Estimates of eigenfunctions of elliptic operators with constant coefficients
Article
Received:
- 26 Downloads
Abstract
The Dirichlet problem for a symmetric elliptic operator with constant coefficients is studied. Estimates of the moduli of normalized eigenfunctions, uniform in a closed region, are obtained. These estimates generalize certain results of Kh. L. Smolitskii, O. A. Ladyzhenskaya, L. N. Slobodetskii, D. M. Éidus, V. A. Il'in, and I. A. Shishmarev.
Keywords
Dirichlet Problem Elliptic Operator Constant Coefficient Closed Region Symmetric Elliptic Operator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Preview
Unable to display preview. Download preview PDF.
Literature cited
- 1.Kh. L. Smolitskii, “Estimates for the derivatives of fundamental functions,” Dokl. Akad. Nauk SSSR,74, No. 2, 205–208 (1950).Google Scholar
- 2.O. A. Ladyzhenskaya, “The Fourier method for hyperbolic equations,” Dokl. Akad. Nauk SSSR,74, No. 3, 417–420 (1950).Google Scholar
- 3.O. A. Ladyzhenskaya, “On the Fourier method for the wave equation,” Dokl. Akad. Nauk SSSR,75, No. 6, 765–768 (1950).Google Scholar
- 4.L. N. Slobodetskii, “Potential theory for parabolic equations,” Dokl. Akad. Nauk SSSR,103, No. 1, 19–22 (1959).Google Scholar
- 5.D. M. Éidus, “Some inequalities for eigenfunctions,” Dokl. Akad. Nauk SSSR,107, No. 6, 796–798 (1956).Google Scholar
- 6.V. A. Il'in and I. A. Shishmarev, “On point estimates of eigenfunctions in a closed region,” in: Proceedings of the Soviet-American Symposium [in Russian], Novosibirsk (1963).Google Scholar
- 7.I. A. Shishmarev, Introduction to the Theory of Elliptic Equations [in Russian], Moscow State Univ. (1979).Google Scholar
- 8.I. A. Shishmarev, “On exact estimates for eigenfunctions of the biharmonic operator,” Dokl. Akad. Nauk SSSR,170, No. 4, 790–793 (1966).Google Scholar
- 9.B. R. Vainberg, “Asymptotic representation of fundamental solutions of hypoelliptic equations and a problem in the whole space with conditions at infinity,” Dokl. Akad. Nauk SSSR,144, No. 5, 958–961 (1962).Google Scholar
- 10.V. V. Grushin, “On conditions of Sommerfeld type for certain classes of partial differential equations,” Mat. Sb.,61, No. 2, 147–174 (1963).Google Scholar
- 11.B. R. Vainberg, Asymptotic Methods in the Equations of Mathematical Physics [in Russian], Moscow State Univ. (1982).Google Scholar
- 12.A. N. Varchenko, “Newton polyhedra and estimates of oscillating integrals,” Funkts. Anal. Prilozhen.,10, No. 3, 13–38 (1976).Google Scholar
Copyright information
© Plenum Publishing Corporation 1989