the nonlocal area functional and, in fact, satisfy a
quantitative isoperimetric inequality with respect to sets that are
-close. The link with local minimizers for the diffuse-interface
Ohta-Kawasaki energy is also discussed. As a byproduct of the quantitative
estimate, we get new results concerning periodic local minimizers of the area
functional and a proof, via second variation, of the sharp quantitative
isoperimetric inequality in the standard Euclidean case. As a further
application, we address the global and local minimality of certain lamellar
configurations.Comment: 35 page
Local calibrations for minimizers of the Mumford-Shah functional with rectilinear discontinuity sets
Using a calibration method, we prove that, if is a function which
satisfies all Euler conditions for the Mumford-Shah functional on a
two-dimensional open set , and the discontinuity set of is a
segment connecting two boundary points, then for every point of
there exists a neighbourhood of such that is a
minimizer of the Mumford-Shah functional on with respect to its own
boundary values on .Comment: 22 pages, 4 figure
Motion of three-dimensional elastic films by anisotropic surface diffusion with curvature regularization
Short time existence for a surface diffusion evolution equation with
curvature regularization is proved in the context of epitaxially strained
three-dimensional films. This is achieved by implementing a minimizing movement
scheme, which is hinged on the -gradient flow structure underpinning
the evolution law. Long-time behavior and Liapunov stability in the case of
initial data close to a flat configuration are also addressed.Comment: 44 page
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