Author | Egorov, Yuri. author |
---|---|
Title | On Spectral Theory of Elliptic Operators [electronic resource] / by Yuri Egorov, Vladimir Kondratiev |
Imprint | Basel : Birkhรคuser Basel, 1996 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-9029-8 |
Descript | X, 334 p. online resource |
1 Hilbert Spaces -- 1.1 Definition and basic properties -- 1.2 Examples -- 1.3 Orthonormal base -- 1.4 Fourier series -- 1.5 Subspaces, orthogonal sums -- 1.6 Linear functionals -- 1.7 Weak convergence -- 1.8 Linear operators -- 1.9 Adjoint operators -- 1.10 The spectrum of an operator -- 1.11 Compact operators -- 1.12 Compact self-adjoint operators -- 1.13 Integral operators -- 1.14 The Lax-Milgram theorem -- 2 Functional Spaces -- 2.1 Notation and definitions -- 2.2 Lebesgue integral -- 2.3 Level sets of functions of a real variable -- 2.4 Symmetrization -- 2.5 The space L1(?) -- 2.6 The space L2(?) -- 2.7 The space Lp(?), p > 1 -- 2.8 Density of the set of continuous functions in L1(?) -- 2.9 Density of the set of continuous functions in Lp(?), p > 1 -- 2.10 Separability of Lp(?) -- 2.11 Global continuity of functions of Lp(?) -- 2.12 Averaging -- 2.13 Compactness of a subset in Lp(?) -- 2.14 Fourier transform -- 2.15 The spaces WPm(?) -- 2.16 The averaging and generalized derivatives -- 2.17 Continuation of functions of WPm(?) -- 2.18 The Sobolev integral representation -- 2.19 The space WP1(?, E) -- 2.20 Properties of the space WP1,0(?) -- 2.21 Sobolevโs embedding theorems -- 2.22 Poincarรฉโs inequality -- 2.23 Interpolation inequalities -- 2.24 Compactness of the embedding -- 2.25 Invariance of Wpm(?) under change of variables -- 2.26 The spaces Wpm(?) for a smooth domain ? -- 2.27 The traces of functions of WP1(?) -- 2.28 The space Hs -- 2.29 The traces of functions of W2k(Rn) -- 2.30 The Hardy inequalities -- 2.31 The Morrey embedding theorem -- 3 Elliptic Operators -- 3.1 Strongly elliptic equations -- 3.2 Elliptic equations -- 3.3 Regularity of solutions -- 3.4 Boundary problems for elliptic equations -- 3.5 Smoothness of solutions up to boundary -- 4 Spectral Properties of Elliptic Operators -- 4.1 Variational principle -- 4.2 The spectrum of a self-adjoint operator -- 4.3 The Friedrichs extension -- 4.4 Examples of linear unbounded operators -- 4.5 Self-adjointness of the Schrรถdinger operator -- 5 The Sturm-Liouville Problem -- 5.1 Elementary properties -- 5.2 On the first eigenvalue of a Sturm-Liouville problem -- 5.3 On other estimates of the first eigenvalue -- 5.4 On a more general estimate of the first eigenvalue of the Sturm-Liouville operator -- 5.5 On estimates of all eigenvalues -- 6 Differential Operators of Any Order -- 6.1 Oscillation of solutions of an equation of any order -- 6.2 On estimates of the first eigenvalue for operators of higher order -- 6.3 Introduction to a Lagrange problem -- 6.4 Preliminary estimates -- 6.5 Precise results -- 7 Eigenfunctions of Elliptic Operators in Bounded Domains -- 7.1 On the Dirichlet problem for strongly elliptic equations -- 7.2 Estimates of eigenfunctions of strongly elliptic operators -- 7.3 Equations of second order -- 7.4 Estimates of eigenfunctions of operator pencils -- 7.5 The method of stationary phase -- 7.6 Asymptotics of a fundamental solution of an elliptic operator with constant coefficients -- 7.7 Estimates of the eigenfunctions of an elliptic operator with constant coefficients -- 7.8 Estimates of the first eigenvalue of an elliptic operator in a multi-connected domain -- 7.9 Estimates of the first eigenvalue of the Schrรถdinger operator in a bounded domain -- 8 Negative Spectra of Elliptic Operators -- 8.1 Introduction -- 8.2 One-dimensional case -- 8.3 Some inequalities and embedding theorems -- 8.4 Estimates of the number of points of the negative spectrum -- 8.5 Some generalizations -- 8.6 Lower estimates for the number N -- 8.7 Other results -- 8.8 On moments of negative eigenvalues of an elliptic operator