Bilinear eigenfunction estimates and the nonlinear Schr�dinger equation on surfaces

2003/08/22 by Nicolas Burq, N. Burq, P. Gerard +4 · 164 citations
Mathematics · #Advanced Mathematical Physics Problems #Bilinear interpolation #Eigenfunction #Eigenvalues and eigenvectors #Geometry #Laplace operator #Laplace transform #Mathematical Analysis and Transform Methods #Mathematical analysis #Mathematical physics #Mathematics #Physics #Quantum mechanics #Spectral Theory in Mathematical Physics #Surface (topology) #math.AP #math.SP

paper · pdf · doi:10.1007/s00222-004-0388-x

published in Inventiones mathematicae 159(1), 187-223 (Springer Science+Business Media) · 34 pages, 2 figures

arxiv created 2003/08/22 · openalex publication_date 2004/07/21 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We study the cubic non linear Schrödinger equation (NLS) on compact surfaces. On the sphere \mathbbS² and more generally on Zoll surfaces, we prove that, for s>1/4, NLS is uniformly well-posed in H^s, which is sharp on the sphere. The main ingredient in our proof is a sharp bilinear estimate for Laplace spectral projectors on compact surfaces. On étudie l'équation de Schrödinger non linéaire (NLS) sur une surface compacte.Sur la sphère \mathbbS² et plus généralement sur toute surface de Zoll, on démontre que pour s>1/4, NLS est uniformément bien posée dans H^s, ce qui est optimalsur la sphère. Le principal ingrédient de notre démonstration est une estimation bilinéaire pour les projecteurs spectraux du laplacien sur une surface compacte.

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