Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence Theorems.- Uniqueness, Generation of Contraction Semigroups, Error Estimates.- Gradient Flow in the Space of Probability Measures.- Preliminary Results on Measure Theory.- The Optimal Transportation Problem.- The Wasserstein Distance and its Behaviour along Geodesics.- Absolutely Continuous Curves in p(X) and the Continuity Equation.- Convex Functionals in p(X).- Metric Slope and Subdifferential Calculus in (X).- Gradient Flows and Curves of Maximal Slope in p(X).

Gradient Flows: In Metric Spaces and in the Space of Probability Measures

Preface Through many centuries physics has been one of the most fruitful sources of inspiration for mathematics. As a consequence, mathematics has become an economic language providing a few basic principles which allow to explain a large variety of physical phenomena. Many of them are described in terms of partial diierential equations (PDEs). In recent years, however, mathematics also has been stimulated by other novel elds such as image processing. Goals like image segmentation, multiscale image representation, or image restoration cause a lot of challenging mathematical questions. Nevertheless, these problems frequently have been tackled with a pool of heuristical recipes. Since the treatment of digital images requires very much computing power, these methods had to be fairly simple. With the tremendous advances in computer technology in the last decade, it has become possible to apply more sophisticated techniques such as PDE-based methods which have been inspired by physical processes. Among these techniques, parabolic PDEs have found a lot of attention for smoothing and restoration purposes, see e.g. 113]. To restore images these equations frequently arise from gradient descent methods applied to variational problems. Image smoothing by parabolic PDEs is closely related to the scale-space concept where one embeds the original image into a family of subsequently simpler , more global representations of it. This idea plays a fundamental role for extracting semantically important information. The pioneering work of Alvarez, Guichard, Lions and Morel 11] has demonstrated that all scale-spaces fulllling a few fairly natural axioms are governed by parabolic PDEs with the original image as initial condition. Within this framework, two classes can be justiied in a rigorous way as scale-spaces: the linear diiusion equation with constant dif-fusivity and nonlinear so-called morphological PDEs. All these methods satisfy a monotony axiom as smoothing requirement which states that, if one image is brighter than another, then this order is preserved during the entire scale-space evolution. An interesting class of parabolic equations which pursue both scale-space and restoration intentions is given by nonlinear diiusion lters. Methods of this type have been proposed for the rst time by Perona and Malik in 1987 190]. In v vi PREFACE order to smooth the image and to simultaneously enhance semantically important features such as edges, they apply a diiusion process whose diiusivity is steered by local image properties. These lters are diicult to analyse mathematically , as they may act locally like a backward diiusion process. …

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Anisotropic diffusion in image processing

A variational model of quasistatic crack evolution is proposed. Although close in spirit to Griffith’s theory of brittle fracture, the proposed model however frees itself of the usual constraints of that theory : a preexisting crack and a well-defined crack path. In contrast, crack initiation as well as crack path can be quantified, as demonstrated on explicitly computable examples. Furthermore the model lends itself to numerical implementation in more complex settings.

Revisiting brittle fracture as an energy minimization problem

The novel concept of total generalized variation of a function $u$ is introduced, and some of its essential properties are proved. Differently from the bounded variation seminorm, the new concept involves higher-order derivatives of $u$. Numerical examples illustrate the high quality of this functional as a regularization term for mathematical imaging problems. In particular this functional selectively regularizes on different regularity levels and, as a side effect, does not lead to a staircasing effect.

http://gpu4vision.icg.tugraz.at/papers/2009/pock_tgv.pdf

Total Generalized Variation

1 Introduction 2 Going variational 2.1 Griffith's theory 2.2 The 1-homogeneous case - A variational equivalence 2.3 Smoothness - The soft belly of Griffith's formulation 2.4 The non 1-homogeneous case - A discrete variational evolution 2.5 Functional framework - A weak variational evolution 2.6 Cohesiveness and the variational evolution 3 Stationarity versus local or global minimality - A comparison 3.1 1d traction 3.1.1 The Griffith case - Soft device 3.1.2 The Griffith case - Hard device 3.1.3 Cohesive case - Soft device 3.1.4 Cohesive case - Hard device 3.2 A tearing experiment 4 Initiation 4.1 Initiation - The Griffith case 4.1.1 Initiation - The Griffith case - Global minimality 4.1.2 Initiation - The Griffith case - Local minimality 4.2 Initiation - The cohesive case 4.2.1 Initiation - The cohesive 1d case - Stationarity 4.2.2 Initiation - The cohesive 3d case - Stationarity 4.2.3 Initiation - The cohesive case - Global minimality 5 Irreversibility 5.1 Irreversibility - The Griffith case - Well-posedness of the variational evolution 5.1.1 Irreversibility - The Griffith case - Discrete evolution 5.1.2 Irreversibility - The Griffith case - Global minimality in the limit 5.1.3 Irreversibility - The Griffith case - Energy balance in the limit 5.1.4 Irreversibility - The Griffith case - The time-continuous evolution 5.2 Irreversibility - The cohesive case 6 Path 7 Griffith vs. Barenblatt 8 Numerics and Griffith 8.1 Numerical approximation of the energy 8.1.1 The first time step 8.1.2 Quasi-static evolution 8.2 Minimization algorithm 8.2.1 The alternate minimization algorithm 8.2.2 The backtracking algorithm 8.3 Numerical experiments 8.3.1 The 1D traction (hard device) 8.3.2 The Tearing experiment 8.3.3 Revisiting the 2D traction experiment on a fiber reinforced matrix 9 Fatigue 9.1 Peeling Evolution 9.2 The limitfatigue law when d tends to 0 9.3 A variational formulation for fatigue 9.3.1 Peeling revisited 9.3.2 Generalization Appendix Glossary References.

The Variational Approach to Fracture

Measure Theory Basic Geometric Measure Theory Functions of bounded variation Special functions of bounded variation Semicontinuity in BV The Mumford-Shah functional Minimisers of free continuity problems Regularity of the free discontinuity set References Index

Functions of Bounded Variation and Free Discontinuity Problems

Soit f:R 3 →R une fonction deux fois continument differentiable et pour u∈W 1,1 (a, b): F(u; a, b)=∫ a b f(x, u(x), u'(x))dx. Si f: R n ×R m ×R nm →R satisfait: 1) f est mesurable en x, continue en Cs, ξ, et 2) 0≤f(x, s, ξ)≤a(x)+C(|s| p +|ξ| p ), ou p≥1, a est une fonction localement sommable non negative, et c une constante non negative, alors la fonctionnelle F(u; a, b) est sequentiellement faiblement semi continue inferieurement sur W 1,p (Ω; R m ) si et seulement si ξ→f(x, s, ξ) est quasi convexe, pour chaque s et presque chaque x

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Semicontinuity problems in the calculus of variations

A quantitative sharp form of the classical isoperimetric inequality is proved, thus giving a positive answer to a conjecture by Hall.

The sharp quantitative isoperimetric inequality

Regularity for Minimizers of Non-quadratic Functionals: The Case 1 <p<2+

We obtain a sharp quantitative isoperimetric inequality for nonlocal s-perimeters, uniform with respect to s bounded away from 0. This allows us to address local and global minimality properties of balls with respect to the volume-constrained minimization of a free energy consisting of a nonlocal s-perimeter plus a non-local repulsive interaction term. In the particular case s = 1, the s-perimeter coincides with the classical perimeter, and our results improve the ones of Knuepfer and Muratov (Comm. Pure Appl. Math. 66(7):1129–1162, 2013; Comm. Pure Appl. Math., 2014) concerning minimality of balls of small volume in isoperimetric problems with a competition between perimeter and a nonlocal potential term. More precisely, their result is extended to its maximal range of validity concerning the type of nonlocal potentials considered, and is also generalized to the case where local perimeters are replaced by their nonlocal counterparts.

Nicola Fusco

Papers

Functions of Bounded Variation and Free Discontinuity Problems

Semicontinuity problems in the calculus of variations

The sharp quantitative isoperimetric inequality

Regularity for Minimizers of Non-quadratic Functionals: The Case 1 <p<2+

Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies