Geodesic
Solvability of the Dirichlet problem for an~inhomogeneous second-order elliptic equation
Sbornik. Mathematics, Tome 206 (2015) no. 10, pp. 1410-1439

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We consider a statement of the Dirichlet problem which generalizes the notions of classical and weak solutions, in which a solution belongs to the space of $(n-1)$-dimensionally continuous functions with values in the space $L_p$. The property of $(n-1)$-dimensional continuity is similar to the classical definition of uniform continuity; however, instead of the value of a function at a point, it looks at the trace of the function on measures in a special class, that is, elements of the space $L_p$ with respect to these measures. Up to now, the problem in the statement under consideration has not been studied in sufficient detail. This relates first to the question of conditions on the right-hand side of the equation which ensure the solvability of the problem. The main results of the paper are devoted to just this question. We discuss the terms in which these conditions can be expressed. In addition, the way the behaviour of a solution near the boundary depends on the right-hand side is investigated. Bibliography: 47 titles.
Keywords: Dirichlet problem, boundary value.
Mots-clés : elliptic equation 
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     author = {A. K. Gushchin},
     title = {Solvability of the {Dirichlet} problem for an~inhomogeneous second-order elliptic equation},
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A. K. Gushchin. Solvability of the Dirichlet problem for an~inhomogeneous second-order elliptic equation. Sbornik. Mathematics, Tome 206 (2015) no. 10, pp. 1410-1439. http://geodesic.mathdoc.fr/item/SM_2015_206_10_a2/