Convex bodies: the brunn–minkowski theory

Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

The porous medium equation

We study the nonlinear monotone elliptic problem  −div (a(x,∇u)) = μ on Ω, u = 0 on ∂Ω, when Ω ⊂ R , μ is a Radon measure with bounded total variation on Ω, 1 < p ≤ N , and u 7→ −div (a(x,∇u)) is a monotone operator acting on W 1,p 0 (Ω). We introduce a new definition of solution (the renormalized solution) in four equivalent ways. We prove the existence of a renormalized solution by an approximation procedure, where the key point is a stability result (the strong convergence in W 1,p 0 (Ω) of the truncates). We also prove partial uniqueness results. SISSA, via Beirut 4, 34014 Trieste, Italy Laboratoire d’Analyse Numerique, Universite Paris VI, Case 187, 75252 Paris cedex 05, France Dipartimento di Matematica, Universita di Roma I, P.le A. Moro 2, 00185 Roma, Italy Universite d’Orleans, rue de Chartres, B.P. 6759, 45067 Orleans cedex 2, France

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Renormalized solutions of elliptic equations with general measure data

1.1. Presentation of the problem. The water-waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect, incompressible, irrotational fluid under the influence of gravity. In this paper, we restrict our attention to the case when the surface is a graph parameterized by a function ?(?,X), where t denotes the time variable and X = (Xi,... ,Xd) Rd the horizontal spatial variables. The method developed here works equally well for any integer d > 1, but the only physically relevant cases are of course d = 1 and d ? 2. The layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time-independent function b(X). We denote by Q,t the fluid domain at time t. The incompressibility of the fluids is expressed by (1.1) div V = 0 in Qu t > 0,

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Well-posedness of the water-waves equations

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An $L^1$ -theory of existence and uniqueness of solutions of nonlinear elliptic equations

The goal of this paper is to describe the state of the art on the question of global existence of solutions to reaction-diffusion systems for which two main properties hold: on one hand, the positivity of the solutions is preserved for all time; on the other hand, the total mass of the components is uniformly controlled in time. This uniform control on the mass (or – in mathematical terms- on the L1-norm of the solution) suggests that no blow up should occur in finite time. It turns out that the situation is not so simple. This explains why so many partial results in different directions are found in the literature on this topic, and why also the general question of global existence is still open, while lots of systems arise in applications with these two natural properties. We recall here the main positive and negative results on global existence, together with many references, a description of the still open problems and a few new results as well.

Global Existence in Reaction-Diffusion Systems with Control of Mass: a Survey

: This work establishes existence of solutions of the initial-value problem u(t) = delta (determinant u to m-1 power), u(x,0), = u(0)(x), where m 1, under the most general conditions on u(0). Namely, u(0) need only be such that R to the minus (2 divided by m-1 +N) sum (determinant x or = R) to the power of u(0)(x)) dx is bounded independently of R or = 1. Aronson and Caffarelli have shown this requirement to be necessary. Many auxiliary results are given in the form of estimates on solutions, uniqueness and continuous dependence theorems, etc. While the results may be viewed as 'technical' in that the main points consist of estimates of various sorts, the equation treated is of broad practical interest and the estimates reflect basic properties of the equation. The results obtained are the only ones known to the authors wherein the solvability of a realistic nonlinear initial value problem for a partial differential equation is now understood as completely as in the case of the heat equation.

Solutions of the Porous Medium Equation in R(N) under Optimal Conditions on Initial Values.

We prove global existence in time of weak solutions to a class of quadratic reaction-diffusion systems for which a Lyapounov structure of LlogL-entropy type holds. The approach relies on an a priori dimension-independent L-2-estimate, valid for a wider class of systems includingalso some classical Lotka-Volterra systems, and which provides an L-1-bound on the nonlinearities, at least for not too degenerate diffusions. In the more degenerate case, some global existence may be stated with the use of a weaker notion of renormalized solution with defect measure, arising in the theory of kinetic equations.

Global existence for quadratic systems of reaction-diffusion

In this paper, we describe the precise structure of second "shape derivatives", that is derivatives of functions whose argument is a variable subset of $ \mathbb{R}^N $. This is done for Frechet derivatives in adequate Banach spaces. Besides the structure itself, interest lies in the way it is derived: the starting point is a "functional analytic" statement of the well-known fact that small regular perturbations of a given regular domain may be "uniquely" represented through normal deformations of the boundary of this domain. The approach involves the implicit function theorem in a convenient functional space. A consequence of this "normal representation" property is that any shape functional may be described through a functional depending on functions defined only on the boundary of the given domain. Differentiating twice this representation leads to the structure theorem. We recover the fact that, at critical shapes, the second derivative around the given domain depends only on the normal component of the deformation vector-field at its boundary. Some examples are explicitly computed.

Michel Pierre

Papers

An $L^1$ -theory of existence and uniqueness of solutions of nonlinear elliptic equations

Global Existence in Reaction-Diffusion Systems with Control of Mass: a Survey

Solutions of the Porous Medium Equation in R(N) under Optimal Conditions on Initial Values.

Global existence for quadratic systems of reaction-diffusion

Structure of shape derivatives