×

zbMATH — the first resource for mathematics

Vishik-Lyusternik method for elliptic boundary-value problems in domains with conical point. III. Problem with degeneracy at a conical point. (English. Russian original) Zbl 0593.35049
Sib. Math. J. 25, 917-925 (1984); translation from Sib. Mat. Zh. 25, No. 6(148), 106-115 (1984).
[For part II see ibid. 22, 753-769 (1982); translation from Sib. Math. Zh. 22, 132-152 (1981; Zbl 0479.35033).]
The author studies elliptic boundary value problems degenerating at a conical point of the boundary into a boundary value problem of lower order. A modified Vishik-Lyusternik method is used for the construction of a boundary layer which is induced by the degeneracy. The problem is formulated as an operator equation in certain weighted spaces. Necessary and sufficient conditions are given under which Noether theorems for the corresponding operator hold.
Reviewer: M.Kučera

MSC:
35J70 Degenerate elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] V. A. Kondrat’ev, ?Boundary-value problems for elliptic equations in domains with conical or corner points,? Tr. Mosk. Mat. Ob-va,16, 209-292 (1967).
[2] A. Pazy, ?Asymptotic expansions of solutions of ordinary differential equations in Hilbert space,? Arch. Rational Mech. Anal.,24, No. 3, 193-218 (1967). · Zbl 0147.12303 · doi:10.1007/BF00281343
[3] M. I. Vishik and V. V. Grushin, ?Boundary-value problems for elliptic equations degenerating on the domain boundary,? Mat. Sb.,80, No. 4, 455-491 (1967).
[4] M. I. Vishik and V. V. Grushin, ?Elliptic boundary-value problems degenerating in submanifolds of the boundary,? Dokl. Akad. Nauk SSSR,190, No. 2, 255-258 (1970). · Zbl 0238.35032
[5] V. G. Maz’ya and B. A. Plamenevskii, ?Elliptic boundary-value problems in manifolds with singularities,? in: Problems of Mathematical Analysis [in Russian], No. 6, Leningrad State Univ. (1977), pp. 85-142.
[6] V. G. Maz’ya and B. A. Plamenevskii, ?On the coefficients in the asymptotic form of the solution of an elliptic boundary-value problem in a domain with conical points,? Math. Nachrichten,76, 29-60 (1977). · Zbl 0359.35024 · doi:10.1002/mana.19770760103
[7] V. G. Maz’ya and B. A. Plamenevskii, ?Estimates in Lp and in Hölder classes and the Miranda-Agmon principle for solutions of elliptic boundary-value problems in domains with singular points on the boundary,? Math. Nachrichten,81, 25-82 (1978). · Zbl 0371.35018 · doi:10.1002/mana.19780810103
[8] M. I. Vishik and L. A. Lyusternik, ?Regular degeneration and boundary layer for linear differential equations with a small parameter,? Usp. Mat. Nauk,12, No. 5, 3-122 (1957). · Zbl 0087.29602
[9] S. A. Nazarov, ?Vishik-Lyusternik method in domains with conical points,? Dokl. Akad. Nauk SSSR,245, No. 5, 1307-1311 (1979). · Zbl 0424.35017
[10] S. A. Nazarov, ?Vishik-Lyusternik method for elliptic boundary-value problems in domains with conical points. I. Problem in the cone,? Sib. Mat. Zh.,22, No. 4, 142-163 (1981). · Zbl 0479.60091 · doi:10.1007/BF00968210
[11] L. A. Bagirov and V. A. Kondrat’ev, ?On a class of elliptic equations in Rn,? in: Partial Differential Equations [in Russian], No. 2, Trudy Seminara S. L. Soboleva, Izd. IM Sib. Otd. Akad. Nauk SSSR, Novosibirsk (1978), pp. 5-16.
[12] S. A. Nazarov, ?Elliptic boundary-value problems degenerating at a conical point,? Vestn. Leningr. Univ., No. 19, 108-109 (1980). · Zbl 0446.35042
[13] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag (1972).
[14] S. A. Nazarov, ?On the index of a boundary-value problem with a small parameter in the highest derivatives,? in: Differential Equations and Their Applications [in Russian], No. 24, Izd. Akad. Nauk LitSSR, Vilnius (1979) pp. 75-86. · Zbl 0425.35050
[15] M. S. Agranovich and M. I. Vishik, ?Elliptic problems with a parameter and parabolic problems of general type,? Usp. Mat. Nauk,19, No. 3, 53-160 (1964). · Zbl 0137.29602
[16] S. Agmon, A. Douglas, and L. Niremberg, ?Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary-value conditions. I,? Commun. Pure Appl. Math.,12, 457-486 (1959). · Zbl 0093.10401 · doi:10.1002/cpa.3160120405
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.