This book is meant to give an account of recent developments in
the theory of Plateau's problem for parametric minimal surfaces and
surfaces of prescribed constant mean curvature ("H-surfaces") and
its analytical framework. A comprehensive overview of the classical
existence and regularity theory for disc-type minimal and
H-surfaces is given and recent advances toward general structure
theorems concerning the existence of multiple solutions are
explored in full detail.
The book focuses on the author's derivation of the
Morse-inequalities and in particular the mountain-pass-lemma of
Morse-Tompkins and Shiffman for minimal surfaces and the proof of
the existence of large (unstable) H-surfaces (Rellich's conjecture)
due to Brezis-Coron, Steffen, and the author. Many related results
are covered as well. More than the geometric aspects of Plateau's
problem (which have been exhaustively covered elsewhere), the
author stresses the analytic side. The emphasis lies on the
variational method.
Originally published in 1989.
The Princeton Legacy Library uses the latest print-on-demand
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from the distinguished backlist of Princeton University Press.
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