Asymptotic Methods in Equations of Mathematical Physics

art 1 The stationary phase method: on asymptotic expansions; the stationary phase method ; the stationary phase method, the multidimensional case; the problem of waves on the surface of a liquid; the asymptotic behaviour of the Fourier transform of a function concentrated on a smooth closed surface. Part 2 The WKB method for ordinary differential equations: the asymptotic behaviour of solutions of a homogeneous equation; the scattering problem; the asymptotic behaviour of solutions of boundary-value problems. Part 3 Partial differential equations of the first order and characteristics for equations of higher order: quasilinear partial differential equations of the first order; general partial differential equations of the first order; the Hamilton-Jacobi equation; example propagation of light waves in an inhomogenous medium; characteristic surfaces for differential operators of high order, connection with the well-posedness of the Cauchy problem; the search for characteristic surfaces. Part 4 Propagation of discontinuities, problems with rapidly oscillating data: the Leibniz formula; problems with rapidly oscillating initial data; discontinuous solutions of equations. Part 5 The Maslov canonical operator: the problem of scattering of a plane wave in an inhomogenous medium; the Lagrange maniford; the precanonical operator; the canonical operator, construction of a formal asymptotic solution; a field in an isotropic medium with parabolic wave front; more general problems. Part 6 Elliptic problems in a bounded domain: Sobolev-Slobodetskii spaces; elliptic problems; elliptic problems with a parameter; inversion of a finitely-meromorphic Fredholm family of operators. Part 7 Equations and systems with constant coefficients in Rn: equations with a non-zero characteristic polynomial; equations and systems of the type of the Helmholtz equation radiation conditions; the principle of limiting absorption. Part 8 Elliptic equations with variable coefficients and boundary-value problems in the exterior of a bounded domain: solubility and a priori estimated of solutions of exterior boundary-value problems; the principle of limiting absorption for exterior problems. Part 9 Analytic properties of the resolvent of operators that depend polynomially on a parameter: equations with constant coefficients; equations with variable coefficients and problems in the interior of a bounded domain; the asymptotic behaviour of solutions of exterior problems for small frequencies. Part 10 Short-wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour of solutions of hyperbolic equations as t ; introduction; short-wave asymptotic behaviour as t of solutions of mixed problems. Part 11 Quasiclasical approximations in stationary scattering problems: the asymptotic behaviour of the solution of the scattering problem and the amplitude of the scattering; proof of theorems 1 and 2. Part contents...