Abstract
In this paper we prove a variant of the Stokes formula for differential forms of a finite codimension in a locally convex space (LCS). The main tool used by us for proving the mentioned formula is the surface layer theorem for surfaces of codimension 1 in a locally convex space which was proved earlier by the first author. Moreover, on some subspace of differential forms of the Sobolev type with respect to a differentiable measure we establish a formula expressing the operator adjoint to the exterior differential via standard operations of the calculus of differential forms and the logarithmic derivative. This connection was established earlier under stronger constraints imposed either on the LCS or on the measure, or on differential forms (the smoothness condition).
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References
Smolyanov, O. G. “De Rham Currents and Stokes’ Formula in a Hilbert Space”, Sov. Math., Dokl. 33, 140–144 (1986).
von Weizsäcker, H. and Smolyanov, O. G. “Differential Forms on Infinite-Dimensional Spaces and an Axiomatic Approach to the Stokes Formula”, Dokl. Akad. Nauk 367, No. 2, 151–154 (1999) [in Russian].
von Weizsäcker, H., Lándre, R., and Smolyanov, O. G. “Algebraic Properties of Infinite-Dimensional Differential Forms of Finite Codegree”, Dokl. Akad. Nauk 369 (6), 727–731 (1999) [in Russian].
Shamarova, E. Yu. “On the Approximation of Surface Measures in a Locally Convex Space”, Math. Notes 72, No. 3–4, 551–568 (2002).
Uglanov, A. V. “Surface Integrals in a Banach Space”, Mat. Sb. (N. S.) 110(152), No. 2, 189–217 (1979).
Cartan, H. Differential Calculus. Differential Forms (Nauka, Moscow, 1970) [Russ. transl.].
Sternberg, S. Lectures on Differential Geometry (Prentice-Hall, Englewood Cliffs, N. J., 1964; Mir, Moscow, 1970).
Diestel, J. and Uhl, J.J. Vector Measures (Mathematical Surveys and Monographs. Vol. 15. Amer. Math. Soc., 1977).
Averbukh, V. I., Smolyanov, O. G. and Fomin, S. V. “Generalized Functions and Differential Equations in Linear Spaces I. Differentiable Measures”, Tr.MMO, 24, 133–174 (1971) [in Russian].
Smolyanov, O. G. and von Weizsäcker, H. “Differentiable Families ofMeasures”, J. Funct. Anal. 118, No. 2, 454–476 (1993).
Uglanov, A. V. “Surface Integrals in Linear Topological Spaces”, Dokl. Akad. Nauk 344 (4), 450–453 (1995) [in Russian].
Uglanov, A. V. Integration on Infinite-Dimensional Surfaces and Its Applications (Math. and Appl., Kluwer Academic Publ., 2000).
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Original Russian Text © E.Yu. Shamarova, N.N. Shamarov, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 8, pp. 84–97.
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Shamarova, E.Y., Shamarov, N.N. Differential forms on locally convex spaces and the stokes formula. Russ Math. 60, 74–85 (2016). https://doi.org/10.3103/S1066369X16080090
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DOI: https://doi.org/10.3103/S1066369X16080090