1,721,054 research outputs found
On the minimization of area among chord-convex sets
No full textIn 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
A survey on the existence of isoperimetric sets in the space R<sup>N</sup> with density
No full textThe aim of this survey is to give a precise idea of the recent results on existence of isoperimetric sets in double-struck RN 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 65th birthday, and to Ula, because she will become my wife in few days
On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation
Full text linkThis 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
On the bi-Sobolev planar homeomorphisms and their approximation
No full textThe first goal of this paper is to give a short description of the planar bi-Sobolev homeomorphisms, providing simple and self-contained proofs for some already known properties. In particular, for any such homeomorphism u:Ω→Δ, one has Du(x)=0 for almost every point x for which Ju(x)=0. As a consequence, one can prove that ∫Ω|Du|=∫Δ|Du−1|. Notice that this estimate holds trivially if one is allowed to use the change of variables formula, but this is not always the case for a bi-Sobolev homeomorphism. As a corollary of our construction, we will show that any W1,1homeomorphism u with W1,1inverse can be approximated with smooth diffeomorphisms (or piecewise affine homeomorphisms) unin such a way that unconverges to u in W1,1and, at the same time, un−1converges to u−1in W1,1. This positively answers an open conjecture (see for instance Iwaniec et al. (2011), Question 4) for the case p=1
On the Cheeger sets in strips and non-convex domains
Full text linkIn 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 a recent
result by D. Krejcirik and the second-named author, 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
The sharp quantitative barycentric isoperimetric inequality for bounded sets
No full textWe prove the sharp quantitative isoperimetric inequality in the case of the barycentric asymmetry, for bounded sets. This generalizes the 2-D case recently proved in [2]
Equivalence between some definitions for the optimal mass transport problem and for the transport density on manifolds
No full text
New trends in shape optimization
No full textThis volume reflects “New Trends in Shape Optimization” and is based on a workshop of the same name organized at the Friedrich-Alexander University Erlangen-Nürnberg in September 2013. During the workshop senior mathematicians and young scientists alike presented their latest findings. The format of the meeting allowed fruitful discussions on challenging open problems, and triggered a number of new and spontaneous collaborations. As such, the idea was born to produce this book, each chapter of which was written by a workshop participant, often with a collaborator. The content of the individual chapters ranges from survey papers to original articles; some focus on the topics discussed at the Workshop, while others involve arguments outside its scope but which are no less relevant for the field today. As such, the book offers readers a balanced introduction to the emerging field of shape optimization
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