Abstract
We propose a new approach to investigation of solvability of the Dirichlet problem for differential equations of infinite order. Namely, by using the embedding theorems obtained by the author in previous papers for the energy spaces, corresponding to operators of infinite order, the initial differential operator of infinite order is expressed as a sum of the main and subordinate operators of infinite order. The conditions, under which the above Dirichlet problems are soluble, are established by using the main term of the corresponding differential operator.
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Dubinskij, Ju. A. Sobolev Spaces of Infinite Order and Differential Equations (Leipzig, 1986).
Dubinskii, Yu. A. “Sobolev Spaces of Infinite Order”, Russian Math. Surveys 46, No. 6, 107–147 (1991).
Balashova, G. S. and Dubinskii, Ju. A. “Uniform Well-posedness of a Family of Nonlinear Boundary Value Problems of Infinite Order”, Differential Equations 30 (4), 559–569 (1994).
Balashova, G. S. “Embedding Theorems for Banach Spaces of Infinitely Differentiable Functions of Several Variables”, Math.Notes 47, No. 5–6, 525–533 (1990).
Balashova, G. S. “Conditions for the Extension of a Trace and an Embedding for Banach Spaces of Infinitely Differentiable Functions”, Russian Acad. Sci. Sb.Math. 78, No. 1, 91–112 (1994).
Mandelbrojt, S. Séries adhérentes, Régularisation des suites, Applications (Gauthier-Villars, Paris, 1952; Inost. Lit.,Moscow, 1955).
Vainberg, M. M. Variation Method and Method of Monotone Operators (Nauka, Moscow, 1972) [in Russian].
Vladimirov, V. S. Equations of Mathematical Physics (Nauka, Moscow, 1976) [in Russian].
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Original Russian Text © G.S. Balashova, 2018, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2018, No. 4, pp. 16–20.
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Balashova, G.S. On Fredholm Solvability of the Dirichet Problem for Linear Differential Equations of Infinite Order. Russ Math. 62, 13–17 (2018). https://doi.org/10.3103/S1066369X18040023
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DOI: https://doi.org/10.3103/S1066369X18040023