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 Mat. Sb. (N.S.), 1971, Volume 85(127), Number 4(8), Pages 586–609 (Mi msb3280)

On positive solutions of elliptic equations

T. G. Pletneva, S. D. Èidel'man, V. A. Kondrat'ev

Abstract: In this paper the authors study weak solutions of elliptic equations of the form
$$Pu\equiv\sum_{|k|\leqslant m}(-1)^kD_x^k(a_k(x)u(x))=f(x)$$
in a bounded domain $\Omega$. It is assumed known about these solutions either that they are positive, or that estimates in certain norms hold for their negative parts. It is assumed moreover that an estimate on the $L_1$-norm of the solution holds on some subdomain $\Omega'\subset\Omega$. Summability of such solutions with a weight function that vanishes at the boundary is established, and with the use of the results of Ya. A. Roitberg integral representations are given in terms of the Green's function for the Dirichlet problem.
Bibliography: 8 titles.

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English version:
Mathematics of the USSR-Sbornik, 1971, 14:4, 587–613

Bibliographic databases:

UDC: 517.946
MSC: Primary 35J30; Secondary 35C15, 35D10

Citation: T. G. Pletneva, S. D. Èidel'man, V. A. Kondrat'ev, “On positive solutions of elliptic equations”, Mat. Sb. (N.S.), 85(127):4(8) (1971), 586–609; Math. USSR-Sb., 14:4 (1971), 587–613

Citation in format AMSBIB
\Bibitem{PleEidKon71} \by T.~G.~Pletneva, S.~D.~\Eidel'man, V.~A.~Kondrat'ev \paper On positive solutions of elliptic equations \jour Mat. Sb. (N.S.) \yr 1971 \vol 85(127) \issue 4(8) \pages 586--609 \mathnet{http://mi.mathnet.ru/msb3280} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=283372} \zmath{https://zbmath.org/?q=an:0222.35019} \transl \jour Math. USSR-Sb. \yr 1971 \vol 14 \issue 4 \pages 587--613 \crossref{https://doi.org/10.1070/SM1971v014n04ABEH002823} `

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. V. A. Kondrat'ev, T. G. Pletneva, S. D. Èidel'man, “Positive solutions of linear quasielliptic evolution equations”, Math. USSR-Sb., 18:1 (1972), 15–44
2. V. A. Kondrat'ev, S. D. Èidel'man, “On the summability of positive solutions of partial differential equations of arbitrary order in a neighborhood of a characteristic manifold”, Math. USSR-Sb., 28:4 (1976), 521–531
3. Khung N., “The Absence of Positive Solutions of Second-Order Nonlinear Elliptic Equations in Conical Domains”, Differ. Equ., 34:4 (1998), 532–539
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