QUELQUES PROBLEMES DE LA THEORIE DES SYSTEMES PARABOLIQUES DEGENERES NON-LINEAIRES ET DES LOIS DE CONSERVATION

BESANCON-BU Sciences Staps (250562103) / SudocSudocFranceF

Sur un problème parabolique-elliptique

We prove existence (uniqueness is easy) of a weak solution to a boundary value problem for an equation like $(v-1)^+_t = v_{xx} + F(v)_x$ where the function $F: \Bbb R\rightarrow\Bbb R$ is only supposed to be locally lipschitz continuous. In order to replace the lack of compactness in t on v<1, we use nonlinear semigroup theory

Quelques résultats d'unicité pour des problèmes elliptiques et paraboliques

BESANCON-BU Sciences Staps (250562103) / SudocSudocFranceF

Enchères asymétriques: contribution à la détermination numérique des stratégies d'équilibre bayésien

In this paper, we drop the symmetry assumption in a model of first price procurement auction. We consider the case of two groups of bidders whose costs are drawn from two different uniform distributions. Conditions of existence of a common minimum bid are exhibited and bayesian equilibrium strategies of firms in both groups are computed. We show that these strategies can be written as the symmetric equilibrium strategies more or less a mark-up resulting from the asymmetry.

Comparison of solutions of nonlinear evolution problems with different nonlinear terms

The authors study the nonlinear porous media type equation ut(t,x)−Δφ(u(t,x))=0 for (t,x)∈(0,∞)×Ω, φ(u(t,x))=0 for (t,x)∈(0,∞)×∂Ω, u(0,x)=u0(x) for x∈Ω, with Ω an open set in Rn, and φ a regular, real, continuous, nondecreasing function. In the classical framework, the following theorem is proved: Let φi∈C2(R) with φi′>0 and u0i∈C(Ω¯¯¯)∩L∞(Ω), for i=1,2. Then if (i) φ1(u01)≤φ2(u02) on Ω, (ii) ψ′1≤ψ′2 on R, where ψi=φ−1i, and (iii) Δφ2(u02)≤0 on Ω, we have φ1(u1)≤φ2(u2) on (0,∞)×Ω. A counterexample shows the necessity of (iii). The theorem is proved by an application of the maximum principle. In a more abstract framework, a similar theorem is proved for the abstract Cauchy problem du/dt+Au∋f, u(0)=u0, where A operates as a multiapplication in a Banach space X, u0∈X, and f∈L1(0,T:X). The abstract result is applied to well-posed Cauchy problems in L1(Ω). Generalizations are given, including nonlinear boundary conditions and replacing the Laplacian operator Δ by a generalized (nonlinear) Laplacian

On a doubly nonlinear parabolic obstacle problem modelling ice sheet dynamics

This paper deals with the weak formulation of a free (moving) boundary problem arising in theoretical glaciology. Considering shallow ice sheet flow, we present the mathematical analysis and the numerical solution of the second order nonlinear degenerate parabolic equation modelling, in the isothermal case, the ice sheet non-Newtonian dynamics. An obstacle problem is then deduced and analyzed. The existence of a free boundary generated by the support of the solution is proved and its location and evolution are qualitatively described by using a comparison principle and an energy method. Then the solutions are numerically computed with a method of characteristics and a duality algorithm to deal with the resulting variational inequalities. The weak framework we introduce and its analysis (both qualitative and numerical) are not restricted to the simple physics of the ice sheet model we consider nor to the model dimension; they can be successfully applied to more realistic and sophisticated models related to other geophysical settings

An L¹-Theory Of Existence And Uniqueness Of Solutions Of Nonlinear Elliptic Equations

In this paper we study the questions of existence and uniqueness of solutions for equations of the form \GammaAu = F (x; u), posed in \Omega\Gamma an open subset of R N (bounded or unbounded) , with Dirichlet boundary conditions. A is a nonlinear elliptic operator modeled on the p-Laplacian operator \Delta p (u) = div (jDuj p\Gamma2 Du), with p ? 1, and F (x; u) is a Caratheodory function which is nonincreasing in u. Typical cases include F (x; u) = f(x) or F (x) = f(x) \Gamma fi(u), where fi is an increasing function with fi(0) = 0 (or even a maximal monotone graph with 0 2 fi(0)). We use an integrability assumption on F which in these cases means that f 2 L 1 (\Omega\Gamma9 The existence theory offers few difficulties when p ? N . Here we consider the case 1 ! p ! N and establish existence of a weak solution u. For p ? 2 \Gamma (1=N) and\Omega bounded the solution lies in the usual Sobolev space W 1;q 0 (\Omega\Gamma with 1 ! q ! p = N(p \Gamma 1)=(N \Gamma 1). However, when..