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

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Abstract

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.

About the authors

M. S. Smirnov

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

Author for correspondence.
Email: matsmir98@gmail.com
Moscow, 119991 Russia; Moscow, 119333 Russia

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