Seminar on New Results in Nonlinear Partial Differential Equations : A Publication of the Max-Planck-Institut Für Mathematik, Bonn

This book consists almost entirely of papers delivered at the Seminar on partial differential equations held at Max-Planck-Institut in the spring of 1984. They give an insight into important recent research activities. Some further developments are also included. Recent years have seen a tremendous growth of interest in non-linear partial differential equations_ Yau's solution of the Calabi conjecture and more recently Schoen's solution of the Yamabe problem and Wente's counter-example of the Hopf conjecture have been a great stimulus to geometric P.D.E . Applied P.D.E. has also seen great progress. The "year" in P.D.E., January-September 1984 at the Max-Planck-I nstitute was dedicated to bringing together some of the leaders in several fields of P.D.E . We preferred a seminar focused not on a given very specific area but one which cuts across boundaries of research interests. We hoped to have a snapshot of some of the more important recent devel opments in non-linear P.D.E . The current volume contains some of the results reported upon in this seminar. From Henry Wente, in geometric P.D.E., we have new toroidal counter-examples to Hopf's conjecture, and, due to Uwe Abresch, a further development of these ideas. Sergiu Klainerman's global existence result for the non-linear Klein-Gordon equation in four space-time dimensions, achieved while at Max Planck, is included. Vince Moncrief presents his global existence result for the Yang-Mills equation in Min kowski space time. All of these results answer important and outstanding questions Front Matter....Pages I-VI Twisted Tori of Constant Mean Curvature in R 3 ....Pages 1-36 Old and New Doubly Periodic Solutions of the Sinh-Gordon Equation....Pages 37-73 Global Existence of Small Amplitude Solutions to Nonlinear Klein-Gordon Equations in Four Space-Time Dimensions....Pages 75-89 Global Existence of Solutions of the Yang-Mills Equations in Minkowski Spacetime....Pages 91-100 Nonlinear Stability in Fluids and Plasmas....Pages 101-134 Lane-Emden Equations and Related Topics in Nonlinear Elliptic and Parabolic Problems....Pages 135-152 Parabolicity and the Liouville Property on Complete Riemannian Manifolds....Pages 153-166 Global Analysis and Teichmüller Theory....Pages 167-198