A. Wasserman - Academia.edu (original) (raw)

Papers by A. Wasserman

ESAIM: Mathematical Modelling and Numerical Analysis, 1989

Remarks on the uniqueness of radial solutions RAIRO-Modélisation mathématique et analyse numériqu... more Remarks on the uniqueness of radial solutions RAIRO-Modélisation mathématique et analyse numérique, tome 23, n o 3 (1989), p. 535-540. <http://www.numdam.org/item?id=M2AN_1989__23_3_535_0> © AFCET, 1989, tous droits réservés. L'accès aux archives de la revue « RAIRO-Modélisation mathématique et analyse numérique » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Nonlinear Analysis: Theory, Methods & Applications, 1980

Journal of Differential Equations, 1989

Journal of Differential Equations, 1984

Brunovsky and Chow [ 1 ] have recently proved that for a generic C2 function with the Whitney top... more Brunovsky and Chow [ 1 ] have recently proved that for a generic C2 function with the Whitney topology, the "time map" r(.,f) (see [4]) associated with the differential equation U" +f(u) = 0 with homogeneous Dirichlet or Neumann boundary conditions, is a Morse function. In this note we give a simpler proof of this result as well as some new applications. Our method of proof is quite elementary, uses only Sard's theorem and the implicit function theorem for functions in C'(lF?', R), and avoids the use of transversality in function spaces. An annoying difficulty that one has to face is that the domain of T varies with f: We get around this by constructing a continuous function H(f) which, if positive, implies that T is a Morse function. Thus our task is to prove that the set off with H(f) > 0 is generic. Of course, openness is trivial. To prove the denseness, we take anyf, perturb it by a monomial CU", and consider the map 8: (u, c) + T'(u,p), wherej(u) =f(u) + CU". We show that 0 is a regular value of 0 by checking explicitly that the relevant derivative has the form ~ln + b, where a # 0, and b is bounded. Thus for large n, the linear term dominates and this yields the density statement. One consequence of this result is that iff(u) < 0 for u > M, then there are a finite number of (positive) stationary solutions of the equation U, = u,, +f(u), with homogeneous Dirichlet boundary conditions and, generically, we can completely describe all solutions of this partial differential equation.

Journal of Differential Equations, 1981

In this paper, we study the bifurcation of steady-state solutions of a reaction-diffusion equatio... more In this paper, we study the bifurcation of steady-state solutions of a reaction-diffusion equation in one space variable. The steady-state solutions satisfy the equation a" +f(u) = 0 on the interval-L < x < L, where we take f(u) to be a cubic polynomial. The solution is assumed to satisfy either homogeneous Dirichlet, or homogeneous Neumann, or periodic boundary conditions. We take as bifurcation parameter the number L, and we obtain global bifurcation diagrams; that is, we count the exact number of solutions. These solutions can be viewed as the "rest points" of the equation U, = u,, + f(u). In order to determine the global flow of this latter equation, the precise knowledge of the number of rest points is a necessary first step. Our technique is a careful analysis of the so-called "time-map" S(a), a function defined by an elliptic integral, which measures the "time" an orbit takes to get from one boundary line to another. The relevant point is that we are able to count the exact number of critical points of S. This is done for both positive and negative solutions of the Dirichlet problem,' by proving estimates of the form S" + cS' > 0, or (0, for some (nonzero) function c = c(a). For the Neumann problem, we use entirely different techniques to prove that S is never critical, for any cubic polynomial J This implies at once that the Neumann problem can have at most one nonconstant solution (having a given number of maxima or minima). This solution is necessarily strongly nondegenerate, in the sense that zero is not contained in the spectrum of the linearized operator (see [2]). For the Dirichlet problem, the situation is far more complicated, and the bifurcation diagrams undergo qualitative changes, depending on the positions of the roots off: For example, Fig. 1 shows the bifurcation diagram (for the positive solutions), for two different cubic functions of the form f(u) = (a-u)(u-b)(u-c).

Classical and Quantum Gravity, 2002