Geodesic
The boundary values of solutions of an elliptic equation
Sbornik. Mathematics, Tome 210 (2019) no. 12, pp. 1724-1752

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The paper is devoted to the study of the boundary behaviour of solutions of a second-order elliptic equation. Criteria are established for the existence of a boundary value of a solution of the homogeneous equation under the same conditions on the coefficients of the equation as were used to establish that the Dirichlet problem with a boundary function in $L_p$, $p>1$, has a unique solution. In particular, an analogue of Riesz's well-known theorem (on the boundary values of an analytic function) is proved: if a family of norms in the space $L_p$ of the traces of a solution on surfaces ‘parallel’ to the boundary is bounded, then this family of traces converges in $L_p$. This means that the solution of the equation under consideration is a solution of the Dirichlet problem with a certain boundary value in $L_p$. Estimates of the nontangential maximal function and of an analogue of the Luzin area integral hold for such a solution, which make it possible to claim that the boundary value is taken in a substantially stronger sense. Bibliography: 57 titles.
Keywords: boundary value, Dirichlet problem.
Mots-clés : elliptic equation 
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     title = {The boundary values of solutions of an elliptic equation},
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A. K. Gushchin. The boundary values of solutions of an elliptic equation. Sbornik. Mathematics, Tome 210 (2019) no. 12, pp. 1724-1752. http://geodesic.mathdoc.fr/item/SM_2019_210_12_a3/