A method is presented for obtaining explicit upper and lower pointwise bounds for the solution of rather general interior boundary value problems. The differential equations associated with these problems are of the elliptic type in certain sections while both linear and non-linear parabolic equations are the subject of investigation in other sections. The bounds which are obtained are in terms of the integrals of the squares of known functions and hence, in the linear case, improvement is possible using the Rayleigh-Ritz technique.

This Research Note collects reports of the invited plenary addresses given during the conference Elliptic and Parabolic Partial Differential Equations and Applications held in Capri, Italy, 19-23 September 1994. The conference was devoted to new developments in partial differential equations of elliptic and parabolic type and to their applications in various fields.

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.