Geodesic
Solvability of the~Dirichlet problem for second-order elliptic equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 180 (2014) no. 2, pp. 189-205

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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 $Q\subset\mathbb R_n$ 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 $W_2^{1/2}(\partial Q)$.
Keywords: Dirichlet problem
Mots-clés : elliptic equation. 
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     author = {V. Zh. Dumanyan},
     title = {Solvability of {the~Dirichlet} problem for second-order elliptic equations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {189--205},
     publisher = {mathdoc},
     volume = {180},
     number = {2},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2014_180_2_a2/}
}
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V. Zh. Dumanyan. Solvability of the~Dirichlet problem for second-order elliptic equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 180 (2014) no. 2, pp. 189-205. http://geodesic.mathdoc.fr/item/TMF_2014_180_2_a2/