Abstract
The paper deals with the distribution of eigenvalues of some degenerate elliptic operators in the unit square defined via closable symmetric singular bilinear forms. The approach relies mainly on entropy numbers for Sobolev spaces with mixed integrability.
Similar content being viewed by others
References
Birman, M.S., Solomyak, M.Z.: Spectral asymptotics of non-smooth elliptic operators. I. Trans. Moscow Math. Soc. 27, 1–52 (1972)
Birman, M.S., Solomyak, M.Z.: Spectral asymptotics of non-smooth elliptic operators. II. Trans. Moscow Math. Soc. 28, 1–32 (1973)
Birman, M.S., Solomyak, M.Z.: Quantitative analysis in Sobolev imbedding theorems and applications to spectral theory. Am. Math. Soc. Transl. 114(2), 1–132 (1980)
Carl, B.: Entropy numbers, s-numbers and eigenvalue problems. J. Funct. Anal. 41, 290–306 (1981)
Carl, B., Stephani, I.: Entropy, Compactness and the Approximation of Operators. Cambridge Univ. Press, Cambridge (1990)
Carl, B., Triebel, H.: Inequalities between eigenvalues, entropy numbers and related quantities in Banach spaces. Math. Ann. 251, 129–133 (1980)
Edmunds, D.E., Triebel, H.: Eigenvalue distributions of some degenerate elliptic operators: An approach via entropy numbers. Math. Ann. 299, 311–340 (1994)
Edmunds, D.E., Triebel, H.: Function Spaces, Entropy Numbers, Differential Operators. Cambridge Univ. Press, Cambridge (1996)
Haroske, D.D.: Envelopes and Sharp Embeddings of Function Spaces. Chapman & Hall/CRS, Boca Raton (2007)
Haroske, D.D., Triebel, H.: Distributions, Sobolev Spaces, Elliptic Equations. European Math. Soc. Publishing House, Zürich (2008)
Ma, Z.-M., Röckner, M.: Introduction to the Theory of (Non-symmetric) Dirichlet Forms. Springer, Berlin (1992)
Skrzypczak, L.: Entropy numbers of Trudinger-Strichartz embeddings of radial Besov spaces and applications. J. Lond. Math. Soc. 69(2), 465–488 (2004)
Skrzypczak, L.: Wavelet frames, Sobolev embeddings and negative spectrum of Schrödinger operators on manifolds with bounded geometry. J. Fourier Anal. Appl. 14, 415–442 (2008)
Skrzypczak, L., Tomasz, B.: Entropy of Sobolev embeddings of radial functions and radial eigenvalues of Schrödinger operators on isotropic manifolds. Math. Nachr. 280, 654–675 (2007)
Triebel, H.: Higher Analysis. Barth, Leipzig (1992)
Triebel, H.: Fractals and Spectra Related to Fourier Analysis and Function Spaces. Birkhäuser, Basel (1997)
Triebel, H.: Entropy numbers in function spaces with mixed integrability. Rev. Mat. Complut. (to appear), doi:10.1007/s13163-010-0034-7
Ziemer, W.P.: Weakly Differentiable Functions. Springer, New York (1989)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Triebel, H. Eigenvalue distributions of some non-isotropic degenerate elliptic operators. Rev Mat Complut 24, 343–355 (2011). https://doi.org/10.1007/s13163-010-0042-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13163-010-0042-7
Keywords
- Eigenvalue distribution
- Singular elliptic operators
- Entropy numbers
- Sobolev spaces with mixed integrability