Geodesic
$L_p$-estimates for solutions of second-order elliptic equation Dirichlet problem
Teoretičeskaâ i matematičeskaâ fizika, Tome 174 (2013) no. 2, pp. 243-255

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For solutions of the Dirichlet problem for a second-order elliptic equation, we establish an analogue of the Carleson theorem on $L_p$-estimates. Under the same conditions on the coefficients for which the unique solvability of the considered problem is known, we prove this criterion for the validity of estimate of the solution norm in the space $L_p$ with a measure. We require their Dini continuity on the boundary, but we assume only their measurability and boundedness in the domain under consideration.
Mots-clés : elliptic equation, nontangent maximal function
Keywords: Dirichlet problem, boundary value, Carleson measure. 
@article{TMF_2013_174_2_a4,
     author = {A. K. Gushchin},
     title = {$L_p$-estimates for solutions of second-order elliptic equation {Dirichlet} problem},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {243--255},
     publisher = {mathdoc},
     volume = {174},
     number = {2},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2013_174_2_a4/}
}
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A. K. Gushchin. $L_p$-estimates for solutions of second-order elliptic equation Dirichlet problem. Teoretičeskaâ i matematičeskaâ fizika, Tome 174 (2013) no. 2, pp. 243-255. http://geodesic.mathdoc.fr/item/TMF_2013_174_2_a4/