Staircasing effect for minimizers of the one-dimensional discrete Perona-Malik functional

We consider the one-dimensional Perona-Malik functional, that is the energy associated to the celebrated forward-backward equation introduced by P. Perona and J. Malik in the context of image processing, with the addition of a forcing term. We discretize the functional by restricting its domain to a finite dimensional space of piecewise constant functions, and by replacing the derivative with a difference quotient. We investigate the asymptotic behavior of minima and minimizers as the discretization scale vanishes. In particular, if the forcing term has bounded variation, we show that any sequence of minimizers converges in the sense of varifolds to the graph of the forcing term, but with tangent component which is a combination of the horizontal and vertical directions. If the forcing term is more regular, we also prove that minimizers actually develop a microstructure that looks like a piecewise constant function at a suitable scale, which is intermediate between the macroscopic scale and the scale of the discretization.Comment: 43 pages. In the second version some typos have been correcte

The Perona-Malik problem: singular perturbation and semi-discrete approximation

We consider the Perona-Malik equation, which is a forward-backward parabolic equation that can be seen as the formal gradient-flow of a non-convex functional, and we regularize the problem in two different ways: by adding an higher order term or by space discretization.Concerning the higher order regularization, we present some recent results concerning the asymptotic behavior of minimizers of the regularized functional when a fidelity term is added. We show that these minimizers exhibit a multi-scale structure by characterizing the possible blow-up at different scales. The main results are obtained in the one-dimensional case, but some partial generalizations to any dimension are also provided.In the final chapter, we consider the discrete approximation of the dynamic problem in the one-dimensional case and we study the evolution of the maximum and of the total variation of all limits of discrete evolutions. We provide an example in which these quantities do not pass to the limit from the discrete to the continuous setting. Nevertheless, we show that these quantities inherit the same monotonicity properties that hold at the discrete level. These monotonicity results actually hold for a general class of one-dimensional evolution curves that we call uvw-evolutions

Gamma-liminf estimate for a class of non-local approximations of Sobolev and BV norms

We consider a family of non-local and non-convex functionals, and we prove that their Gamma-liminf is bounded from below by a positive multiple of the Sobolev norm or the total variation. As a by-product, we answer some open questions concerning the limiting behavior of these functionals. The proof relies on the analysis of a discretized version of these functionals.Comment: 22 pages. In the second version we expanded the introduction, we added some references, and we corrected a few typo

Monotonicity properties of limits of solutions to the semi-discrete scheme for the Perona-Malik equation

We consider generalized solutions of the Perona-Malik equation in dimension one, defined as all possible limits of solutions to the semi-discrete approximation in which derivatives with respect to the space variable are replaced by difference quotients. Our first result is a pathological example in which the initial data converge strictly as bounded variation functions, but strict convergence is not preserved for all positive times, and in particular many basic quantities, such as the supremum or the total variation, do not pass to the limit. Nevertheless, in our second result we show that all our generalized solutions satisfy some of the properties of classical smooth solutions, namely the maximum principle and the monotonicity of the total variation. The verification of the counterexample relies on a comparison result with suitable sub/supersolutions. The monotonicity results are proved for a more general class of evolution curves, that we call $uv$-evolutions.Comment: 33 page

On the characterization of constant functions through nonlocal functionals

We provide a counterexample to an open question concerning a characterization of constant functions through double integrals that involve different quotients. This counterexample requires the construction of an unbounded function whose difference quotients avoid a sequence of intervals with endpoints that diverge to infinity.Comment: 6 page

A quantitative variational analysis of the staircasing phenomenon for a second order regularization of the Perona-Malik functional

We consider the Perona-Malik functional in dimension one, namely an integral functional whose Lagrangian is convex-concave with respect to the derivative, with a convexification that is identically zero. We approximate and regularize the functional by adding a term that depends on second order derivatives multiplied by a small coefficient. We investigate the asymptotic behavior of minima and minimizers as this small parameter vanishes. In particular, we show that minimizers exhibit the so-called staircasing phenomenon, namely they develop a sort of microstructure that looks like a piecewise constant function at a suitable scale. Our analysis relies on Gamma-convergence results for a rescaled functional, blow-up techniques, and a characterization of local minimizers for the limit problem. This approach can be extended to more general models

On the shape factor of interaction laws for a non-local approximation of the Sobolev norm and the total variation

We consider the family of non-local and non-convex functionals introduced by H. Brezis and H.-M. Nguyen in a recent paper. These functionals Gamma-converge to a multiple of the Sobolev norm or the total variation, depending on a summability exponent, but the exact values of the constants are unknown in many cases. We describe a new approach to the Gamma-convergence result that leads in some special cases to the exact value of the constants, and to the existence of smooth recovery families.Comment: Compte-rendu that summarizes the strategy developed in ArXiv:1708.01231 and ArXiv:1712.04413. This version extends the previous one keeping into account the changes in the above papers. 9 page

Nonlocal characterizations of Sobolev spaces and functions of bounded variation in dimension one

In the thesis we study two kinds of families of nonlocal functionals, whose limits (in the appropriate sense) are multiples either of the Sobolev norm or of the total variation, depending on the value of a parameter p. This provides a characterization of the Sobolev spaces and of the space of functions of bounded variation as the set of functions for which the (Γ-)limit of those families is finite. For the sake of simplicity, we consider only the one dimensional case, even if almost all the results are valid in every dimension. In the first family, derivatives are replaced by finite differences weighted by a family of mollifiers. As shown in 2001 by J. Bourgain, H. Brezis and P. Mironescu, this family converges to the Sobolev norm or the total variation both in the sense of pointwise convergence and in the sense of De Giorgi’s Γ-convergence. The second family has been studied in a series of paper (Nguyen (JFA 2006, Duke 2011), Bourgain-Nguyen (CRAS 2006), Brezis-Nguyen (ArXiv 2016)). In these papers the authors proved that (if p>1) the pointwise limit is equal to a multiple of the Sobolev norm, and that the Γ-limit is also a multiple of the Sobolev norm (or the total variation if p=1), but with a strictly lower constant. Nevertheless, many problems remained open. In this thesis we answer two of them: • we compute the exact value of the constant that appears in the Γ-limit (that was only conjectured until now), • we show the existence of smooth recovery families. The techniques we use provide also a simpler proof of the Γ-convergence result. Some parts of this thesis are based on a joint work with C. Antonucci, M. Gobbino and M. Migliorini