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 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 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.
