9783642116971
Minimal Surfaces - Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny
Springer (2010)
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#7335

Read It:
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Mathematics / Calculus, Mathematics / Differential Equations, Mathematics / Geometry / Differential, Mathematics / Mathematical Analysis, Science / Mathematical Physics

Minimal Surfaces is the first volume of a three volume treatise on minimal surfaces (Grundlehren Nr. 339-341). Each volume which can be read and studied independently of the others. The central theme is boundary value problems for minimal surfaces.The treatise is a greatly revised and extended version of the monograph Minimal Surfaces I, II (Grundlehren Nr. 295 & 296).The first volume begins with an exposition of basic ideas of the theory of surfaces in three-dimensional Euclidean space, followed by an introduction of minimal surfaces as stationary points of area, or equivalently, as surfacesof zero mean curvature. The final definition of a minimal surface is that of a non-constant harmónic mapping X: O-> R3 which is conformally parametrized on O-> R2 and may have branch points. Thereafter the classical theroy of minimal surfaces is surveyed, comprising many examples, a treatment of Björling ´s initial value problem, reflection principles, a formula of the second variation of area, the theorems of Bernstein, Heinz, Osserman, and Fujimoto.The second part of this volume begins with a survey of Plateau ´s problem and of some of its modifications. One of the main features is a new completely elementary proof of the fact that area A and Dirichlet integral D have the same infimum in the class C(G) of admissible surfaces spanning a prescribed contour G. This leads to a new, simplified solution of the simultaneous problem of minimizing A and D in C(G), as well as to new proofs of the mapping theorem of Riemann and Korn-Lichtenstein, and to a new solution of the simultaneous Douglas problem for A and D where G consists of several closed components. Then basic facts of stable minimal suurfaces are derived; this is done in the context of stable H-surfaces (i.e. of stable surfaces of prescribed mean curvature H), especially of cmc-surfaces (H = const), and leads to curvature estimates for stable, immersed cmc-surfaces and to Nitsche ´s uniqueness theorem and Tomi ´s finiteness result.In addition, a theory of unstable solutions of Plateau ´s problems is developed which is based on Courant ´s mountain pass lemma. Furthermore, Dirichlet ´s problem for nonparametric H-surfaces is solved, using the solution of Plateau ´s problem dor H-surfaces and the pertinent estimates.

Product Details
Dewey 516.362
No. of Pages 680
Height x Width 250 mm