{"product_id":"9783642116971","title":"Grundlehren der mathematischen Wissenschaften","description":"\u003ch1\u003eGrundlehren der mathematischen Wissenschaften\u003c\/h1\u003e \u003ch2\u003eDierkes, Ulrich; Jakob, Ruben; Hildebrandt, Stefan; Küster, Albrecht; Sauvigny, Friedrich\u003c\/h2\u003e \u003cp\u003eMinimal Surfaces is the first volume of a three volume treatise on minimal surfaces (Grundlehren Nr. 339-341). Each volume can be read and studied independently of the others. The central theme is boundary value problems for minimal surfaces.\nThe treatise is a substantially revised and extended version of the monograph Minimal Surfaces I, II (Grundlehren Nr. 295 \u0026amp; 296).\nThe 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 surfaces of zero mean curvature. The final definition of a minimal surface is that of a nonconstant harmonic mapping X: \\Omega\\to\\R^3 which is conformally parametrized on \\Omega\\subset\\R^2 and may have branch points. Thereafter the classical theory 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.\nThe 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 theorems 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. \nThen basic facts of stable minimal surfaces 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 andTomi´s finiteness result.\nIn 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 for H-surfaces and the pertinent estimates.\u003c\/p\u003e \u003ch3\u003eDetails\u003c\/h3\u003e \u003cp\u003ePublished by: Springer\u003c\/p\u003e \u003cp\u003ePublication Date: 2010-10-01\u003c\/p\u003e \u003cp\u003eFormat: Hardcover\u003c\/p\u003e \u003cp\u003eISBN-13: 9783642116971\u003c\/p\u003e \u003cp\u003eDOI: 10.1007\/978-3-642-11698-8\u003c\/p\u003e \u003cp\u003eDimensions: 235cm x155cm\u003c\/p\u003e \u003cp\u003ePages: 692\u003c\/p\u003e ","brand":"Springer Berlin Heidelberg","offers":[{"title":"Default Title","offer_id":45379164110988,"sku":"9783642116971","price":143.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0710\/9545\/1788\/files\/9783642116971.jpg?v=1775731630","url":"https:\/\/fh90cf-fv.myshopify.com\/products\/9783642116971","provider":"Late Knight Books and Services, LLC","version":"1.0","type":"link"}