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On solvability of the Dirichlet problem with the boundary function in \(L_2\) for a second-order elliptic equation. (English. Russian original) Zbl 1336.35137

J. Contemp. Math. Anal., Armen. Acad. Sci. 50, No. 4, 153-166 (2015); translation from Izv. Nats. Akad. Nauk Armen., Mat. 50, No. 4, 3-22 (2015).
Summary: We consider Dirichlet problems in a bounded domain \(Q \subset \mathbb R_n\) for a general second-order 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 the 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 \(W_2^{1/2}(\partial Q)\), then the obtained conditions transform into solvability conditions in the space \(W_2^1(Q)\).

MSC:

35J25 Boundary value problems for second-order elliptic equations
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