On the Solvability of Linear Differential Operators on Vector Bundles over a Manifold

封面

如何引用文章

全文:

开放存取 开放存取
受限制的访问 ##reader.subscriptionAccessGranted##
受限制的访问 订阅存取

详细

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.

作者简介

M. Smirnov

Lomonosov Moscow State University; Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences

编辑信件的主要联系方式.
Email: matsmir98@gmail.com
Moscow, 119991 Russia; Moscow, 119333 Russia

参考

  1. Sagraloff B. Normal solvability of linear partial differential operators in Cinfty(Omega) // Geometrical Approaches to Differential Equations. Lect. Not. in Math. 2006. V. 810. P. 290-305.
  2. Хермандер Л. Анализ линейных дифференциальных операторов. М., 1986.
  3. Duistermaat J.J., H"ormander L. Fourier integral operators. II // Acta Math. 1972. V. 128. № 3-4. P. 183-269.
  4. H"ormander L. Propagation of singularities and semiglobal existence theorems for (pseudo)differential operators of principal type // Ann. of Math. 1978. V. 108 (3). № 2. P. 569-609.
  5. Hounie J. A note on global solvability of vector fields // Proc. Amer. Math. Soc. 1985. V. 94. № 1. P. 61-64.
  6. Rauch J., Wigner D. Global solvability of the Casimir operator // Ann. of Math. 1976. V. 103. № 2. P. 229-236.
  7. Helgason S. Solvability of invariant differential operators on homogeneous manifolds // Centro Internaz. Mat. Estivo. 1975. P. 281-310.
  8. Ara'ujo G. Regularity and solvability of linear differential operators in Gevrey spaces // Math. Nachr. 2018. V. 291. № 5-6. P. 729-758.
  9. Bunke U., Olbrich M. Gamma-cohomology and the Selberg zeta function // J. Reine Angew. Math. 1995. V. 467. P. 199-219.
  10. Rinehart L. Elliptic operators on non-compact manifolds have closed range // https://arxiv.org/abs/ 2203.07534.
  11. Kazdan J.L. Unique continuation in geometry // Comm. Pure Appl. Math. 1988. V. 41. № 5. P. 667-681.
  12. Grosser M., Kunzinger M., Oberguggenberger M., Steinbauer R. Geometric Theory of Generalized Functions with Applications to General Relativity. Dordrecht, 2001.
  13. Grubb G. Distributions and Operators. New York, 2009.
  14. Шефер Х. Топологические векторные пространства. М., 1971.
  15. Palais R.S. Seminar on the Atiyah-Singer Index Theorem. Princeton, 1965.
  16. Уэллс Р. Дифференциальное исчисление на комплексных многообразиях. М., 1976.
  17. Шубин М.А. Псевдодифференциальные операторы и спектральная теория. М., 2005.
  18. Antoni'c N., Burazin K. On certain properties of spaces of locally Sobolev functions // Proc. of the Conf. on Appl. Math. and Sci. Comp. Dordrecht, 2005. P. 109-120.
  19. Aronszajn N. A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order // J. Math. Pures. Appl. 1957. V. 36. № 9. P. 235-249.
  20. Munkres J.R. Topology. Upper Saddle River, 2000.
  21. Kriegl A., Michor P.W. The Convenient Setting of Global Analysis. Providence, 1997.
  22. Petrowsky I.G. Sur l'analyticit'e des solutions des syst'emes d''equations diff'erentielles // Rec. Math. N. S. 1939. V. 5. № 47. P. 3-70.
  23. Форстер О. Римановы поверхности. М., 1980.
  24. Smirnov M. On the rate of polynomial approximations of holomorphic functions on convex compact sets // Complex Anal. Oper. Theory. 2023. V. 17. Art. 129.
  25. Lee J.M. Introduction to Smooth Manifolds. Graduate Texts in Math. V. 218. New York, 2013.

补充文件

附件文件
动作
1. JATS XML
下载

版权所有 © Russian Academy of Sciences, 2023