Selected works of Eberhard Hopf with commentaries

著者

書誌事項

Selected works of Eberhard Hopf with commentaries

Cathleen S. Morawetz, James B. Serrin, Yakov G. Sinai, editors

American Mathematical Society, c2002

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注記

Includes bibliographical references

Text in English and German

内容説明・目次

内容説明

This work celebrates the work of Eberhard Hopf, a founding father of ergodic theory, a mathematician who produced many beautiful, elegantly written, and now classical, results in integral equations and partial differential equations. Hopf's results remain at the core of these fields, and the title includes Hopf's original mathematical papers, still notable for their elegance and clarity of the writing, with accompanying summaries and commentary by well-known mathematicians. Today, ergodic theory and P.D.E. continue to be active, important areas of mathematics. In this volume the reader will find the roots of many ergodic theory concepts and theorems. Hopf authored fundamental results for P.D.E., such as the maximum principle of elliptic equations and the complete solution of Burger's equation.The familiar properties of elliptic equations were proved for the first time in his earliest work and are included here. His bifurcation theorem, still used over and over again, is a particular gem. The proof of the Wiener-Hopf Theorem is a stunning application of deep analysis. The volume is presented in two main parts. The first section is dedicated to classical papers in analysis and fluid dynamics, and the second to ergodic theory. These works and all the others in the Selected Works carry commentaries by a stellar group of mathematicians who write of the origin of the problems, the important results that followed. Many a mathematical researcher and graduate student will find these collected works to be an excellent resource.

目次

Part I: Elementare Bemerkungen uber die Losungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus by E. Hopf Commentary by J. B. Serrin A remark on linear elliptic differential equations of second order by E. Hopf Commentary by J. B. Serrin Zum analytischen Charakter der Losungen regularer zweidimensionaler Variationsprobleme by E. Hopf Commentary by H. Weinberger Uber eine Klasse singularer Integralgleichungen by N. Wiener and E. Hopf Commentary by H. Widom Uber den funktionalen, insbesondere den analytischen Charakter der Losungen elliptscher Differentialgleichungen zweiter Ordnung by E. Hopf Commentary by H. Weinberger Abzweigung einer periodischen Losung von einer stationaren Losung eines Differentialsystems by E. Hopf Commentary by M. Golubitsky and P. H. Rabinowitz Repeated branching through loss of stability. An example by E. Hopf A mathematical example displaying features of turbulence by E. Hopf Commentary by R. Temam On S. Bernstein's theorem on surfaces $z(x,y)$ of nonpositive curvature by E. Hopf Commentary by L. Nirenberg The partial differential equation $u_t+uu_x=\mu_{xx}$ by E. Hopf Commentary by P. D. Lax Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen by E. Hopf Commentary by J. B. Serrin Hamilton's theory and generalized solutions of the Hamilton-Jacobi equation by E. D. Conway and E. Hopf Commentary by C. S. Morawetz Part II: Statistik der geodatischen Linien in Mannigfaltigkeiten negativer Krummung by E. Hopf Statistik der Losungen geodatischer Probleme vom unstabilen Typus. II by E. Hopf Commentary by Ya. G. Sinai Closed surfaces without conjugate points by E. Hopf Commentary by Ya. G. Sinai Statistical hydromechanics and functional calculus by E. Hopf Commentary by Ya. G. Sinai On the ergodic theorem for positive linear operators by E. Hopf Commentary by D. Ornstein Acknowledgments.

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