A metric approach to a class of doubly nonlinear evolution equations and applications

Riccarda Rossi; Alexander Mielke; Giuseppe Savaré

Displaying similar documents to “A metric approach to a class of doubly nonlinear evolution equations and applications”

Scaling laws for non-euclidean plates and the $W^{2, 2}$ isometric immersions of riemannian metrics

Marta Lewicka, Mohammad Reza Pakzad (2011)

ESAIM: Control, Optimisation and Calculus of Variations

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Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its -convergence under...

A generalization of boundedly compact metric spaces

Gerald Beer, Anna Di Concilio (1991)

Commentationes Mathematicae Universitatis Carolinae

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A metric space $〈 X, d 〉$ is called a $UC$ space provided each continuous function on $X$ into a metric target space is uniformly continuous. We introduce a class of metric spaces that play, relative to the boundedly compact metric spaces, the same role that $UC$ spaces play relative to the compact metric spaces.

The Banach contraction mapping principle and cohomology

Ludvík Janoš (2000)

Commentationes Mathematicae Universitatis Carolinae

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By a dynamical system $(X, T)$ we mean the action of the semigroup $(ℤ^{+}, +)$ on a metrizable topological space $X$ induced by a continuous selfmap $T : X \to X$ . Let $M (X)$ denote the set of all compatible metrics on the space $X$ . Our main objective is to show that a selfmap $T$ of a compact space $X$ is a Banach contraction relative to some $d_{1} \in M (X)$ if and only if there exists some $d_{2} \in M (X)$ which, regarded as a $1$ -cocycle of the system $(X, T) \times (X, T)$ , is a coboundary.

On the metric reflection of a pseudometric space in ZF

Horst Herrlich, Kyriakos Keremedis (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show: (i) The countable axiom of choice $𝐂𝐀𝐂$ is equivalent to each one of the statements: (a) a pseudometric space is sequentially compact iff its metric reflection is sequentially compact, (b) a pseudometric space is complete iff its metric reflection is complete. (ii) The countable multiple choice axiom $𝐂𝐌𝐂$ is equivalent to the statement: (a) a pseudometric space is Weierstrass-compact iff its metric reflection is Weierstrass-compact. (iii) The axiom of choice $𝐀𝐂$ is equivalent to each...

Fractal star bodies

Irmina Herburt, Maria Moszyńska, Dorette Pronk (2009)

Banach Center Publications

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In 1989 R. Arnold proved that for every pair (A,B) of compact convex subsets of ℝ there is an Euclidean isometry optimal with respect to L₂ metric and if f₀ is such an isometry, then the Steiner points of f₀(A) and B coincide. In the present paper we solve related problems for metrics topologically equivalent to the Hausdorff metric, in particular for $L_{p}$ metrics for all p ≥ 2 and the symmetric difference metric.

Metric enrichment, finite generation, and the path coreflection

Alexandru Chirvasitu (2024)

Archivum Mathematicum

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We prove a number of results involving categories enriched over CMet, the category of complete metric spaces with possibly infinite distances. The category CPMet of path complete metric spaces is locally $ℵ_{1}$ -presentable, closed monoidal, and coreflective in CMet. We also prove that the category CCMet of convex complete metric spaces is not closed monoidal and characterize the isometry- $ℵ_{0}$ -generated objects in CMet, CPMet and CCMet, answering questions by Di Liberti and Rosický. Other results...

Constructive quantization: approximation by empirical measures

Steffen Dereich, Michael Scheutzow, Reik Schottstedt (2013)

Annales de l'I.H.P. Probabilités et statistiques

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In this article, we study the approximation of a probability measure $μ$ on $ℝ^{d}$ by its empirical measure ${\hat{μ}}_{N}$ interpreted as a random quantization. As error criterion we consider an averaged $p$ th moment Wasserstein metric. In the case where $2 p l t; d$ , we establish fine upper and lower bounds for the error, a. Moreover, we provide a universal estimate based on moments, a . In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.

A new kind of augmentation of filtrations

Joseph Najnudel, Ashkan Nikeghbali (2011)

ESAIM: Probability and Statistics

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Let (Ω, $ℱ$ , ( $ℱ_{t}$ ), $ℙ$ ) be a filtered probability space satisfying the usual assumptions: it is usually not possible to extend to $ℱ_{\infty}$ (the-algebra generated by ( $ℱ_{t}$ )) a coherent family of probability measures ( $ℚ_{t}$ ) indexed by , each of them being defined on $ℱ_{t}$ . It is known that for instance, on the Wiener space, this extension problem has a positive answer if one takes the filtration generated by the coordinate process, made right-continuous, but can have a negative...

Characterizing metric spaces whose hyperspaces are homeomorphic to ℓ₂

T. Banakh, R. Voytsitskyy (2008)

Colloquium Mathematicae

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It is shown that the hyperspace $C l d_{H} (X)$ (resp. $B d d_{H} (X)$ ) of non-empty closed (resp. closed and bounded) subsets of a metric space (X,d) is homeomorphic to ℓ₂ if and only if the completion X̅ of X is connected and locally connected, X is topologically complete and nowhere locally compact, and each subset (resp. each bounded subset) of X is totally bounded.

Multi-Harnack smoothings of real plane branches

Pedro Daniel González Pérez, Jean-Jacques Risler (2010)

Annales scientifiques de l'École Normale Supérieure

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Let $Δ \subset 𝐑^{2}$ be an integral convex polygon. G. Mikhalkin introduced the notion of, a class of real algebraic curves, defined by polynomials supported on $Δ$ and contained in the corresponding toric surface. He proved their existence, viamethod, and that the topological type of their real parts is unique (and determined by $Δ$ ). This paper is concerned with the description of the analogous statement in the case of a smoothing of a real plane branch $(C, 0)$ . We introduce the class ofsmoothings of $(C, 0)$ by...

The systolic constant of orientable Bieberbach 3-manifolds

Chady El Mir, Jacques Lafontaine (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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A compact manifold is called if it carries a flat Riemannian metric. Bieberbach manifolds are aspherical, therefore the supremum of their systolic ratio, over the set of Riemannian metrics, is finite by a fundamental result of M. Gromov. We study the optimal systolic ratio of compact $3$ -dimensional orientable Bieberbach manifolds which are not tori, and prove that it cannot be realized by a flat metric. We also highlight a metric that we construct on one type of such manifolds ( $C_{2}$ ) which...