1,721,003 research outputs found
Multiphase free discontinuity problems: Monotonicity formula and regularity results
No full textThe purpose of this paper is to analyze regularity properties of local solutions to free discontinuity problems characterized by the presence of multiple phases. The key feature of the problem is related to the way in which two neighboring phases interact: the contact is penalized at jump points, while no cost is assigned to no-jump interfaces which may occur at the zero level of the corresponding state functions. Our main results state that the phases are open and the jump set (globally considered for all the phases) is essentially closed and Ahlfors regular. The proof relies on a multiphase monotonicity formula and on a sharp collective Sobolev extension result for functions with disjoint supports on a sphere, which may be of independent interest
The multiphase mumford-shah problem
No full textWe perform a rigorous analysis of the multiphase version of the Mumford--Shah functional. A characteristic property of the formulation is the presence of a true partition of the image (so in two dimensions of closed contours), each cell of the partition possibly containing inner jumps. The nontrivial partitioning naturally occurs as a consequence of the presence in the energy functional of statistical terms or of phase dependent weights. In particular, we prove a multiphase version of the De Giorgi--Carriero--Leaci result
Topological equivalence of some variational problems involving distances
No full textTo every distance d on a given open set \Omega\subseteq\mathbb R^n, we may associate several kinds of variational problems. We show that, on the class of all geodesic distances d on \Omega which are bounded from above and from below by fixed multiples of the Euclidean one, the uniform convergence on compact sets turns out to be equivalent to the \Gamma-convergence of each of the corresponding variational problems under consideration
On a long-standing conjecture by Polya-Szego and related topics
No full textThe electrostatic capacity of a convex body is usually not simple
to compute. Almost one century ago, Aichi-Russell
suggested a simple approximate formula which
involves the 2-dimensional measure of the boundary of the convex
body. This approximation is estimated by what physicists usually
call ``shape factor'', roughly speaking the ratio between the
capacity and the approximate capacity. We discuss a long-standing
conjecture by P\'olya-Szeg\"o. It says that the
minimum of the ratio is attained by the 2-dimensional disk
Some estimates of the torsional rigidity of heterogeneous rods
No full textA well-known problem in elasticity consists in placing two
linearly elastic materials (of different shear moduli) in a given
plane domain , so as to maximize the torsional rigidity of
the resulting rod; moreover, the proportion of these materials is
prescribed. Such a problem may not have a classical solution as
the optimal design may contain homogenization regions, where the
two materials are mixed in a microscopic scale. Then, the optimal
torsional rigidity becomes difficult to compute. In this paper we
give some different theoretical upper and lower bounds for the
optimal torsional rigidity, and we compare them on some
significant domains
On a geometric combination of functions related to Prekopa-Leindler inequality
Full text linkWe introduce a new operation between nonnegative integrable functions on Rn\mathbb {R}<^>n, that we call geometric combination; it is obtained via a mass transportation approach, playing with inverse distribution functions. The main feature of this operation is that the Lebesgue integral of the geometric combination equals the geometric mean of the two separate integrals; as a natural consequence, we derive a new functional inequality of Prekopa-Leindler type. When applied to the characteristic functions of two measurable sets, their geometric combination provides a set whose volume equals the geometric mean of the two separate volumes. In the framework of convex bodies, by comparing the geometric combination with the 0-sum, we get an alternative proof of the log-Brunn-Minkowski inequality for unconditional convex bodies and for convex bodies with n symmetries
Elliptic approximations to prescribed mean curvature surfaces in Finsler geometry
No full textWe approximate a hypersurface Sigma with prescribed anisotropic mean curvature with solutions u(epsilon), of suitable nonlinear elliptic equations depending on a small parameter epsilon > O. We work in relative geometry, by endowing R-N with a Finsler norm phi describing the anisotropy. The main result states that Sigma and (u(epsilon) = 0) are close of order epsilon(2)/log epsilon/(2), and this estimate is optimal. This is obtained for two different elliptic equations by sub- and supersolutions technique, under smoothness and nondegeneracy assumptions on Sigma. Basic steps are: (i) an explicit computation of the second variation of the phi-Minkowski content along geodesics; (ii) the definition of a Laplace-Beltrami operator on Sigma; (iii) the expansion of the phi-mean curvature of Sigma in a suitable tubular neighbourhood
Continuity of an optimal transport in Monge Problem
No full textGiven two absolutely continuous probability measures f ± in R2 , we consider the classical Monge
transport problem, with the Euclidean distance as cost function. We prove the existence of a contin-
uous optimal transport, under the assumptions that (the densities of) f ± are continuous and strictly
positive in the interior part of their supports, and that such supports are convex, compact, and disjoint.
We show through several examples that our statement is nearly optimal. Moreover, under the same
hypotheses, we also obtain the continuity of the transport density
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