Nonelliptic Partial Differential Equations

This book provides a very readable description of a technique, developed by the author years ago but as current as ever, for proving that solutions to certain (non-elliptic) partial differential equations only have real analytic solutions when the data are real analytic (locally). The technique is completely elementary but relies on a construction, a kind of a non-commutative power series, to localize the analysis of high powers of derivatives in the so-called bad direction. It is hoped that this work will permit a far greater audience of researchers to come to a deep understanding of this technique and its power and flexibility.

1. What this book is and is not. - 2. Brief Introduction. - 3. Overview of Proofs. - 4. Full Proof for the Heisenberg Group. - 5. Coefficients. - 6. Pseudo-differential Problems. - 7. Sums of Squares and Real Vector Fields. - 8. \bar{\partial}-Neumann and the Boundary Laplacian. - 9. Symmetric Degeneracies. - 10. Details of the Previous Chapter. -11. Non-symplectic Strategem ahe. - 12. Operators of Kohn Type Which Lose Derivatives. - 13. Non-linear Problems. - 14. Treves' Approach. - 15. Appendix. - Bibliography.