000			04883nam a22005775i 4500
001			978-3-642-11698-8
003			DE-He213
005			20140220084531.0
007			cr nn 008mamaa
008			100825s2010 gw | s |||| 0|eng d
020			_a9783642116988 _9978-3-642-11698-8
024	7		_a10.1007/978-3-642-11698-8 _2doi
050		4	_aQA315-316
050		4	_aQA402.3
050		4	_aQA402.5-QA402.6
072		7	_aPBKQ _2bicssc
072		7	_aPBU _2bicssc
072		7	_aMAT005000 _2bisacsh
072		7	_aMAT029020 _2bisacsh
082	0	4	_a515.64 _223
100	1		_aDierkes, Ulrich. _eauthor.
245	1	0	_aMinimal Surfaces _h[electronic resource] / _cby Ulrich Dierkes, Stefan Hildebrandt, Friedrich Sauvigny.
264		1	_aBerlin, Heidelberg : _bSpringer Berlin Heidelberg, _c2010.
300			_aXVI, 692 p. _bonline resource.
336			_atext _btxt _2rdacontent
337			_acomputer _bc _2rdamedia
338			_aonline resource _bcr _2rdacarrier
347			_atext file _bPDF _2rda
490	1		_aGrundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics, _x0072-7830 ; _v339
505	0		_ato the Geometry of Surfaces and to Minimal Surfaces -- Differential Geometry of Surfaces in Three-Dimensional Euclidean Space -- Minimal Surfaces -- Representation Formulas and Examples of Minimal Surfaces -- Plateau's Problem -- The Plateau Problem and the Partially Free Boundary Problem -- Stable Minimal- and H-Surfaces -- Unstable Minimal Surfaces -- Graphs with Prescribed Mean Curvature -- to the Douglas Problem -- Problems.
520			_aMinimal 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. The treatise is a substantially 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 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. 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 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. Then 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 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 for H-surfaces and the pertinent estimates.
650		0	_aMathematics.
650		0	_aFunctions of complex variables.
650		0	_aDifferential equations, partial.
650		0	_aGlobal differential geometry.
650	1	4	_aMathematics.
650	2	4	_aCalculus of Variations and Optimal Control, Optimization.
650	2	4	_aDifferential Geometry.
650	2	4	_aPartial Differential Equations.
650	2	4	_aFunctions of a Complex Variable.
650	2	4	_aTheoretical, Mathematical and Computational Physics.
700	1		_aHildebrandt, Stefan. _eauthor.
700	1		_aSauvigny, Friedrich. _eauthor.
710	2		_aSpringerLink (Online service)
773	0		_tSpringer eBooks
776	0	8	_iPrinted edition: _z9783642116971
830		0	_aGrundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics, _x0072-7830 ; _v339
856	4	0	_uhttp://dx.doi.org/10.1007/978-3-642-11698-8
912			_aZDB-2-SMA
999			_c111915 _d111915