% Encoding: UTF-8
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@phdthesis{faucris.117325604,
author = {Pratelli, Aldo},
faupublication = {no},
peerreviewed = {automatic},
school = {Friedrich-Alexander-Universität Erlangen-Nürnberg},
title = {{Existence} of optimal transport maps and regularity of the transport density in mass transportation problems},
year = {2003}
}
@article{faucris.115373984,
abstract = {A variational model introduced by Spencer and Tersoff (Appl. Phys. Lett. 96:073114, 2010) to describe optimal faceted shapes of epitaxially deposited films is studied analytically in the case in which there are a non-vanishing crystallographic miscut and a lattice incompatibility between the film and the substrate. The existence of faceted minimizers for every volume of the deposited film is established. In particular, it is shown that there is no wetting effect for small volumes. Geometric properties including a faceted version of the zero contact angle are derived, and the explicit shapes of minimizers for small volumes are identified.},
author = {Fonseca, Irene and Pratelli, Aldo and Zwicknagl, Barbara},
doi = {10.1007/s00205-014-0767-4},
faupublication = {yes},
journal = {Archive for Rational Mechanics and Analysis},
pages = {359-401},
peerreviewed = {Yes},
title = {{Shapes} of {Epitaxially} {Grown} {Quantum} {Dots}},
volume = {214},
year = {2014}
}
@article{faucris.117192284,
abstract = {A quantitative version of the sharp Sobolev inequality in W (ℝ), 1 < p < n, is established with a remainder term involving the distance from the family of extremals. © European Mathematical Society 2009.},
author = {Cianchi, Andrea and Fusco, Nicola and Maggi, Francesco and Pratelli, Aldo},
faupublication = {no},
journal = {Journal of the European Mathematical Society},
pages = {1105-1139},
peerreviewed = {Yes},
title = {{The} sharp {Sobolev} inequality in quantitative form},
url = {https://www.scopus.com/inward/record.url?partnerID=HzOxMe3b&scp=74249085442&origin=inward},
volume = {11},
year = {2009}
}
@article{faucris.109593484,
abstract = {The integration-by-parts methods introduced in this paper improve upon the L estimates on transport densities given in the recent paper by L. De Pascale and A. Pratelli (Calc. Var. Partial Differential Equations 14 (2002) 249-274).},
author = {de Pascale, Luigi and Evans, Lawrence C. and Pratelli, Aldo},
doi = {10.1112/S0024609303003035},
faupublication = {no},
journal = {Bulletin of the London Mathematical Society},
pages = {383-395},
peerreviewed = {Yes},
title = {{Integral} estimates for transport densities},
volume = {36},
year = {2004}
}
@article{faucris.117325384,
author = {Maggi, Francesco and Ponsiglione, Marcello and Pratelli, Aldo},
faupublication = {yes},
journal = {Transactions of the American Mathematical Society},
pages = {1141-1160},
peerreviewed = {Yes},
title = {{Quantitative} stability in the isodiametric inequality via the isoperimetric inequality},
volume = {366},
year = {2013}
}
@article{faucris.117323844,
abstract = {We show that every isoperimetric set in ℝ^{N} with density is bounded if the density is continuous and bounded above and below. This improves the previously known boundedness results, which basically needed a Lipschitz assumption; on the other hand, the present assumption is sharp, as we show with an explicit example. To obtain our result, we observe that the main tool which is often used, namely a classical “ε - ε” property already discussed by Allard, Almgren and Bombieri, admits a weaker counterpart which is still sufficient for the boundedness, namely, an “ε - ε^{β}” version of the property. And in turn, while for the validity of the first property the Lipschitz assumption is essential, for the latter the continuity alone is enough. We conclude by deriving some consequences of our result for the existence and almost-everywhere regularity of isoperimetric sets.},
author = {Cinti, Eleonora and Pratelli, Aldo},
doi = {10.1515/crelle-2014-0120},
faupublication = {yes},
journal = {Journal für die reine und angewandte Mathematik},
peerreviewed = {unknown},
title = {{The} ε−εβ property, the boundedness of isoperimetric sets in {RN} with density, and some applications},
year = {2015}
}
@incollection{faucris.118665844,
abstract = {In this paper we study the problem of minimizing the area for the chord-convex sets of given size, that is, the sets for which each bisecting chord is a segment of length at least 2. This problem has been already studied and solved in the framework of convex sets, though nothing has been said in the non-convex case. We introduce here the relevant concepts and show some first properties.},
author = {Acciaio, Beatrice and Pratelli, Aldo},
booktitle = {New Trends in Shape Optimization},
doi = {10.1007/978-3-319-17563-8_1},
editor = {Günther Leugering, Aldo Pratelli},
faupublication = {yes},
keywords = {Area-minimizing sets Chord convex},
pages = {1-17},
peerreviewed = {unknown},
series = {International Series of Numerical Mathematics},
title = {{On} the minimization of area among chord-convex sets},
volume = {166},
year = {2015}
}
@article{faucris.121364364,
abstract = {We prove the existence of isoperimetric regions in with density under various hypotheses on the growth of the density. Along the way, we prove results on the boundedness of isoperimetric regions.},
author = {Morgan, Frank and Pratelli, Aldo},
doi = {10.1007/s10455-012-9348-7},
faupublication = {no},
journal = {Annals of Global Analysis and Geometry},
keywords = {Isoperimetric sets;R-n with density;Existence of optimal sets;Boundedness of optimal sets},
pages = {331-365},
peerreviewed = {Yes},
title = {{Existence} of isoperimetric regions in with density},
volume = {43},
year = {2013}
}
@article{faucris.121209264,
abstract = {In this paper we prove a result, extending Lemma 8.1 in [2], which allows to proof that a set of segments is the set of the maximal transport rays for a transport problem. This is particularly useful to build non-trivial examples of transport maps, then in particular to provide specific examples (or counter-examples) in mass transportation. We also give some of these examples.},
author = {Pratelli, Aldo},
faupublication = {no},
journal = {Rendiconti Del Seminario Matematico Della Universita Di Padova},
month = {Jan},
pages = {179-201},
peerreviewed = {Yes},
title = {{How} to show that some rays are maximal transport rays in {Monge} problem.},
volume = {113},
year = {2005}
}
@article{faucris.119112224,
abstract = {We prove a sharp quantitative version of the isoperimetric inequality in the space R(n) endowed with the Gaussian measure.},
author = {Cianchi, Andrea and Fusco, Nicola and Maggi, Francesco and Pratelli, Aldo},
doi = {10.1353/ajm.2011.0005},
faupublication = {no},
journal = {American Journal of Mathematics},
pages = {131-186},
peerreviewed = {Yes},
title = {{ON} {THE} {ISOPERIMETRIC} {DEFICIT} {IN} {GAUSS} {SPACE}},
volume = {133},
year = {2011}
}
@article{faucris.119904884,
author = {Pratelli, Aldo},
doi = {10.1016/j.na.2016.07.006},
faupublication = {yes},
journal = {Nonlinear Analysis - Theory Methods & Applications},
keywords = {Bi-Sobolev homeomorphisms; Smooth approximation},
pages = {258-268},
peerreviewed = {Yes},
title = {{On} the bi-{Sobolev} planar homeomorphisms and their approximation},
volume = {154},
year = {2017}
}
@article{faucris.116121324,
abstract = {The classical Sobolev embedding theorem of the space of functions of bounded variation BV(R-n) into L-n' (R-n) is proved in a sharp quantitative form. (c) 2006 Elsevier Inc. All rights reserved.},
author = {Fusco, Nicola and Maggi, Francesco and Pratelli, Aldo},
doi = {10.1016/j.jfa.2006.10.015},
faupublication = {no},
journal = {Journal of Functional Analysis},
keywords = {Sobolev inequality;BV functions;sharp estimate;isoperimetric inequality},
pages = {315-341},
peerreviewed = {Yes},
title = {{The} sharp quantitative {Sobolev} inequality for functions of bounded variation},
volume = {244},
year = {2007}
}
@article{faucris.123895244,
abstract = {Starting from a mass transportation proof of the Brunn-Minkowski inequality on convex sets, we improve the inequality showing a sharp estimate about the stability property of optimal sets. This is based on a Poincare-type trace inequality on convex sets that is also proved in sharp form. (C) 2009 Elsevier Masson SAS. All rights reserved.},
author = {Figalli, Alessio and Maggi, Francesco and Pratelli, Aldo},
doi = {10.1016/j.anihpc.2009.07.004},
faupublication = {no},
journal = {Annales de l'Institut Henri Poincaré - Analyse Non Linéaire},
keywords = {Brunn-Minkowski inequality;Sharp estimates;Stability results},
pages = {2511-2519},
peerreviewed = {Yes},
title = {{A} refined {Brunn}-{Minkowski} inequality for convex sets},
volume = {26},
year = {2009}
}
@article{faucris.123898764,
abstract = {This paper concerns the Monge's transport problem in a general Polish space. We find optimal conditions to establish the equality between the infimum of Monge's problem and the minimum of the Kantorovich's relaxed version of the problem. A preliminary version of the results of this paper is contained in the Ph.D. thesis [A. Pratelli, Existence of optimal transport maps and regularity of the transport density in mass transportation problems, Ph.D. Thesis, Scuola Normale Superiore, Pisa, Italy, 2003. Available on http://cvamt.sns.it/]. (c) 2006 Elsevier Masson SAS. All rights reserved.},
author = {Pratelli, Aldo},
doi = {10.1016/j.anihpb.2005.12.001},
faupublication = {no},
journal = {Annales de l'Institut Henri Poincaré - Probabilités Et Statistiques},
keywords = {Monge problem;optimal transport},
month = {Jan},
pages = {1-13},
peerreviewed = {Yes},
title = {{On} the equality between {Monge}'s infimum and {Kantorovich}'s minimum in optimal mass transportation},
volume = {43},
year = {2007}
}
@article{faucris.108548264,
abstract = {By elementary geometric arguments, correlation inequalities for radially symmetric probability measures are proved in the plane. Precisely, it is shown that the correlation ratio for pairs of width-decreasing sets is minimized within the class of infinite strips. Since open convex sets which are symmetric with respect to the origin turn out to be width-decreasing sets, Pitt's Gaussian correlation inequality (the two-dimensional case of the long-standing Gaussian correlation conjecture) is derived as a corollary, and it is in fact extended to a wide class of radially symmetric measures.},
author = {Figalli, Alessio and Maggi, Francesco and Pratelli, Aldo},
doi = {10.1214/12-AIHP494},
faupublication = {yes},
journal = {Annales de l'Institut Henri Poincaré - Probabilités Et Statistiques},
keywords = {Correlation inequalities;Gaussian correlation conjecture;Radially symmetric measures},
pages = {1-14},
peerreviewed = {Yes},
title = {{A} geometric approach to correlation inequalities in the plane},
volume = {50},
year = {2014}
}
@article{faucris.123902504,
abstract = {In this paper we study the problem of finding an optimal pricing policy for the use of the public transportation network in a given populated area. The transportation network, modeled by a Borel set ∑ ⊂ ℝ of finite length, the densities of the population and of the services (or workplaces), modeled by the respective finite Borel measures φ and φ, and the effective cost A(t) for a citizen to cover a distance t without the use of the transportation network are assumed to be given. The pricing policy to be found is then a cost B(t) to cover a distance t with the use of the transportation network (i.e., the "price of the ticket for a distance t"), and it has to provide an equilibrium between the needs of the population (hence minimizing the total cost of transportation of the population to the services/workplaces) and that of the owner of the transportation network (hence maximizing the total income of the latter). We present a model for such a choice and discuss the existence as well as some qualitative properties of the resulting optimal pricing policies. © 2006 Society for Industrial and Applied Mathematics.},
author = {Buttazzo, Giuseppe and Pratelli, Aldo and Stepanov, Eugene},
doi = {10.1137/040619831},
faupublication = {no},
journal = {SIAM Journal on Optimization},
keywords = {Monge-Kantorovich problem; Nash equilibrium; Optimal pricing; Optimal transportation; Transportation network},
pages = {826-853},
peerreviewed = {Yes},
title = {{Optimal} pricing policies for public transportation networks},
volume = {16},
year = {2006}
}
@article{faucris.122837924,
abstract = {The aim of this survey is to give a precise idea of the recent results on existence of isoperimetric sets in double-struck R^{N} with density. We will mainly focus on the overall ideas, leaving away some technical details of the proofs, which can be found in the cited papers. No previous knowledge on the subject is assumed from the reader. This survey originates from a talk of the author at the conference "New Trends in Nonlinear PDE's" held at the Accademia dei Lincei on November 26th, 2013. I wish to dedicate this paper to Carlo Sbordone, because of his recent 65^{th} birthday, and to Ula, because she will become my wife in few days.},
author = {Pratelli, Aldo},
doi = {10.4171/RLM/696},
faupublication = {yes},
journal = {Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni},
keywords = {Existence of optimal sets; Isoperimetric problem; Perimeter with density},
pages = {99-118},
peerreviewed = {unknown},
title = {{A} survey on the existence of isoperimetric sets in the space {ℝ}
^{{N}} with density},
volume = {26},
year = {2015}
}
@article{faucris.121015884,
abstract = {In this paper we consider three problems, which are related to the classical Monge's optimal mass transport problem and which are known to be equivalent when the ambient space is an open, convex and bounded subset of R-n; to these problems there correspond different definitions of particular measures (often called transport densities), which are also known to be equivalent. Here We will generalize the setting of these problems and the resulting definitions of transport densities to the case of a Riemannian manifold endowed with a finslerian semidistance, and we will see that the equivalences still hold.},
author = {Pratelli, Aldo},
doi = {10.1007/s10231-004-0109-5},
faupublication = {no},
journal = {Annali Di Matematica Pura Ed Applicata},
keywords = {mass transportation;transport density;shape optimization},
pages = {215-238},
peerreviewed = {Yes},
title = {{Equivalence} between some definitions for the optimal mass transport problem and for the transport density on manifolds},
volume = {184},
year = {2005}
}
@article{faucris.122942424,
abstract = {In this paper we study the dimension of some measures which are related to the classical Monge's optimal mass transport problem and are solutions of a scalar shape optimization problem. Moreover in the case of maximal dimension we will study the summability of the associate densities.},
author = {de Pascale, Luigi and Pratelli, Aldo},
faupublication = {no},
journal = {Calculus of Variations and Partial Differential Equations},
pages = {249-274},
peerreviewed = {Yes},
title = {{Regularity} properties for {Monge} {Transport} {Density} and for {Solutions} of some {Shape} {Optimization} {Problem}},
volume = {14},
year = {2002}
}
@article{faucris.116106584,
abstract = {A sharp quantitative version of the anisotropic isoperimetric inequality is established, corresponding to a stability estimate for the Wulff shape of a given surface tension energy. This is achieved by exploiting mass transportation theory, especially Gromov's proof of the isoperimetric inequality and the Brenier-McCann Theorem. A sharp quantitative version of the Brunn-Minkowski inequality for convex sets is proved as a corollary.},
author = {Figalli, Alessio and Maggi, Francesco and Pratelli, Aldo},
doi = {10.1007/s00222-010-0261-z},
faupublication = {no},
journal = {Inventiones Mathematicae},
pages = {167-211},
peerreviewed = {Yes},
title = {{A} mass transportation approach to quantitative isoperimetric inequalities},
volume = {182},
year = {2010}
}
@article{faucris.107071844,
abstract = {We show the existence of optimal transport maps in the case when the cost function is the distance induced by a crystalline norm in ℝ, assuming that the initial distribution of mass is absolutely continuous with respect to ℒ. The proof is based on a careful decomposition of the space in transport rays induced by a secondary variational problem having the Euclidean distance as cost function. Moreover, improving a construction by Larman, we show the existence of a Nikodym set in ℝ having full measure in the unit cube, intersecting each element of a family of pairwise disjoint open lines only in one point. This example can be used to show that the regularity of the decomposition in transport rays plays an essential role in Sudakov-type arguments for proving the existence of optimal transport maps.},
author = {Pratelli, Aldo and Ambrosio, Luigi and Kirchheim, Bernd},
faupublication = {yes},
journal = {Duke Mathematical Journal},
pages = {201-241},
peerreviewed = {Yes},
title = {{Existence} of optimal transport maps for crystalline norms},
url = {https://www.scopus.com/inward/record.url?partnerID=HzOxMe3b&scp=10044256416&origin=inward},
volume = {125},
year = {2004}
}
@article{faucris.114017024,
abstract = {We show that a planar bi-Lipschitz orientation-preserving homeomorphism can be approximated in the W-1,W-P norm, together with its inverse, with an orientation-preserving homeomorphism which is piecewise affine or smooth. (c) 2013 Elsevier Masson SAS. All rights reserved.},
author = {Daneri, Sara and Pratelli, Aldo},
doi = {10.1016/j.anihpc.2013.04.007},
faupublication = {yes},
journal = {Annales de l'Institut Henri Poincaré - Analyse Non Linéaire},
pages = {567-589},
peerreviewed = {Yes},
title = {{Smooth} approximation of bi-{Lipschitz} orientation-preserving homeomorphisms},
volume = {31},
year = {2014}
}
@article{faucris.119159304,
abstract = {Starting from the quantitative isoperimetric inequality, we prove a sharp quantitative version of the Cheeger inequality.},
author = {Figalli, Alessio and Maggi, Francesco and Pratelli, Aldo},
doi = {10.1090/S0002-9939-09-09795-0},
faupublication = {no},
journal = {Proceedings of the American Mathematical Society},
month = {Jan},
pages = {2057-2062},
peerreviewed = {Yes},
title = {{A} {NOTE} {ON} {CHEEGER} {SETS}},
volume = {137},
year = {2009}
}
@article{faucris.121444444,
abstract = {We prove the Lipschitz regularity of the eigenfunctions u(1), ... , u(p) of the Dirichlet Laplacian on the optimal set Omega* and, as a corollary, we deduce that Omega* is open. For functionals depending only on a generic subset of the spectrum, as for example lambda(k)(Omega), our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved.},
author = {Bucur, Dorin and Mazzoleni, Dario and Pratelli, Aldo and Velichkov, Bozhidar},
doi = {10.1007/s00205-014-0801-6},
faupublication = {yes},
journal = {Archive for Rational Mechanics and Analysis},
pages = {117-151},
peerreviewed = {Yes},
title = {{Lipschitz} {Regularity} of the {Eigenfunctions} on {Optimal} {Domains}},
volume = {216},
year = {2015}
}
@book{faucris.119359504,
author = {Pratelli, Aldo and Leugering, Günter},
doi = {10.1007/978-3-319-17563-8},
faupublication = {yes},
isbn = {978-3-319-17563-8},
month = {Jan},
pages = {VII-VIII},
peerreviewed = {unknown},
publisher = {Springer International Publishing},
series = {International Series of Numerical Mathematics},
title = {{New} {Trends} in {Shape} {Optimization}},
url = {http://link.springer.com/book/10.1007/978-3-319-17563-8},
volume = {166},
year = {2015}
}
@article{faucris.115768884,
abstract = {We study the Cheeger constant and Cheeger set for domains obtained as strip-like neighborhoods of curves in the plane. If the reference curve is complete and finite (a "curved annulus"), then the strip itself is a Cheeger set and the Cheeger constant equals the inverse of the half-width of the strip. The latter holds true for unbounded strips as well, but there is no Cheeger set. Finally, for strips about noncomplete finite curves, we derive lower and upper bounds to the Cheeger set, which become sharp for infinite curves. The paper is concluded by numerical results for circular sectors.},
author = {Krejcirik, David and Pratelli, Aldo},
faupublication = {no},
journal = {Pacific Journal of Mathematics},
keywords = {Cheeger sets;Cheeger constant;curved strips},
pages = {309-333},
peerreviewed = {Yes},
title = {{THE} {CHEEGER} {CONSTANT} {OF} {CURVED} {STRIPS}},
volume = {254},
year = {2011}
}
@article{faucris.122838584,
abstract = {In this paper we consider the Cheeger problem for non-convex domains, with a particular interest in the case of planar strips, which have been extensively studied in recent years. Our main results are an estimate on the Cheeger constant of strips, which is stronger than the previous one known from Krejčiřík and Pratelli (Pac J Math 254(2):309–333, 2011), and the proof that strips share with convex domains a number of crucial properties with respect to the Cheeger problem. Moreover, we present several counterexamples showing that the same properties are not valid for generic non-convex domains.},
author = {Leonardi, Gian Paolo and Pratelli, Aldo},
doi = {10.1007/s00526-016-0953-3},
faupublication = {yes},
journal = {Calculus of Variations and Partial Differential Equations},
keywords = {Cheeger sets; Isoperimetric; Prescribed mean curvature},
pages = {1-28},
peerreviewed = {Yes},
title = {{On} the {Cheeger} sets in strips and non-convex domains},
volume = {55},
year = {2016}
}
@article{faucris.115626104,
abstract = {It is well-known that duality in the Monge-Kantorovich transport problem holds true provided that the cost function c : X x Y -> [0, a] is lower semi-continuous or finitely valued, but it may fail otherwise. We present a suitable notion of rectification c (r) of the cost c, so that the Monge-Kantorovich duality holds true replacing c by c (r) . In particular, passing from c to c (r) only changes the value of the primal Monge-Kantorovich problem. Finally, the rectified function c (r) is lower semi-continuous as soon as X and Y are endowed with proper topologies, thus emphasizing the role of lower semi-continuity in the duality-theory of optimal transport.},
author = {Beiglböck, Mathias and Pratelli, Aldo},
doi = {10.1007/s00526-011-0449-0},
faupublication = {no},
journal = {Calculus of Variations and Partial Differential Equations},
pages = {27-41},
peerreviewed = {Yes},
title = {{Duality} for rectified cost functions},
volume = {45},
year = {2012}
}
@article{faucris.115625664,
abstract = {In this work we review two classical isoperimetric inequalities involving eigenvalues of the Laplacian, both with Dirichlet and Neumann boundary conditions. The first one is classically attributed to Krahn and P. Szego and asserts that among sets of given measure, the disjoint union of two balls with the same radius minimizes the second eigenvalue of the Dirichlet-Laplacian, while the second one is due to G. SzegA and Weinberger and deals with the maximization of the first non-trivial eigenvalue of the Neumann-Laplacian. New stability estimates are provided for both of them.},
author = {Brasco, Lorenzo and Pratelli, Aldo},
doi = {10.1007/s00039-012-0148-9},
faupublication = {no},
journal = {Geometric and Functional Analysis},
keywords = {Stability for eigenvalues;Krahn-Szego inequality;Szego-Weinberger inequality;isoperimetric inequalities},
pages = {107-135},
peerreviewed = {Yes},
title = {{Sharp} {Stability} of {Some} {Spectral} {Inequalities}},
volume = {22},
year = {2012}
}
@article{faucris.117190964,
abstract = {We prove that, given a planar bi-Lipschitz map u defined on the boundary of the unit square, it is possible to extend it to a function v of the whole square, in such a way that v is still bi-Lipschitz. In particular, denoting by L and L the bi-Lipschitz constants of u and v, with our construction one has L ≤ CL^{4} (C being an explicit geometric constant). The same result was proved in 1980 by Tukia (see [Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 49-72]), using a completely different argument, but without any estimate on the constant L. In particular, the function v can be taken either smooth or (countably) piecewise affine.},
author = {Pratelli, Aldo and Daneri, Sara},
doi = {10.1515/acv-2012-0013},
faupublication = {yes},
journal = {Advances in Calculus of Variations},
keywords = {Bi-Lipschitz; extension; piecewise affine},
pages = {221-266},
peerreviewed = {unknown},
title = {{A} planar bi-{Lipschitz} extension theorem},
volume = {8},
year = {2015}
}
@article{faucris.122839024,
abstract = {We consider the isoperimetric problem in (Formula presented.) with density. We show that, if the density is (Formula presented.), then the boundary of any isoperimetric set is of class (Formula presented.). This improves the previously known regularity.},
author = {Cinti, Eleonora and Pratelli, Aldo},
doi = {10.1007/s00208-016-1409-y},
faupublication = {yes},
journal = {Mathematische Annalen},
pages = {1-14},
peerreviewed = {Yes},
title = {{Regularity} of isoperimetric sets in {R2} with density},
volume = {365},
year = {2016}
}
@article{faucris.119904664,
author = {De Philippis, Guido and Franzina, Giovanni and Pratelli, Aldo},
doi = {10.1007/s12220-016-9711-1},
faupublication = {yes},
journal = {Journal of Geometric Analysis},
keywords = {Existence of optimal sets; Isoperimetric problem; Perimeter with density},
pages = {1086-1105},
peerreviewed = {Yes},
title = {{Existence} of {Isoperimetric} {Sets} with {Densities} “{Converging} from {Below}” on {RN}},
volume = {27},
year = {2017}
}
@incollection{faucris.107439904,
author = {Ambrosio, Luigi and Pratelli, Aldo},
booktitle = {Optimal Transportation and Applications},
doi = {10.1007/978-3-540-44857-0_5},
editor = {L.A. Caffarelli, S. Salsa},
faupublication = {no},
pages = {123-160},
peerreviewed = {unknown},
publisher = {Springer},
series = {Lecture Notes in Mathematics},
title = {{Existence} and stability results in the {L1} theory of optimal transportation},
year = {2003}
}
@incollection{faucris.117364764,
abstract = {G. Alberti, F. Ancona, G. Crippa, S. Bianchini, C. De Lellis, A. Marson, C. Mascia},
author = {Pratelli, Aldo and Puglisi, Simona},
booktitle = {HCDTE Lecture Notes. Part II. Nonlinear HYperboliC PDEs, Dispersive and Transport Equations},
faupublication = {yes},
isbn = {10: 1-60133-015-4},
pages = {51-127},
peerreviewed = {unknown},
publisher = {American Institute of Mathematical Sciences},
series = {AIMS on Applied Mathematics},
title = {{Elastic} deformations on the plane and approximations},
volume = {7},
year = {2016}
}
@article{faucris.109004764,
abstract = {Given two absolutely continuous probability measures f(+/-) in R-2, we consider the classical Monge transport problem, with the Euclidean distance as cost function. We prove the existence of a continuous optimal transport, under the assumptions that (the densities of) f(+/-) are continuous and strictly AA 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. (c) 2005 Elsevier SAS. All rights reserved.},
author = {Fragala, Ilaria and Gelli, Maria Stella and Pratelli, Aldo},
doi = {10.1016/j.matpur.2005.02.002},
faupublication = {no},
journal = {Journal De Mathematiques Pures Et Appliquees},
keywords = {Monge problem;optimal transport;continuity;maximal transport rays},
pages = {1261-1294},
peerreviewed = {Yes},
title = {{Continuity} of an optimal transport in {Monge} problem},
volume = {84},
year = {2005}
}
@masterthesis{faucris.123618924,
author = {Pratelli, Aldo},
faupublication = {no},
peerreviewed = {automatic},
school = {Friedrich-Alexander-Universität Erlangen-Nürnberg},
title = {{Regolarità} delle misure ottimali nel problema di trasporto di {Monge}--{Kantorovich}},
year = {2000}
}
@article{faucris.114276844,
abstract = {We prove that any countably piecewise affine homeomorphism from an open set of R-2 can be approximated, together with its inverse, by diffeomorphisms in the W-1,W-p and the L-infinity norms.},
author = {Mora-Corral, Carlos and Pratelli, Aldo},
doi = {10.1007/s12220-012-9378-1},
faupublication = {yes},
journal = {Journal of Geometric Analysis},
keywords = {Approximation of homeomorphisms;Piecewise affine homeomorphisms;Diffeomorphisms},
pages = {1398-1424},
peerreviewed = {Yes},
title = {{Approximation} of {Piecewise} {Affine} {Homeomorphisms} by {Diffeomorphisms}},
volume = {24},
year = {2014}
}
@article{faucris.115754804,
abstract = {Denoting by S the sharp constant in the Sobolev inequality in W(0)(1,2) (B), being B the unit ball in R(3), and denoting by S(h) its approximation in a suitable finite element space, we show that S(h) converges to S as h SE arrow 0 with a polynomial rate of convergence. We provide both an upper and a lower bound on the rate of convergence, and present some numerical results.},
author = {Antonietti, Paola and Pratelli, Aldo},
doi = {10.1007/s00211-010-0347-y},
faupublication = {no},
journal = {Numerische Mathematik},
month = {Jan},
pages = {37-64},
peerreviewed = {Yes},
title = {{Finite} element approximation of the {Sobolev} constant},
volume = {117},
year = {2011}
}
@article{faucris.211503128,
abstract = {We show that a planar BV homeomorphism can be approximated in the area strict sense, together with its inverse, with smooth or piecewise affine homeomorphisms. (C) 2018 Elsevier Inc. All rights reserved.},
author = {Pratelli, Aldo and Radici, Emanuela},
doi = {10.1016/j.jfa.2018.10.022},
faupublication = {yes},
journal = {Journal of Functional Analysis},
note = {CRIS-Team WoS Importer:2019-02-21},
pages = {659-686},
peerreviewed = {Yes},
title = {{Approximation} of planar {BV} homeomorphisms by diffeomorphisms},
volume = {276},
year = {2019}
}
@article{faucris.108865504,
author = {Figalli, Alessio and Maggi, Francesco and Pratelli, Aldo},
doi = {10.1016/j.aim.2013.04.007},
faupublication = {yes},
journal = {Advances in Mathematics},
pages = {80-101},
peerreviewed = {Yes},
title = {{Sharp} stability theorems for the anisotropic {Sobolev} and log-{Sobolev} inequalities on functions of bounded variation},
volume = {242},
year = {2013}
}
@misc{faucris.123065624,
abstract = {In this paper we deal with the task of uniformly approximating an L-biLipschitz curve by means of piecewise linear ones. This is rather simple if one is satisfied to have approximating functions which are L-biLipschitz, for instance this was already done with L=4L in [Daneri-Pratelli, Lemma 5.5]. The main result of this paper is to do the same with L=L+ε (which is of course the best possible result); in the end, we generalize the result to the case of closed curves.},
author = {Pratelli, Aldo and Radici, Emanuela},
faupublication = {yes},
peerreviewed = {automatic},
title = {{On} the piecewise approximation of bi {Lipschitz} curves},
url = {http://arxiv.org/abs/1505.06510},
year = {2015}
}
@article{faucris.115767124,
abstract = {The isoperimetric problem with respect to the product-type density e-vertical bar x vertical bar/2 dx dy on the Euclidean space R(h) x R(k) is studied. In particular, existence, symmetry and regularity of minimizers is proved. In the special case k = 1, also the shape of all the minimizers is derived. Finally, a conjecture about the minimality of large cylinders in the case k > 1 is formulated. (C) 2011 Elsevier Inc. All rights reserved.},
author = {Fusco, Nicola and Maggi, Francesco and Pratelli, Aldo},
doi = {10.1016/j.jfa.2011.01.007},
faupublication = {no},
journal = {Journal of Functional Analysis},
keywords = {Isoperimetric problem;Gaussian density;Existence of minimizers},
pages = {3678-3717},
peerreviewed = {Yes},
title = {{On} the isoperimetric problem with respect to a mixed {Euclidean}-{Gaussian} density},
volume = {260},
year = {2011}
}
@article{faucris.116188204,
abstract = {We consider a mass transport problem among Polish spaces, and we show that a transport plan concentrated in a c-cyclical monotone set is optimal if the cost function is continuous and possibly +infinity valued; the same result is proved for a generic l.s.c. cost function in the case when the measures are purely atomic. This generalizes the previously known results.},
author = {Pratelli, Aldo},
doi = {10.1007/s00209-007-0191-7},
faupublication = {no},
journal = {Mathematische Zeitschrift},
pages = {677-690},
peerreviewed = {Yes},
title = {{On} the sufficiency of c-cyclical monotonicity for optimality of transport plans},
volume = {258},
year = {2008}
}
@article{faucris.119114204,
abstract = {In 1938 Herman Auerbach published a paper where he showed a deep connection between the solutions of the Ulam problem of floating bodies and a class of sets studied by Zindler, which are the planar sets whose bisecting chords all have the same length. In the same paper he conjectured that among Zindler sets the one with minimal area, as well as with maximal perimeter, is the so-called "Auerbach triangle". We prove this conjecture.},
author = {Fusco, Nicola and Pratelli, Aldo},
doi = {10.4171/JEMS/290},
faupublication = {no},
journal = {Journal of the European Mathematical Society},
month = {Jan},
pages = {1633-1676},
peerreviewed = {Yes},
title = {{On} a conjecture by {Auerbach}},
volume = {13},
year = {2011}
}
@article{faucris.115305784,
author = {Mazzoleni, Dario and Pratelli, Aldo},
doi = {10.1016/j.matpur.2013.01.008},
faupublication = {yes},
journal = {Journal De Mathematiques Pures Et Appliquees},
month = {Jan},
pages = {433-453},
peerreviewed = {Yes},
title = {{Existence} of minimizers for spectral problems},
volume = {100},
year = {2013}
}
@article{faucris.116115164,
abstract = {Consider a distribution of citizens in an urban area in which some services (supermarkets, post offices...) are present. Each citizen, in order to use a service, spends an amount of time which is due both to the travel time to the service and to the queue time waiting in the service. The choice of the service to be used is made by every citizen in order to be served more quickly. Two types of problems can be considered: a global optimization of the total time spent by the citizens of the whole city (we define a global optimum and we study it with techniques from optimal mass transportation) and an individual optimization, in which each citizen chooses the service trying to minimize just his own time expense (we define the concept of equilibrium and we study it with techniques from game theory). In this framework we are also able to exhibit two time-dependent strategies (based on the notions of prudence and memory respectively) which converge to the equilibrium.},
author = {Crippa, Gianluca and Jimenez, Chloe and Pratelli, Aldo},
doi = {10.1515/ACV.2009.009},
faupublication = {no},
journal = {Advances in Calculus of Variations},
keywords = {Optimal mass transportation;game theory;Nash equilibrium;Pareto optimum;volutionary strategies},
pages = {207-246},
peerreviewed = {Yes},
title = {{Optimum} and equilibrium in a transport problem with queue penalization effect},
volume = {2},
year = {2009}
}
@book{faucris.122791064,
author = {Buttazzo, Giuseppe and Pratelli, Aldo and Solimini, Sergio and Stepanov, Eugene},
doi = {10.1007/978-3-540-85799-0},
faupublication = {no},
isbn = {9783540857983},
pages = {1-50},
peerreviewed = {unknown},
series = {Springer Lecture Notes in Mathematics},
title = {{Optimal} urban networks via mass transportation},
volume = {1961},
year = {2008}
}
@article{faucris.123469324,
abstract = {Using some results proved in De Pascale and Pratelli [Calc. Var. Partial Differ. Equ. 14 (2002) 249-274] (and De Pascale et al. [Bull. London Math. Soc. 36 (2004) 383-395]) and a suitable interpolation technique, we show that the transport density relative to an L source is also an L function for any 1 ≤ p ≤ +∞.},
author = {de Pascale, Luigi and Pratelli, Aldo},
doi = {10.1051/cocv:2004019},
faupublication = {no},
journal = {Esaim-Control Optimisation and Calculus of Variations},
keywords = {Interpolation; Summability; Transport density},
pages = {549-552},
peerreviewed = {Yes},
title = {{Sharp} summability for monge transport density via interpolation},
year = {2004}
}
@article{faucris.108969784,
abstract = {The first eigenvalue of the p-Laplacian on an open set of given measure attains its minimum value if and only if the set is a ball. We provide a quantitative version of this statement by an argument that can be easily adapted to treat also certain isocapacitary and Cheeger inequalities.},
author = {Fusco, Nicola and Maggi, Francesco and Pratelli, Aldo},
faupublication = {no},
journal = {Annali della Scuola Normale Superiore di Pisa-Classe di Scienze},
month = {Jan},
pages = {51-71},
peerreviewed = {Yes},
title = {{Stability} estimates for certain {Faber}-{Krahn}, isocapacitary and {Cheeger} inequalities},
volume = {8},
year = {2009}
}
@article{faucris.114591004,
abstract = {Denoting by the set of the pairs for all the open sets with unit measure, and by the union of two disjoint balls of half measure, we give an elementary proof of the fact that has horizontal tangent at its lowest point .},
author = {Brasco, Lorenzo and Nitsch, Carlo and Pratelli, Aldo},
doi = {10.1007/s00033-012-0250-8},
faupublication = {no},
journal = {Zeitschrift für Angewandte Mathematik und Physik},
keywords = {Dirichlet-Laplacian spectrum;Shape optimization},
pages = {591-597},
peerreviewed = {Yes},
title = {{On} the boundary of the attainable set of the dirichlet spectrum},
volume = {64},
year = {2013}
}
@article{faucris.116118684,
abstract = {A quantitative sharp form of the classical isoperimetric inequality is proved, thus giving a positive answer to a conjecture by Hall.},
author = {Fusco, Nicola and Maggi, Francesco and Pratelli, Aldo},
doi = {10.4007/annals.2008.168.941},
faupublication = {no},
journal = {Annals of Mathematics},
pages = {941-980},
peerreviewed = {Yes},
title = {{The} sharp quantitative isoperimetric inequality},
volume = {168},
year = {2008}
}