arXiv subjects
Publications and source records attributed to Filippo Cagnetti.
In \cite{CMP17} a novel distributional approach has been introduced to provide a well-posed formulation of a class of crystalline mean curvature flows. In this paper, such an approach is extended to the nonlocal setting. Applications include the fractional mean curvature flow and the Minkowski flow; i.e., the geometric flow generated by the $(N-1)$-dimensional Minkowski pre-content.
In this paper we study the deterministic and stochastic homogenisation of free-discontinuity functionals under \emph{linear} growth and coercivity conditions. The main novelty of our deterministic result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Combining this result with the pointwise Subadditive Ergodic Theorem by Akcoglu and Krengel, we prove a stochastic homogenisation result, in the case of stationary random integrands. In particular, we characterise the limit integrands in terms of asymptotic cell formulas, as in the classical case of periodic homogenisation.
Necessary and sufficient conditions for rigidity of the perimeter inequality under spherical symmetrisation are given. That is, a characterisation for the uniqueness (up to orthogonal transformations) of the extremals is provided. This is obtained through a careful analysis of the equality cases, and studying fine properties of the circular symmetrisation, which was firstly introduced by Pólya in 1950.
We study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions and on the fracture energy density, we show that minimizers are $C^{1, 1/2}$, and that near non-degenerate points the fracture set is $C^{1, α}$, for some $α\in (0, 1)$.
We study the $Γ$-convergence of sequences of free-discontinuity functionals depending on vector-valued functions $u$ which can be discontinuous across hypersurfaces whose shape and location are not known a priori. The main novelty of our result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Further, we consider the case of surface integrands which are not bounded from below by the amplitude of the jump of $u$. We obtain three main results: compactness with respect to $Γ$-convergence, representation of the $Γ$-limit in an integral form and identification of its integrands, and homogenisation formulas without periodicity assumptions. In particular, the classical case of periodic homogenisation follows as a by-product of our analysis. Moreover, our result covers also the case of stochastic homogenisation, as we will show in a forthcoming paper.
In this paper we study the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which is deterministic in the ergodic case. Moreover, by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise Subadditive Ergodic Theorem by Akcoglou and Krengel, we characterise the limit volume and surface integrands in terms of asymptotic cell formulas.
We introduce a novel constructive approach to define time evolution of critical points of an energy functional. Our procedure, which is different from other more established approaches based on viscosity approximations in infinite dimension, is prone to efficient and consistent numerical implementations, and allows for an existence proof under very general assumptions. We consider in particular rather nonsmooth and nonconvex energy functionals, provided the domain of the energy is finite dimensional. Nevertheless, in the infinite dimensional case study of a cohesive fracture model, we prove a consistency theorem of a discrete-to-continuum limit. We show that a quasistatic evolution can be indeed recovered as a limit of evolutions of critical points of finite dimensional discretizations of the energy, constructed according to our scheme. To illustrate the results, we provide several numerical experiments both in one and two dimensions. These agree with the crack initiation criterion, which states that a fracture appears only when the stress overcomes a certain threshold, depending on the material.
We introduce a new machinery to study the large time behavior for general classes of Hamilton--Jacobi type equations, which include degenerate parabolic equations and weakly coupled systems. We establish the convergence results by using the nonlinear adjoint method and identifying new long time averaging effects. These methods are robust and can easily be adapted to study the large time behavior of related problems.
Characterization results for equality cases and for rigidity of equality cases in Steiner's perimeter inequality are presented. (By rigidity, we mean the situation when all equality cases are vertical translations of the Steiner's symmetral under consideration.) We achieve this through the introduction of a suitable measure-theoretic notion of connectedness and a fine analysis of barycenter functions for sets of finite perimeter having segments as orthogonal sections with respect to an hyperplane.
We provide a geometric characterization of rigidity of equality cases in Ehrhard's symmetrization inequality for Gaussian perimeter. This condition is formulated in terms of a new measure-theoretic notion of connectedness for Borel sets, inspired by Federer's definition of indecomposable current.
The adjoint method, recently introduced by Evans, is used to study obstacle problems, weakly coupled systems, cell problems for weakly coupled systems of Hamilton-Jacobi equations, and weakly coupled systems of obstacle type. In particular, new results about the speed of convergence of common approximation procedures are derived.
The adjoint method introduced in [Eva] and [Tra] is used, to construct analogs to the Aubry-Mather measures for non convex Hamiltonians. More precisely, a general construction of probability measures, that in the convex setting agree with Mather measures, is provided. These measures may fail to be invariant under the Hamiltonian flow and a dissipation arises, which is described by a positive semidefinite matrix of Borel measures. However, in the important case of uniformly quasiconvex Hamiltonians the dissipation vanishes, and as a consequence the invariance is guaranteed.
We consider a numerical scheme for the one dimensional time dependent Hamilton-Jacobi equation in the periodic setting. We present a new and simple proof of the rate of convergence of the approximations based on the adjoint method recently introduced by Evans.
A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on $H^1_0(Γ)$, $Γ$ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of $Γ$. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.