Topological properties of the range of a second order nonlinear operator

PABLO AMSTER; MARIEL PAULA KUNA

Barcelona

Workshop; Ninth Jornades d' Interacció entre Sistemes Dinamics i a les EDP'; 2011

Universidad Politécnica de Catalunya

Topological properties of the range of a second order nonlinear operator Pablo Amster and M. Paula Kuna Departamento de Matematica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Argentina. Consejo Nacional de Investigaciones Cientcas y Tecnicas, Argentina. pamster@dm.uba.ar | mpkuna@dm.uba.ar Abstract Let us consider the following system for a vector function u : [0; T] R ! R n satisfying u 00 + rG(u) = p(t) u(0) = u(T); u 0 (0) = u 0 (T); (1) where G : R n ! R is a C 2 function. We are interested in the problem of nding the set of all possible values of p 2 L 2 ((0; T); R n ) such that (1) has at least one solution. In other words, we shall study the range of the semilinear operator S : H2 per ! L 2 ((0; T); R n ) given by Su = u 00 + rG(u); where H2 per = u 2 H2 ((0; T); R n )= u(0) = u(T); u 0 (0) = u 0 (T) : Writing p(t) = p + ~p(t), where p := 1 T R T 0 p(t) dt, we shall present several results concerning the topological structure of the set I(~p) = fp 2 R n =p + ~p 2 Im(S)g :