On The Functional Calculus Of Pseudo Differential Boundary Problems

Functional Calculus of Pseudo-Differential Boundary Problems

CHAPTER 1. STANDARD PSEUDO-DIFFERENTIAL BOUNDARY PROBLEMS AND THEIR REALIZATIONS 1. 1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1. 2 The calculus of pseudo-differential boundary problems . . •. 19 1. 3 Green's formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1. 4 Realizations and normal boundary conditions . . . . . . . . . . . . . . 39 1. 5 Parameter-ellipticity and parabolicity . . . . . . . . . . . . . . . . . . . 50 1. 6 Adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 1. 7 Semiboundedness and coerciveness . . . . . . . . •. . . . . . . . . . . •. . . . 96 CHAPTER 2. THE CALCULUS OF PARAMETER-DEPENDENT OPERATORS 2. 1 Parameter-dependent pseudo-differential operators . . •. . . . . 125 2. 2 The transmission property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 2. 3 Parameter-dependent boundary symbol s . . . . . . . . . . . . . . . . . . . . . 179 2. 4 Operators and kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 2. 5 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 2. 6 Composition of xn-independent boundary symbol operators . . 234 2. 7 Compositions in general . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 2. 8 Strictly homogeneous symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 CHAPTER 3. PARAMETRIX AND RESOLVENT CONSTRUCTIONS 3. 1 Ellipticity. Auxiliary elliptic operators . . . . . . . . . . . . . . . . 280 3. 2 The parametrix construction . . . . . . . . . . •. . . . . . . . . . . . . . . . . . . 297 3. 3 The resolvent of arealization . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 3. 4 Other special cases . . . . . . •. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 CHAPTER 4. SOME APPLICATIONS 4. 1 Evolution problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 4. 2 The heat operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 4. 3 An index formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 4. 4 Complex powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 4. 5 Spectral asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 4. 6 Implicit eigenvalue problems . . . . . . . . . . . . . . . . . . . . . . . •. . . . . 437 4. 7 Singular perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 APPENDIX. VARIOUS PREREQUISITES (A. 1 General notation. A. 2 Functions and distributions. A. 3 Sobolev spaces. A. 4 Spaces over sub sets of mn. A. 5 Spaces over manifolds. A. 6 Notions from 473 spectral theory. ) '" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BIBLIOGRAPHY . . . •. . . . . . . •. . . . . . . . . . . . . . . •. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

On the Functional Calculus of Pseudo-differential Boundary Problems

Functional Calculus of Pseudodifferential Boundary Problems

Pseudodifferential methods are central to the study of partial differential equations, because they permit an "algebraization." A replacement of compositions of operators in n-space by simpler product rules for thier symbols. The main purpose of this book is to set up an operational calculus for operators defined from differential and pseudodifferential boundary values problems via a resolvent construction. A secondary purposed is to give a complete treatment of the properties of the calculus of pseudodifferential boundary problems with transmission, both the first version by Boutet de Monvel (brought completely up to date in this edition) and in version containing a parameter running in an unbounded set. And finally, the book presents some applications to evolution problems, index theory, fractional powers, spectral theory and singular perturbation theory. In this second edition the author has extended the scope and applicability of the calculus wit original contributions and perspectives developed in the years since the first edition. A main improvement is the inclusion of globally estimated symbols, allowing a treatment of operators on noncompact manifolds. Many proofs have been replaced by new and simpler arguments, giving better results and clearer insights. The applications to specific problems have been adapted to use these improved and more concrete techniques. Interest continues to increase among geometers and operator theory specialists in the Boutet de Movel calculus and its various generalizations. Thus the book’s improved proofs and modern points of view will be useful to research mathematicians and to graduate students studying partial differential equations and pseudodifferential operators.