Maths Seminar (24 May)

[Math]

M A T H E M A T I C S   S E M I N A R S   2006




Wednesday, 24 May at 14:30 hrs.

Professor Dr. Yuri Bozkhov
(IMECC, State University of Campinas, S.P., Brazil)

Noether symmetries of critical
nonlinear differential equations

Abstract: 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.


VENUE:  SEMINAR ROOM
(Main Building, first floor)