Treatise on Analysis: v. 8
This volume, the eighth out of nine, continues the translation of "Treatise on Analysis" by the French author and mathematician, Jean Dieudonne. The author shows how, for a voluntary restricted class of linear partial differential equations, the use of Lax/Maslov operators and pseudodifferential operators, combined with the spectral theory of operators in Hilbert spaces, leads to solutions that are much more explicit than solutions arrived at through "a priori" inequalities, which are useless applications.
- Hardback | 360 pages
- 165.1 x 241.3 x 19.05mm | 680.39g
- 01 Sep 1993
- Elsevier Science Publishing Co Inc
- Academic Press Inc
- San Diego, United States
- references, index
Table of contents
Part 1 Weyl-Kodaira theory: elliptical differential operators on an interval of R boundary conditions; self-adjoint operators associated with a linear differential equation; the case of second order equations; example - second order equations with periodic-coefficients; example - Gelfand-Levitan equations. Part 2 Multilayer potentials: symbols of rational type; the case of hyperplane multilayers; general case. Part 3 Fine boundary value problems for elliptical differential operators: the Calderon operator; elliptic boundary value problems; ellipticity criteria; the spaces Hs,r (U+) Hs,r - spaces and P-potentials; regularity on the boundary; coercive problems; generalized Green's formula; fine problems associated with coercive problems; examples; extension to some non-Hermitian operators; case of second-order operators; Neumann's problem; the maximum principle. Part 4 Parabolic equations: construction of one-sided local resolvent; the one-sided global Cauchy problem; traces and eigenvalues. Part 5 Evolution distributions - the wave equation: generalized Cauchy problem; propagation and domain of influence; signals, waves and rays. Part 6 Strictly hyperbolic equations: preliminary results; construction of a local approximate resolvent; examples and variations; the Cauchy problem for strictly hyperbolic differential operators; existence and local uniqueness; global problems; extension to manifolds. Part 7 Application to the spectrum of a Hermitian elliptic operator.