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On a class of variational integrals over BV varieties

Primo Brandi, Anna Salvadori (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We present here our most recent results ([1def]) about the definition of non-linear Weiertrass-type integrals over BV varieties, possibly discontinuous and not necessarily Sobolev's.

On a conjecture by Auerbach

Nicola Fusco, A. Pratelli (2011)

Journal of the European Mathematical Society

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.

On a magnetic characterization of spectral minimal partitions

Bernard Helffer, Thomas Hoffmann-Ostenhof (2013)

Journal of the European Mathematical Society

Given a bounded open set Ω in n (or in a Riemannian manifold) and a partition of Ω by k open sets D j , we consider the quantity 𝚖𝚊𝚡 j λ ( D j ) where λ ( D j ) is the ground state energy of the Dirichlet realization of the Laplacian in D j . If we denote by k ( Ω ) the infimum over all the k -partitions of 𝚖𝚊𝚡 j λ ( D j ) , a minimal k -partition is then a partition which realizes the infimum. When k = 2 , we find the two nodal domains of a second eigenfunction, but the analysis of higher k ’s is non trivial and quite interesting. In this paper, we give...

On a variant of Korn’s inequality arising in statistical mechanics

L. Desvillettes, Cédric Villani (2002)

ESAIM: Control, Optimisation and Calculus of Variations

We state and prove a Korn-like inequality for a vector field in a bounded open set of N , satisfying a tangency boundary condition. This inequality, which is crucial in our study of the trend towards equilibrium for dilute gases, holds true if and only if the domain is not axisymmetric. We give quantitative, explicit estimates on how the departure from axisymmetry affects the constants; a Monge–Kantorovich minimization problem naturally arises in this process. Variants in the axisymmetric case are...

On a variant of Korn's inequality arising in statistical mechanics

L. Desvillettes, Cédric Villani (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We state and prove a Korn-like inequality for a vector field in a bounded open set of N , satisfying a tangency boundary condition. This inequality, which is crucial in our study of the trend towards equilibrium for dilute gases, holds true if and only if the domain is not axisymmetric. We give quantitative, explicit estimates on how the departure from axisymmetry affects the constants; a Monge–Kantorovich minimization problem naturally arises in this process. Variants in the axisymmetric case...

On an optimal shape design problem in conduction

José Carlos Bellido (2006)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we analyze a typical shape optimization problem in two-dimensional conductivity. We study relaxation for this problem itself. We also analyze the question of the approximation of this problem by the two-phase optimal design problems obtained when we fill out the holes that we want to design in the original problem by a very poor conductor, that we make to converge to zero.

On Conditions for Unrectifiability of a Metric Space

Piotr Hajłasz, Soheil Malekzadeh (2015)

Analysis and Geometry in Metric Spaces

We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.

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