The Ginzburg-Landau equations of superconductivity and the one-phase Stefan problem
The Indicatrix Method in Free Boundary Problems.
The inhomogeneous dam problem with discontinuous permeability
The method of fictitious domains in the Signorini problem.
The Method of Lines for Poisson's Equation with Nonlinear or Free Boundary Conditions.
The motion of the free surface of a liquid
The obstacle problem for the biharmonic operator
The -Laplacian – mascot of nonlinear analysis
The phase field equations and gradient flows of marginal functions.
The quasi-stationary approximation for the Stefan problem with a convective boundary condition.
The relation between the porous medium and the eikonal equations in several space dimensions.
We study the relation between the porous medium equation ut = Δ(um), m > 1, and the eikonal equation vt = |Dv|2. Under quite general assumtions, we prove that the pressure and the interface of the solution of the Cauchy problem for the porous medium equation converge as m↓1 to the viscosity solution and the interface of the Cauchy problem for the eikonal equation. We also address the same questions for the case of the Dirichlet boundary value problem.
The Stefan problem in heterogeneous media
The Stefan problem with kinetic condition at the free boundary
The support shrinking properties for solutions of quasilinear parabolic equations with strong absorption terms
The use of linear approximation scheme for solving the Stefan problem
This paper deals with the linear approximation scheme to approximate a singular parabolic problem: the two-phase Stefan problem on a domain consisting of two components with imperfect contact. The results of some numerical experiments and comparisons are presented. The method was used to determine the temperature of steel in the process of continuous casting.
The waiting time property for parabolic problems trough the nondiffusion of support for the stationary problems.
In this note we study the waiting time phenomenon for local solutions of the nonlinear diffusion equation through its connection with the nondiffusion of the support property for local solutions of the family of discretized elliptic problems. We show that, under a suitable growth condition on the initial datum near the boundary of its support, a finite part of the family of solutions of the discretized problem maintain unchanged its support.
Theoretical and numerical study of a free boundary problem by boundary integral methods
In this paper we study a free boundary problem appearing in electromagnetism and its numerical approximation by means of boundary integral methods. Once the problem is written in a equivalent integro-differential form, with the arc parametrization of the boundary as unknown, we analyse it in this new setting. Then we consider Galerkin and collocation methods with trigonometric polynomial and spline curves as approximate solutions.
Theoretical and numerical study of a free boundary problem by boundary integral methods
In this paper we study a free boundary problem appearing in electromagnetism and its numerical approximation by means of boundary integral methods. Once the problem is written in a equivalent integro-differential form, with the arc parametrization of the boundary as unknown, we analyse it in this new setting. Then we consider Galerkin and collocation methods with trigonometric polynomial and spline curves as approximate solutions.
Transizioni di fase ed isteresi
L'attività di ricerca di chi scrive si è finora indirizzata principalmente verso l'esame dei modelli di transizione di fase, dei modelli di isteresi, e delle relative equazioni non lineari alle derivate parziali. Qui si illustrano brevemente tali problematiche, indicando alcuni degli elementi che le collegano tra di loro. Il lavoro è organizzato come segue. I paragrafi 1, 2, 3 vertono sulle transizioni di fase: si introducono le formulazioni forte e debole del classico modello di Stefan, e si illustrano...
Two Numerical Methods for the elliptic Monge-Ampère equation
The numerical solution of the elliptic Monge-Ampère Partial Differential Equation has been a subject of increasing interest recently [Glowinski, in 6th International Congress on Industrial and Applied Mathematics, ICIAM 07, Invited Lectures (2009) 155–192; Oliker and Prussner, Numer. Math.54 (1988) 271–293; Oberman, Discrete Contin. Dyn. Syst. Ser. B10 (2008) 221–238; Dean and Glowinski, in Partial differential equations, Comput. Methods Appl. Sci. 16 (2008) 43–63; Glowinski et al., Japan...