Giorgio Fusco

Jacobi matrices and transversality

The paper deals with smooth nonlinear ODE systems in ℝ n , ẋ = f ( x ), such that the derivative f ′( x ) has a matrix representation of Jacobi type (not necessarily symmetric) with positive off diagonal entries. A discrete functional is introduced and is discovered to be nonincreasing along the solutions of the associated linear variational system ẏ = f ′( x ( t )) y . Two families of transversal cones invariant under the flow of that linear system allow us to prove transversality between the stable and unstable manifolds of any two hyperbolic critical points of the given nonlinear system; it is also proved that the nonwandering points are critical points. A new class of Morse–Smale systems in ℝ n is then explicitly constructed.

The Gamma-limit and the related gradient flow for singular perturbation functionals of Perona-Malik type

We consider a class of nonconvex functionals of the gradient in one dimension, which we regularize with a second order derivative term. After a proper rescaling, suggested by the associated dynamical problems, we show that the sequence {F-nu} of regularized functionals Gamma-converges, as nu --> 0(+), to a particular class of free-discontinuity functionals F, concentrated on SBV functions with finite energy and having only the jump part in the derivative. We study the singular dynamic associated with F, using the minimizing movements method. We show that the minimizing movement starting from an initial datum with a finite number of discontinuities has jump positions fixed in space and whose number is nonincreasing with time. Moreover, there are a finite number of singular times at which there is a dropping of the number of discontinuities. In the interval between two subsequent singular times, the vector of the survived jumps is determined by the system of ODEs which expresses the L-2-gradient of the Gamma-limit. Furthermore the minimizing movement turns out to be continuous with respect to the initial datum. Some properties of a minimizing movement starting from a function with an in finite number of discontinuities are also derived.

The Equations of Ostwald Ripening for Dilute Systems

We consider a dilute mixture in 3D of a finite number of particles initially close to spherical, but of varying sizes, and representing one of the phases evolving according to the quasistatic dynamics. Under the scaling hypotheses that (1) typical size/typical distance and (2) deviation from sphericity/typical size are small, we associate centers and radii to each particle for the whole evolution and derive rigorously a set of ODEs fo the radii which we relate to the Lifschitz–Slyosov–Wagner theory of coarsening.

Transversality between Invariant Manifolds of Periodic Orbits for a Class of Monotone Dynamical Systems

We consider a special class of monotone dynamical systems and show that in this special class the stable and unstable manifolds of two hyperbolic periodic orbits always intersect transversally. The proof is based on the existence of a family of positively invariant nested cones.

A maximum principle for systems with variational structure and an application to standing waves

We establish via variational methods the existence of a standing wave together with an estimate on the convergence to its asymptotic states for a bistable system of partial differential equations on a periodic domain. The main tool is a replacement lemma which has as a corollary a maximum principle for local minimizers.

Motion of bubbles towards the boundary for the Cahn–Hilliard equation

In this work we describe some aspects of the dynamics of the Cahn–Hilliard equation. In particular, we consider the dynamics of ‘bubble’ solutions that is spherical interfaces which move superslowly towards the boundary without changing their shape. We show for the Cahn–Hilliard that the bubble drifts towards the closest point on the boundary provided it is sufficiently small. This is contrasted with the related mass conserving Allen–Cahn equation where size is not an issue.

Analysis of the heteroclinic connection in a singularly perturbed system arising from the study of crystalline grain boundaries

Mathematically, the problem considered here is that of heteroclinic connections for a system of two second order differential equations of Hamiltonian type, in which a small parameter conveys a singular perturbation. The motivation comes from a multi-order-parameter phase field model developed by Braun et al. [5] and [22] for the description of crystalline interphase boundaries. In this model, the smallness of is related to large anisotropy. The existence of such a heteroclinic, and its dependence on , is proved. In addition, its robustness is investigated by establishing its spectral stability.

A dynamical systems proof of the Krein–Rutman Theorem and an extension of the Perron Theorem

In this paper we establish Perron and Krein–Rutman-like theorems for an operator mapping a cone into the interior of the cone, by considering the discrete dynamical system for the induced operator on the projective space (= sphere). Existence of a positive eigenvector reduces to showing that the ω-limit set of the induced operator consists of a single equilibrium. A special feature of our approach is that the convexity of the cone is needed only for establishing the non-emptiness of the w-limit set. This allows us in finite dimensions to establish an abstract Perron Theorem for non-convex cones.

Continuum limits of particles interacting via diffusion

We consider a two-phase system mainly in three dimensions and we examine the coarsening of the spatial distribution, driven by the reduction of interface energy and limited by diffusion as described by the quasistatic Stefan free boundary problem. Under the appropriate scaling we pass rigorously to the limit by taking into account the motion of the centers and the deformation of the spherical shape. We distinguish between two different cases and we derive the classical mean-field model and another continuum limit corresponding to critical density which can be related to a continuity equation obtained recently by Niethammer andOtto. So, the theory of Lifshitz, Slyozov, and Wagner is improved by taking into account the geometry of the spatial distribution.

Slow Motion Manifolds For A Class of Singular Perturbation Problems: The Linearized Equations

Publisher Summary This chapter describes the slow motion manifolds for a class of singular perturbation problems. The analysis is divided into three parts: (1) spectral analysis of the linearized operator about the elements of an approximate manifold, (2) refinement of the first approximation via linearization, and (3) existence of the slow motion manifold in a neighborhood of the approximate manifold by a perturbation argument. This method appears to be quite general and applicable to a large class of singular perturbation problems. In addition, the chapter also provides the information on the co-ordinate system, derivation of equation in new co-ordinate system, and refinement of the V-estimate.