Geodesic
A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis, mathematical physics, and applications, Tome 301 (2018), pp. 53-73

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The paper is devoted to the study of the boundary behavior of solutions to a second-order elliptic equation. A criterion is established for the existence in $L_p$, $p>1$, of a boundary value of a solution to a homogeneous equation in the self-adjoint form without lower order terms. Under the conditions of this criterion, the solution belongs to the space of $(n-1)$-dimensionally continuous functions; thus, the boundary value is taken in a much stronger sense. Moreover, for such a solution to the Dirichlet problem, estimates for the nontangential maximal function and for an analog of the Lusin area integral hold.
Mots-clés : elliptic equation
Keywords: boundary value, Dirichlet problem, Lusin area integral. 
@article{TM_2018_301_a4,
     author = {A. K. Gushchin},
     title = {A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {53--73},
     publisher = {mathdoc},
     volume = {301},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2018_301_a4/}
}
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A. K. Gushchin. A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis, mathematical physics, and applications, Tome 301 (2018), pp. 53-73. http://geodesic.mathdoc.fr/item/TM_2018_301_a4/