We investigate the dispersive properties of evolution equations on waveguides with a non-flat shape. More precisely, we consider an operator H=-Delta(x)-Delta(y)+V(x,y) with Dirichlet boundary conditions on an unbounded domain Omega, and we introduce the notion of a repulsive waveguide along the direction of the first group of variables, x. If Omega is a repulsive waveguide, we prove a sharp estimate for the Helmholtz equation Hu-lambda u = f. As consequences, we prove smoothing estimates for the Schrodinger and wave equations associated to H, and Strichartz estimates for the Schrodinger equation. Additionally, we deduce that the operator H does not admit eigenvalues.
Evolution Equations on Non-Flat Waveguides / D'Ancona, Piero Antonio; Reinhard, Racke. - In: ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS. - ISSN 0003-9527. - STAMPA. - 206:1(2012), pp. 81-110. [10.1007/s00205-012-0524-5]
Evolution Equations on Non-Flat Waveguides
D'ANCONA, Piero Antonio;
2012
Abstract
We investigate the dispersive properties of evolution equations on waveguides with a non-flat shape. More precisely, we consider an operator H=-Delta(x)-Delta(y)+V(x,y) with Dirichlet boundary conditions on an unbounded domain Omega, and we introduce the notion of a repulsive waveguide along the direction of the first group of variables, x. If Omega is a repulsive waveguide, we prove a sharp estimate for the Helmholtz equation Hu-lambda u = f. As consequences, we prove smoothing estimates for the Schrodinger and wave equations associated to H, and Strichartz estimates for the Schrodinger equation. Additionally, we deduce that the operator H does not admit eigenvalues.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.