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TMF, 2013, Volume 174, Number 2, Pages 243–255 (Mi tmf8410)  

This article is cited in 8 scientific papers (total in 8 papers)

$L_p$-estimates for solutions of second-order elliptic equation Dirichlet problem

A. K. Gushchin

Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia

Abstract: 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.

Keywords: elliptic equation, Dirichlet problem, boundary value, nontangent maximal function, Carleson measure

Funding Agency Grant Number
Russian Foundation for Basic Research 10-01-00178_а
Ministry of Education and Science of the Russian Federation НШ-2928.2012.1


DOI: https://doi.org/10.4213/tmf8410

Full text: PDF file (458 kB)
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English version:
Theoretical and Mathematical Physics, 2013, 174:2, 209–219

Bibliographic databases:

Received: 10.09.2012

Citation: A. K. Gushchin, “$L_p$-estimates for solutions of second-order elliptic equation Dirichlet problem”, TMF, 174:2 (2013), 243–255; Theoret. and Math. Phys., 174:2 (2013), 209–219

Citation in format AMSBIB
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  • http://mi.mathnet.ru/eng/tmf/v174/i2/p243

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. K. Guschin, “$L_p$-otsenki nekasatelnoi maksimalnoi funktsii dlya reshenii ellipticheskogo uravneniya vtorogo poryadka”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(30) (2013), 53–69  mathnet  crossref
    2. V. Zh. Dumanyan, “Solvability of the Dirichlet problem for second-order elliptic equations”, Theoret. and Math. Phys., 180:2 (2014), 917–931  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    3. A. K. Guschin, “O zadache Dirikhle dlya ellipticheskogo uravneniya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 19:1 (2015), 19–43  mathnet  crossref  zmath  elib
    4. A. K. Gushchin, “V.A. Steklov's work on equations of mathematical physics and development of his results in this field”, Proc. Steklov Inst. Math., 289 (2015), 134–151  mathnet  crossref  crossref  isi  elib
    5. A. K. Gushchin, “Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation”, Sb. Math., 206:10 (2015), 1410–1439  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. A. K. Gushchin, “$L_p$-estimates for the nontangential maximal function of the solution to a second-order elliptic equation”, Sb. Math., 207:10 (2016), 1384–1409  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    7. A. K. Gushchin, “The Luzin area integral and the nontangential maximal function for solutions to a second-order elliptic equation”, Sb. Math., 209:6 (2018), 823–839  mathnet  crossref  crossref  adsnasa  isi  elib
    8. A. K. Gushchin, “A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation”, Proc. Steklov Inst. Math., 301 (2018), 44–64  mathnet  crossref  crossref  isi  elib  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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