Category: List of Bases 2006 Throughout the year RESEARCH IN MATHEMATICS

Title:

Noether symmetries of critical nonlinear differential equations.

Start Time:

14:30

Location:

ICTP (Main Building Seminar Room)

Contact:

A. Bergamo

Speaker(s):

Yuri Bozhkov (IMECC, State University of Campinas, Brazil)

Description:

The purpose of this talk is to discuss a common property of certain classes of quasilinear and semilinear differential equations, namely: "a Lie point symmetry of the considered equation is a Noether symmetry if and only if the equation parameters assume critical values".
By "Noether symmetry" we mean a variational or a divergence symmetry. As it is well known, the so-called critical exponent is found as the critical power for embedding theorems. It is also related to some numbers dividing the existence and nonexistence cases for the solutions of differential equations, in particular of semilinear differential equations involving the Laplace operator. We shall consider the Sobolev case and the Pohozhaev-Trudinger case for some second order ODE: the general class studied by Clement-de Figueiredo-Mitidieri, involving the radial forms of PDEs containing the Laplace, p-Laplace and k-Hessian opertors, the Lane-Emden equation, the Emden-Fowler equation, the Boltzmann equation; the Lane-Emden system, and PDE: the semilinear polyharmonic equation in R^n and the Kohn-Laplace equation on the Heisenberg group H^1.
Since the Noether symmetry allows to reduce by 2 the integration procedure of an ODE one can find explicitly the solutions corresponding to the Sobolev and Pohozhaev-Trudinger cases. Regarding PDE, we find via the Noether Theorem conservation laws for the critical equations.