Mathematical and Numerical Analysis of some Non Linear P D E 's Problems

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Mathematical and Numerical Analysis of some Non-Linear P.D.E.'s Problems Jean R. Roche Institut Elie Cartan UMR 7502, Nancy-Université, CNRS, INRIA . CALVI INRIA Research Team B.P. 239, 54506 Vandoeuvre les Nancy Cedex, France, e-mail: . MATH AmSud, Lima-2008 Jean R. Roche Non-Linear P.D.E.'s Problems

differential geometry

equations

domain decomposition method

fluid-structure problems

vlasov-poisson-maxwell equations

research team

numerical analysis

capes - cofecub bilateral

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Publié par

pefav

Nombre de lectures

Langue

English

Poids de l'ouvrage

2 Mo

Mathematical and Numerical Analysis of some Non-Linear P.D.E.’s Problems

Jean R.

Roche

Institut Elie Cartan UMR 7502, Nancy-Université, CNRS, INRIA . CALVI INRIA Research Team B.P. 239, 54506 Vandoeuvre les Nancy Cedex, France, e-mail: roche@iecn.u-nancy.fr.

MATH AmSud, Lima-2008

Jean R. Roche

Non-Linear P.D.E.’s Problems

I.E.C.N.

TheInstitut Elie Cartan de Nancyis the Pure and Applied Mathematics Laboratory of the three Universities of Nancy: Université Henri Poincaré-Nancy I

Université Nancy 2 Institut National Polytechnique de Lorraine U.M.R. CNRS 7502 and associated to the INRIA Web adress: www.iecn.u-nancy.fr

Jean R. Roche

Non-Linear P.D.E.’s Problems

The Research Teams

Analysis and Complex Geometry Differential Geometry Lie Groups and Harmonic Analysis Number Theory

Probability and Statistics (TOSCA INRIA Research Team) Partial Differential Equations and Applications (CALVI and CORIDA INRIA Research Team)

Jean R. Roche

Non-Linear P.D.E.’s Problems

Partial Differential Equations and Applications

Free boundary problems, shape optimization, inverse problems Non-linear evolution problems, reaction-diffusion systems, asymptotic behavior Control and stabilization of systems governed by PDE’s Integral equations, domain decomposition Fluid-structure problems Vlasov-Poisson-Maxwell equations, applications to plasma physics Previous collaborations with South America : INRIA associated team ANCIF with the University of Chile and a CAPES - COFECUB bilateral agreement with the UFRJ and the LNCC

Jean R. Roche

Non-Linear P.D.E.’s Problems

A domain decomposition method for second order nonlinear equations

Joint work with NourEddine Alaa from the University Cadi Ayyad, Marrakech, Maroc Introduction The aim of this communication is to give a result of existence and present a numerical analysis of weak non-negative solutions for the following quasi-linear elliptic problem in one and two dimensions: u−(Aux()x=)0+oGn(x,∂ΩDu(x)) =F(x,u(x)) +f(x)inΩ,(1)

whereAa second order derivatives operator in oneis dimension and the Laplace operator in two dimensions,G,F are measurable and continuous non negative functions. The functionfis given ﬁnite and non negative. The domain Ω⊂RN,N=1,2is open and bounded.

Jean R. Roche

Non-Linear P.D.E.’s Problems

In the one dimensional case we are particularly interested by situations wherefis irregular and where the growth of bothG with respect tou0andFwith respect touare arbitrary. A model problem is the following: in(0,1)(2)

−u00(t) +|u0(t)|q=|u(t)|p+f u(0) =u(1) =0

wherep,q≥1andf∈MB+(0,1).

Jean R. Roche

Non-Linear P.D.E.’s Problems

In the two dimensional case we assume that the growth ofGis subquadratic. A classical example is the following: u−=Δu0(xi)n+∂Ω|ru(x)|p=|u(x)|q+finΩ(3)

where1<p,q<∞and

f∈L1(Ω),f≥0

Jean R. Roche

Non-Linear P.D.E.’s Problems

Remarks

•Whenfproved by in P.L. Lions that ifis regular, it is (2)has a non negative supersolution inW10,∞then(2)has a solution in W10,∞TW2,p. Here the supersolution vanish at the boundary. •Whenfis irregular andGis subquadratic with respect tou0 namely:

G(t,r)≤c(g(t) +|r|2),

g(t)∈L1(0,1),c>0

(4)

Then(2)has a solutionu∈H01(0,1)if(2)has a supersolution inW1,∞(0,1). Choquet-Bruhat-Leray, P.L. Lions,Boccardo-Murat-Puel, Bensoussan, Gallouët, Brezis-Strauss, Baras-Pierre, Porretta Wk,p(0,1) ={f∈Lp(0,1)such that∀α,|α| ≤k, ∂αf∈Lp(0,1)}. W0k,p(0,1)is the closure ofCc∞(0,1)inWk,p(0,1).

Jean R. Roche

Non-Linear P.D.E.’s Problems

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