13 Novembre, 2015 11:00 in punto

Sezione di Analisi

On the Dirichlet problem of mixed type for lower hybrid waves in axisymmetric cold plasmas

Dario D. Monticelli, Politecnico di Milano

Aula seminari III piano

In this talk we will describe some recent results concerning the Dirichlet problem for a class of second order differential equations of mixed elliptic-hyperbolic type on suitable bounded domains of R^2, which is used as a model to describe possible heating in axisymmetric cold plasmas subjected to high frequency electromagnetic waves near certain frequencies. The presence of hyperbolicity makes the problem overdetermined for classical solutions. We will show that the problem is well-posed for weak solutions belonging to a weighted version of the classical Sobolev space H^1_0, when the datum is chosen in a suitable weighted L^2 space. We will also provide a complete spectral theory for the Dirichlet problem in the setting of weighted Lebesgue and Sobolev spaces, with some applications to semilinear equations and to equations with lower order terms. Finally we will give a variational characterisation of weak solutions, which are shown to be saddle points of an associated strongly indefinite functional.

These results are joint work with D. Lupo (Politecnico di Milano) and K.R. Payne (Università degli Studi di Milano).