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Mat. Sb. (N.S.), 1969, Volume 80(122), Number 4(12), Pages 455–491 (Mi msb3637)  

This article is cited in 23 scientific papers (total in 23 papers)

Boundary value problems for elliptic equations degenerate on the boundary of a domain

M. I. Vishik, V. V. Grushin


Abstract: We investigate the elliptic equation $Lu=f$ of order $2m$, degenerate on the boundary $\Gamma$ of a bounded domain $G$. In local coordinates $(x_1,…,x_n)$, introduced in a neighborhood $U(x_0)$ of the point $x_0\in\Gamma$ and in which $\Gamma\cap U(x_0)$ is given by $x_n=0$, the operator
$$ L(x;x_n;D^\alpha)=\sum_{|\alpha|\leqslant m}\alpha_\alpha(x)x_n^{l_\alpha}D^\alpha, $$
where $l_\alpha=\max(0,q\alpha_n+q'\alpha'-qr)$, $q\geqslant1$, $q'\geqslant0$. For $x_n=0$ the operator $Lu$ degenerates into the quasi-elliptic operator
$$ L_1u=\sum_{\frac rr'|\alpha'|+\alpha_n\leqslant r}\alpha_\alpha(x)D^\alpha\qquad(|\alpha'|\leqslant r'\quad(qr=q'r')). $$

In particular we study the case of transition, for $x_n=0$, of an elliptic operator into a parabolic operator.
Figures: 3.
Bibliography: 19 titles.

Full text: PDF file (3914 kB)
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English version:
Mathematics of the USSR-Sbornik, 1969, 9:4, 423–454

Bibliographic databases:

UDC: 517.946.9
MSC: 35J25, 35Sxx, 35J70
Received: 03.06.1969

Citation: M. I. Vishik, V. V. Grushin, “Boundary value problems for elliptic equations degenerate on the boundary of a domain”, Mat. Sb. (N.S.), 80(122):4(12) (1969), 455–491; Math. USSR-Sb., 9:4 (1969), 423–454

Citation in format AMSBIB
\Bibitem{VisGru69}
\by M.~I.~Vishik, V.~V.~Grushin
\paper Boundary value problems for elliptic equations degenerate on the boundary of a~domain
\jour Mat. Sb. (N.S.)
\yr 1969
\vol 80(122)
\issue 4(12)
\pages 455--491
\mathnet{http://mi.mathnet.ru/msb3637}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=257562}
\zmath{https://zbmath.org/?q=an:0186.17101|0202.11402}
\transl
\jour Math. USSR-Sb.
\yr 1969
\vol 9
\issue 4
\pages 423--454
\crossref{https://doi.org/10.1070/SM1969v009n04ABEH002055}


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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. V. V. Grushin, “On a class of hypoelliptic operators”, Math. USSR-Sb., 12:3 (1970), 458–476  mathnet  crossref  mathscinet  zmath
    2. M. I. Vishik, V. V. Grushin, “Degenerating elliptic differential and psevdo-differential operators”, Russian Math. Surveys, 25:4 (1970), 21–50  mathnet  crossref  mathscinet  zmath
    3. V. V. Grushin, “Pseudodifferential operators on $\mathbf R^n$ with bounded symbols”, Funct. Anal. Appl., 4:3 (1970), 202–212  mathnet  crossref  mathscinet  zmath
    4. V. V. Grushin, “On a class of elliptic pseudodifferential operators degenerate on a submanifold”, Math. USSR-Sb., 13:2 (1971), 155–185  mathnet  crossref  mathscinet  zmath
    5. V. V. Grushin, “Hypoelliptic differential equations and pseudodifferential operators with operator-valued symbols”, Math. USSR-Sb., 17:4 (1972), 497–514  mathnet  crossref  mathscinet  zmath
    6. Pham The Lai, “Probleme de dirichelt dans un cône avec parametre spectral pour une classe d'espaces de sobolev a poids”, Communications in Partial Differential Equations, 4:4 (1979), 389  crossref
    7. S. Z. Levendorskii, “Boundary value problems in a half-space for quasielliptic pseudodifferential operators degenerating on the boundary”, Math. USSR-Sb., 39:4 (1981), 429–447  mathnet  crossref  mathscinet  zmath  isi
    8. V. V. Katrakhov, “General boundary value problems for a class of singular and degenerate elliptic equations”, Math. USSR-Sb., 40:3 (1981), 325–347  mathnet  crossref  mathscinet  zmath  isi
    9. Slutski A., “On the Asymptotics of Solutions of Degenerative Elliptic-Equations with a Small Parameter at Higher Derivatives”, no. 3, 1981, 59–64  isi
    10. S. Z. Levendorskii, “Asymptotic distribution of eigenvalues”, Math. USSR-Izv., 21:1 (1983), 119–160  mathnet  crossref  mathscinet  zmath
    11. Paul Godin, “Perturbations of pseudo-differential operators with double characteristics of constant multiplicity”, Journal of Differential Equations, 50:1 (1983), 20  crossref
    12. A. I. Karol', “The zeta-function of a degenerate elliptic operator”, Math. USSR-Sb., 52:1 (1985), 209–230  mathnet  crossref  mathscinet  zmath
    13. S. Z. Levendorskii, “On the symbols of degenerate elliptic differential and hypoelliptic pseudodifferential operators”, Math. USSR-Izv., 32:3 (1989), 543–561  mathnet  crossref  mathscinet  zmath
    14. S. Z. Levendorskii, “On types of degenerate elliptic operators”, Math. USSR-Sb., 66:2 (1990), 523–540  mathnet  crossref  mathscinet  zmath  isi
    15. Manuel Núñez, Luis A. Tristán, “A Singular Transmission Problem: Alfvenic Resonance in Plasmas”, Math Meth Appl Sci, 20:11 (1997), 891  crossref  mathscinet  zmath
    16. Kyeong-Hun Kim, “Sobolev Space Theory of Parabolic Equations Degenerating on the Boundary of C1 Domains”, Comm. in Partial Differential Equations, 32:8 (2007), 1261  crossref
    17. Kyeong-Hun Kim, “ L p -Theory of Parabolic SPDEs Degenerating on the Boundary of C 1 Domains”, J Theoret Probab, 21:1 (2008), 169  crossref  mathscinet  zmath  isi
    18. P. V. Sadchikov, A. D. Baev, “O nekotorykh kraevykh zadachakh v poluprostranstve dlya odnogo klassa psevdodifferentsialnykh uravnenii s vyrozhdeniem”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 10:2 (2010), 34–41  mathnet  elib
    19. A. D. Baev, S. S. Buneev, “A priori estimates of solutions of boundary value problems in a band for a class of degenerate elliptic equation of higher order”, Russian Math. (Iz. VUZ), 56:7 (2012), 44–46  mathnet  crossref  mathscinet
    20. A. D. Baev, S. S. Buneev, “Teorema o suschestvovanii i edinstvennosti resheniya odnoi kraevoi zadachi v polose dlya vyrozhdayuschegosya ellipticheskogo uravneniya vysokogo poryadka”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 12:3 (2012), 8–17  mathnet
    21. Baev A.D., Buneev S.S., “Apriornaya otsenka reshenii odnoi kraevoi zadachi v polose dlya vyrozhdayuschegosya ellipticheskogo uravneniya vysokogo poryadka”, Vestnik voronezhskogo gosudarstvennogo universiteta. seriya: fizika. matematika, 2012, no. 1, 81–81  elib
    22. B. V. Bazalii, S. P. Degtyarev, “A boundary-value problem in weighted Hölder spaces for elliptic equations which degenerate at the boundary of the domain”, Sb. Math., 204:7 (2013), 958–978  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    23. V. V. Katrakhov, S. M. Sitnik, “Metod operatorov preobrazovaniya i kraevye zadachi dlya singulyarnykh ellipticheskikh uravnenii”, Singulyarnye differentsialnye uravneniya, SMFN, 64, no. 2, Rossiiskii universitet druzhby narodov, M., 2018, 211–426  mathnet  crossref
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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