The Analysis Of Linear Partial Differential Operators Ii: Differential Operators With Constant Coefficients (grundlehren Der Mathematischen Wissenschaften) (v. 2)

This volume is an expanded version of Chapters III, IV, V and VII of my 1963 book "Linear partial differential operators". In addition there is an entirely new chapter on convolution equations, one on scattering theory, and one on methods from the theory of analytic functions of several complex variables. The latter is somewhat limited in scope though since it seems superfluous to duplicate the monographs by Ehrenpreis and by Palamodov on this subject. The reader is assumed to be familiar with distribution theory as presented in Volume I. Most topics discussed here have in fact been encountered in Volume I in special cases, which should provide the necessary motivation and background for a more systematic and precise exposition. The main technical tool in this volume is the Fourier- Laplace transformation. More powerful methods for the study of operators with variable coefficients will be developed in Volume III. However, constant coefficient theory has given the guidance for all that work. Although the field is no longer very active - perhaps because of its advanced state of development - and although it is possible to pass directly from Volume I to Volume III, the material presented here should not be neglected by the serious student who wants to gain a balanced perspective of the theory of linear partial differen­tial equations. This volume is an expanded version of Chapters III, IV, V and VII of my 1963 book "Linear partial differential operators." In addition there is an entirely new chapter on convolution equations, one on scattering theory, and one on methods from the theory of analytic functions of several complex variables. The latter is somewhat limited in scope though since it seems superfluous to duplicate the monographs by Eh- renpreis and by Palamodov on this subject. The reader is assumed to be familiar with distribution theory as presented in Volume I. Most topics discussed here have in fact been encountered in Volume I in special cases, which should provide the necessary motivation and background for a more systematic and pre- cise exposition. The main technical tool in this volume is the Fourier- Laplace transformation. More powerful methods for the study of operators with variable coefficients will be developed in Volume III. However, the constant coefficient theory has given the guidehnes for all that work. Although the field is no longer very active - perhaps because of its advanced state of development - and although it is pos- sible to pass directly from Volume I to Volume III, the material pre- sented here should not be neglected by the serious student who wants to gain a balanced perspective of the theory of linear partial differen- tial equations Vol. I of Lars Hörmander's 4-volume treatisewas an exposition of the theory of distributions and Fourier analysis preparing for the study of linear partial differential operators. The present Vol. II is mainly devoted to operators with constant coefficients. An analysis of the existence and regularity of (fundamental) solutions in the first two chapters is followed by a thorough study of the Cauchy problem. One chapter is devoted to the spectral theory of short range perturbations of operatorswith constant coefficients, and another presents Fourier-Laplace representations of solutions of homogeneous differential equations with constant coefficients. The last chapter is a study of the closely related subject of convolution operators. TOC:Introduction.- Existence and Approximation of Solutions of Differential Equations.- Interior Regularity of Solutions of Differential Equations.- The Cauchy and Mixed Problems.- Differential Operators of Constant Strength.- Scattering Theory.- Analytic Function Theory and Differential Equations.- Convolution Equations.- Appendix A: Some Algebraic Lemmas.- Bibliography.- Index.- Index of Notation

Vol. I of Lars Hörmander's 4-volume treatise was an exposition of the theory of distributions and Fourier analysis preparing for the study of linear partial differential operators.

The present Vol. II is mainly devoted to operators with constant coefficients. An analysis of the existence and regularity of (fundamental) solutions in the first two chapters is followed by a thorough study of the Cauchy problem. One chapter is devoted to the spectral theory of short range perturbations of operators with constant coefficients, and another presents Fourier-Laplace representations of solutions of homogeneous differential equations with constant coefficients. The last chapter is a study of the closely related subject of convolution operators.

Vol. I of Lars Hormander's 4-volume treatise was an exposition of the theory of distributions and Fourier analysis preparing for the study of linear partial differential operators. The present Vol. II is mainly devoted to operators with constant coefficients. An analysis of the existence and regularity of (fundamental) solutions in the first two chapters is followed by a thorough study of the Cauchy problem. One chapter is devoted to the spectral theory of short range perturbations of operators with constant coefficients, and another presents Fourier-Laplace representations of solutions of homogeneous differential equations with constant coefficients. The last chapter is a study of the closely related subject of convolution operators. Front Matter....Pages I-VIII Introduction....Pages 1-2 Existence and Approximation of Solutions of Differential Equations....Pages 3-59 Interior Regularity of Solutions of Differential Equations....Pages 60-93 The Cauchy and Mixed Problems....Pages 94-181 Differential Operators of Constant Strength....Pages 182-224 Scattering Theory....Pages 225-269 Analytic Function Theory and Differential Equations....Pages 270-301 Convolution Equations....Pages 302-361 Back Matter....Pages 362-392 V. 1. Distribution Theory And Fourier Analysis -- V. 2. Differential Operators With Constant Coefficients -- V. 3. Pseudo-differential Operators -- V. 4. Fourier Integral Operators. Lars Hörmander. Vols. 1-2 Are An Expanded Version Of The Author's 1 Vol. Work: Linear Partial Differential Operators. Includes Bibliographies And Indexes. Author received the 1962 Fields Medal Author received the 1988 Wolf Prize (honoring achievemnets of a lifetime) Author is leading expert in partial differential equations In the preceding chapters we have constructed a number of explicit fundamental solutions.