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Numerical methods for phase transition problems

Claudio Verdi (1998)

Bollettino dell'Unione Matematica Italiana

Nel presente articolo si illustrano alcuni dei principali metodi numerici per l'approssimazione di modelli matematici legati ai fenomeni di transizione di fase. Per semplificare e contenere l'esposizione ci siamo limitati a discutere con un certo dettaglio i metodi più recenti, presentandoli nel caso di problemi modello, quali il classico problema di Stefan e l'evoluzione di superficie per curvatura media, solo accennando alle applicazioni e modelli più generali.

Obstacle problems for scalar conservation laws

Laurent Levi (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper we are interested in bilateral obstacle problems for quasilinear scalar conservation laws associated with Dirichlet boundary conditions. Firstly, we provide a suitable entropy formulation which ensures uniqueness. Then, we justify the existence of a solution through the method of penalization and by referring to the notion of entropy process solution due to specific properties of bounded sequences in L ∞ . Lastly, we study the behaviour of this solution and its stability properties with...

Obstacle problems for scalar conservation laws

Laurent Levi (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we are interested in bilateral obstacle problems for quasilinear scalar conservation laws associated with Dirichlet boundary conditions. Firstly, we provide a suitable entropy formulation which ensures uniqueness. Then, we justify the existence of a solution through the method of penalization and by referring to the notion of entropy process solution due to specific properties of bounded sequences in L∞. Lastly, we study the behaviour of this solution and its stability properties...

On a Bernoulli problem with geometric constraints

Antoine Laurain, Yannick Privat (2012)

ESAIM: Control, Optimisation and Calculus of Variations

A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by x1 ≥ 0 and its boundary to contain a segment of the hyperplane  {x1 = 0}  where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed...

On a Liouville type theorem for isotropic homogeneous fully nonlinear elliptic equations in dimension two

Jean Dolbeault, Régis Monneau (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper we establish a Liouville type theorem for fully nonlinear elliptic equations related to a conjecture of De Giorgi in ℝ 2 . We prove that if the level lines of a solution have bounded curvature, then these level lines are straight lines. As a consequence, the solution is one-dimensional. The method also provides a result on free boundary problems of Serrin type.

On a mathematical model for the crystallization of polymers

Maria Pia Gualdani (2003)

Bollettino dell'Unione Matematica Italiana

We consider a mathematical model proposed in [1] for the cristallization of polymers, describing the evolution of temperature, crystalline volume fraction, number and average size of crystals. The model includes a constraint W e ⁢ q on the crystal volume fraction. Essentially, the model is a system of both second order and first order evolutionary partial differential equations with nonlinear terms which are Lipschitz continuous, as in [1], or Hölder continuous, as in [3]. The main novelty here is the...

On a non-stationary free boundary transmission problem with continuous extraction and convection, arising in industrial processes

Bui Ton, Grzegorz Łukaszewicz (1992)

Banach Center Publications

The existence of a weak solution of a non-stationary free boundary transmission problem arising in the production of industrial materials is established. The process is governed by a coupled system involving the Navier--Stokes equations and a non-linear heat equation. The stationary case was studied in [7].

On a quasi-variational inequality arising in semiconductor theory.

José-Francisco Rodrigues (1992)

Revista Matemática de la Universidad Complutense de Madrid

Some new mathematical results of existence and uniqueness of solutions are obtained for a class of quasi-variational inequalities modeling the free boundary problem for the determination of the depletion zone in reverse biased semiconductor diodes. The corresponding one (or two) obstacle implicit problems are solved by direct methods with weak regularity estimates for mixed boundary value elliptic problems of second order.

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