{"title":"On the Solvability of Linear Differential Operators on Vector Bundles over a Manifold","authors":"M. S. Smirnov","doi":"10.1134/s0012266123120078","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A necessary and sufficient condition is established for the closedness of the range or\nsurjectivity of a differential operator acting on smooth sections of vector bundles. For connected\nnoncompact manifolds it is shown that these conditions are derived from the regularity conditions\nand the unique continuation property of solutions. An application of these results to elliptic\noperators (more precisely, to operators with a surjective principal symbol) with analytic\ncoefficients, to second-order elliptic operators on line bundles with a real leading part, and to the\nHodge–Laplace–de Rham operator is given. It is shown that the top de Rham (respectively,\nDolbeault) cohomology group on a connected noncompact smooth (respectively, complex-analytic)\nmanifold vanishes. For elliptic operators, we prove that solvability in smooth sections implies\nsolvability in generalized sections.\n</p>","PeriodicalId":50580,"journal":{"name":"Differential Equations","volume":"39 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0012266123120078","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A necessary and sufficient condition is established for the closedness of the range or
surjectivity of a differential operator acting on smooth sections of vector bundles. For connected
noncompact manifolds it is shown that these conditions are derived from the regularity conditions
and the unique continuation property of solutions. An application of these results to elliptic
operators (more precisely, to operators with a surjective principal symbol) with analytic
coefficients, to second-order elliptic operators on line bundles with a real leading part, and to the
Hodge–Laplace–de Rham operator is given. It is shown that the top de Rham (respectively,
Dolbeault) cohomology group on a connected noncompact smooth (respectively, complex-analytic)
manifold vanishes. For elliptic operators, we prove that solvability in smooth sections implies
solvability in generalized sections.
期刊介绍：
Differential Equations is a journal devoted to differential equations and the associated integral equations. The journal publishes original articles by authors from all countries and accepts manuscripts in English and Russian. The topics of the journal cover ordinary differential equations, partial differential equations, spectral theory of differential operators, integral and integral–differential equations, difference equations and their applications in control theory, mathematical modeling, shell theory, informatics, and oscillation theory. The journal is published in collaboration with the Department of Mathematics and the Division of Nanotechnologies and Information Technologies of the Russian Academy of Sciences and the Institute of Mathematics of the National Academy of Sciences of Belarus.
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