It is well known that a wealth of problems of different nature, applied as well as purely theoretic, can be reduced to the study of elliptic equations and their eigen-values. During the years many books and articles have been published on this topic, considering spectral properties of elliptic differential operators from different points of view. This is one more book on these properties. This book is devoted to the study of some classical problems of the spectral theory of elliptic differential equations. The reader will find hardly any intersections with the books of Shubin [Sh] or Rempel-Schulze [ReSch] or with the works cited there. This book also has no general information in common with the books by Egorov and Shubin [EgShu], which also deal with spectral properties of elliptic operators. There is nothing here on oblique derivative problems; the reader will meet no pseudodifferential operators. The main subject of the book is the estimates of eigenvalues, especially of the first one, and of eigenfunctions of elliptic operators. The considered problems have in common the approach consisting of the application of the variational principle and some a priori estimates, usually in Sobolev spaces. In many cases, impor tant for physics and mechanics, as well as for geometry and analysis, this rather elementary approach allows one to obtain sharp results.

Inhalt
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 Sobolev's embedding theorems.- 2.22 Poincaré's 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 EllipticOperators.- 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 Schrödinger 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 Schrödinger 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.