1,296 research outputs found
Local calibrations for minimizers of the Mumford-Shah functional with a triple junction
We prove that, if u is a function satisfying all Euler conditions for the
Mumford-Shah functional and the discontinuity set of u is given by three line
segments meeting at the origin with equal angles, then there exists a
neighbourhood U of the origin such that u is a minimizer of the Mumford-Shah
functional on U with respect to its own boundary conditions on the boundary of
U. The proof is obtained by using the calibration method.Comment: 28 pages, 4 figure
Discrete stochastic approximations of the Mumford-Shah functional
We propose a -convergent discrete approximation of the Mumford-Shah
functional. The discrete functionals act on functions defined on stationary
stochastic lattices and take into account general finite differences through a
non-convex potential. In this setting the geometry of the lattice strongly
influences the anisotropy of the limit functional. Thus we can use
statistically isotropic lattices and stochastic homogenization techniques to
approximate the vectorial Mumford-Shah functional in any dimension.Comment: 47 pages, reorganized versio
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
A second order local minimality criterion for the triple junction singularity of the Mumford-Shah functional
This paper is the first part of an ongoing project aimed at providing a local
minimality criterion, based on a second variation approach, for the triple
point configurations of the Mumford-Shah functional
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
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