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Cancer as Multifaceted Disease

A. Friedman (2012)

Mathematical Modelling of Natural Phenomena

Cancer has recently overtaken heart disease as the world’s biggest killer. Cancer is initiated by gene mutations that result in local proliferation of abnormal cells and their migration to other parts of the human body, a process called metastasis. The metastasized cancer cells then interfere with the normal functions of the body, eventually leading to death. There are two hundred types of cancer, classified by their point of origin. Most of them...

Computation of bifurcated branches in a free boundary problem arising in combustion theory

Olivier Baconneau, Claude-Michel Brauner, Alessandra Lunardi (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a parabolic 2D Free Boundary Problem, with jump conditions at the interface. Its planar travelling-wave solutions are orbitally stable provided the bifurcation parameter u * does not exceed a critical value u * c . The latter is the limit of a decreasing sequence ( u * k ) of bifurcation points. The paper deals with the study of the 2D bifurcated branches from the planar branch, for small k. Our technique is based on the elimination of the unknown front, turning the problem into a fully nonlinear...

Conformal Geometry and the Composite Membrane Problem

Sagun Chanillo (2013)

Analysis and Geometry in Metric Spaces

We show that a certain eigenvalue minimization problem in two dimensions for the Laplace operator in conformal classes is equivalent to the composite membrane problem. We again establish such a link in higher dimensions for eigenvalue problems stemming from the critical GJMS operators. New free boundary problems of unstable type arise in higher dimensions linked to the critical GJMS operator. In dimension four, the critical GJMS operator is exactly the Paneitz operator.

Continuity for bounded solutions of multiphase Stefan problem

Emmanuele DiBenedetto, Vincenzo Vespri (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We establish the continuity of bounded local solutions of the equation β ⁢ u t = Δ ⁢ u . Here β is any coercive maximal monotone graph in R × R , bounded for bounded values of its argument. The multiphase Stefan problem and the Buckley-Leverett model of two immiscible fluids in a porous medium give rise to such singular equations.

Contrôle par les coefficients dans le modèle Elrod-Adams

Mohamed El Alaoui Talibi, Abdellah El Kacimi (2001)

ESAIM: Control, Optimisation and Calculus of Variations

Dans ce papier, nous étudions un problème de contrôle par les coefficients issu de la lubrification élastohydrodynamique. La variable de contrôle est l’épaisseur du fluide. Le phénomène de cavitation est pris en compte par le modèle Elrod-Adams, connu pour ses performances dans la conservation des débits d’entrée et de sortie. L’idée est de régulariser dans l’équation d’état le graphe d’Heaviside, en l’approchant par une suite de fonctions monotones et régulières. Nous dérivons les conditions d’optimalité...

Contrôle par les coefficients dans le modèle elrod-adams

Mohamed El Alaoui Talibi, Abdellah El Kacimi (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The purpose of this paper is to study a control by coefficients problem issued from the elastohydrodynamic lubrication. The control variable is the film thickness.The cavitation phenomenon takes place and described by the Elrod-Adams model, suggested in preference to the classical variational inequality due to its ability to describe input and output flow. The idea is to use the penalization in the state equation  by approximating the Heaviside graph whith a sequence of monotone and regular functions....

Convexity and uniqueness in a free boundary problem arising in combustion theory.

Arshak Petrosyan (2001)

Revista Matemática Iberoamericana

We consider solutions to a free boundary problem for the heat equation, describing the propagation of flames. Suppose there is a bounded domain Ω ⊂ QT = Rn x (0,T) for some T > 0 and a function u > 0 in Ω such thatut = Δu,    in Ω,u = 0 and |∇u| = 1,   on Γ := ∂Ω ∩ QT,u(·,0) = u0,     on Ω0,where Ω0 is a given domain in Rn and u0 is a positive and continuous function in Ω0, vanishing on ∂Ω0. If Ω0 is convex and u0 is concave in Ω0, then we show that (u,Ω) is unique and the time sections...

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