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Solvability of the Dirichlet problem for second-order elliptic equations

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Abstract

In our preceding papers, we obtained necessary and sufficient conditions for the existence of an (n−1)-dimensionally continuous solution of the Dirichlet problem in a bounded domain Q ⊂ ℝ n under natural restrictions imposed on the coefficients of the general second-order elliptic equation, but these conditions were formulated in terms of an auxiliary operator equation in a special Hilbert space and are difficult to verify. We here obtain necessary and sufficient conditions for the problem solvability in terms of the initial problem for a somewhat narrower class of right-hand sides of the equation and also prove that the obtained conditions become the solvability conditions in the space W 2 1 (Q) under the additional requirement that the boundary function belongs to the space W 2 1/2 (∂Q).

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Correspondence to V. Zh. Dumanyan.

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__________

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 180, No. 2, pp. 189–205, August, 2014.

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Dumanyan, V.Z. Solvability of the Dirichlet problem for second-order elliptic equations. Theor Math Phys 180, 917–931 (2014). https://doi.org/10.1007/s11232-014-0188-4

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Keywords

  • Dirichlet problem
  • elliptic equation