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On solvability of the Dirichlet problem with the boundary function in L 2 for a second-order elliptic equation

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Abstract

We consider Dirichlet problems in a bounded domain Q ⊂ R n for a general secondorder elliptic equation with the boundary function in L 2. In the author’s previous papers necessary and sufficient conditions for the existence of an (n − 1)-dimensionally continuous solution were obtained under some natural assumptions on the coefficients of equation. Those assumptions are formulated in terms of an auxiliary operator equation in a special Hilbert space and are difficult to verify. In the present paper we obtain necessary and sufficient conditions for the existence of a solution in terms of the original problem for a more narrow class of the right-hand sides. It is shown that if, in addition, the boundary function is assumed to be in the space W 1/22 (∂Q), then the obtained conditions transform into solvability conditions in the space W 12 (Q).

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Authors and Affiliations

  1. Yerevan State University, Yerevan, Armenia

    V. Zh. Dumanyan

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  1. V. Zh. Dumanyan
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Correspondence to V. Zh. Dumanyan.

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Original Russian Text © V. Zh. Dumanyan, 2015, published in Izvestiya NAN Armenii. Matematika, 2015, No. 4, pp. 3–22.

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Dumanyan, V.Z. On solvability of the Dirichlet problem with the boundary function in L 2 for a second-order elliptic equation. J. Contemp. Mathemat. Anal. 50, 153–166 (2015). https://doi.org/10.3103/S1068362315040019

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  • DOI: https://doi.org/10.3103/S1068362315040019

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