2005/01/25 by Detlef Mueller, Mueller, Detlef
Mathematics · Physics and Astronomy · #14P05 #35A07 #43A80 #Advanced Algebra and Geometry #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Waves and Solitons #math.AP #math.CA #msc:14P05 #msc:35A07 #msc:43A80
paper · pdf · doi:10.48550/arxiv.math/0501452
46 pages; replaces an erroneously uploaded file by the correct one
arxiv created 2005/01/25 · openalex publication_date 2005/01/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic second order differential operators. For a large class of such operators, we show that local solvability at a given point implies "essential dissipativity" of the operator at this point. By means of Hoermander's classical necessary condition for local solvability, the proof is reduced to the following question, whose answer forms the core of the paper: Suppose that QA and QB are two real quadratic forms on a finite dimensional symplectic vector space, and let QC:=\QA,QB\ be given by the Poisson bracket of QA and QB. Then QC is again a quadratic form, and we may ask: When can we find a common zero of QA and QB at which QC does not vanish? The second paper, in combination with the first one, will give a fairly comprehensive picture of what rules local solvability of invariant second order operators on the Heisenberg group.