1,721,027 research outputs found
Regularity of optimal transport maps and applications
Get PDFIn this thesis, we study the regularity of optimal transport maps and its applications to the semi-geostrophic system. The first two chapters survey the known theory, in particular there is a self-contained proof of Brenier’ theorem on existence of optimal transport maps and of Caffarelli’s Theorem on Holder continuity of optimal maps. In the third and fourth chapter we start investigating Sobolev regularity of optimal transport maps, while in Chapter 5 we show how the above mentioned results allows to prove the existence of Eulerian solution to the semi-geostrophic equation. In Chapter 6 we prove partial regularity of optimal maps with respect to a generic cost functions (it is well known that in this case global regularity can not be expected). More precisely we show that if the target and source measure have smooth densities the optimal map is always smooth outside a closed set of measure zero
Rigidity and stability of Caffarelli's log-concave perturbation theorem
No full textIn this note we establish some rigidity and stability results for Caffarelli's log-concave perturbation theorem. As an application we show that if a 1-log-concave measure has almost the same Poincaré constant as the Gaussian measure, then it almost splits off a Gaussian factor
Higher Integrability for Minimizers of the Mumford-Shah Functional
No full textWe prove higher integrability for the gradient of local minimizers of the Mumford-Shah energy functional, providing a positive answer to a conjecture of De Giorgi (Free discontinuity problems in calculus of variations. Frontiers in pure and applied mathematics, North-Holland, Amsterdam, pp 55-62, 1991). © 2014 Springer-Verlag Berlin Heidelberg
From volume cone to metric cone in the nonsmooth setting
No full textWe prove that ‘volume cone implies metric cone’ in the setting of RCD spaces, thus generalising to this class of spaces a well known result of Cheeger–Colding valid in Ricci-limit spaces. © 2016, Springer International Publishing
A Note on Petty's Theorem
No full textIn this short note we show how, by exploiting the regularity theory for solutions to the Monge-Ampère equation, Petty’s equation characterizes ellipsoids without assuming any a priori regularity assumption. © 2014, Tokyo Institute of Technology. All rights reserved
W^{2,1}- regularity for solutions of the Monge-Ampere equation
No full textIn this paper we prove that a strictly convex Alexandrov solution u of the Monge-Amp\`ere equation, with right hand side bounded away from zero and infinity, is W2,1loc. This is obtained by showing higher integrability a-priori estimates for D2u, namely D2u∈LlogkL for any k∈N
Weak notions of jacobian determinant and relaxation
Get PDFIn this paper we study two weak notions of Jacobian determinant for Sobolev maps, namely the distributional Jacobian and the relaxed total variation, which in general could be different. We show some cases of equality and use them to give an explicit expression for the relaxation of some polyconvex functionals
Sharp stability inequalities for the Plateau problem
No full textThe validity of global quadratic stability inequalities for uniquely regular area minimizing hypersurfaces is proved to be equivalent to the uniform positivity of the second variation of the area. Concerning singular area minimizing hypersurfaces, by a "quantitative calibration" argument we prove quadratic stability inequalities with explicit constants for all the Lawson's cones, excluding six exceptional cases. As a by-product of these results, explicit lower bounds for the first eigenvalues of the second variation of the area on these cones are derived
A Note on the Dimension of the Singular Set In Free Interface Problems
No full textThe aim of this note is to investigate the size of the singular set of a general class of free interface problems. We show porosity of the singular set, obtaining as a corollary that both its Hausdorff and Minkowski dimensions are strictly smaller than n−1
On the two-state problem for general differential operators
Get PDFIn this note we generalize the Ball-James rigidity theorem for gradient differential inclusions to the setting of a general linear differential constraint. In particular, we prove the rigidity for approximate solutions to the two-state inclusion with incompatible states for merely -bounded sequences. In this way, our theorem can be seen as a result of compensated compactness in the linear-growth setting
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