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1.

図書

図書
Veli-Matti Hokkanen, Gheorghe Moroșanu
出版情報: Boca Raton, Fla. : Chapman & Hall/CRC, c2002  xiii, 243 p. ; 24 cm
シリーズ名: Research notes in mathematics ; 432
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Introduction
Preliminaries / 1:
Function and distribution spaces / 1.1:
Monotone operators, convex functions, and subdifferentials / 1.2:
Some elements of spectral theory / 1.3:
Linear evolution equations and semigroups / 1.4:
Nonlinear evolution equations / 1.5:
Elliptic boundary value problems / 2:
Nondegenerate elliptic boundary value problems / 2.1:
Degenerate elliptic boundary value problems / 2.2:
Parabolic problems with algebraic boundary conditions / 3:
Homogeneous boundary conditions / 3.1:
Nonhomogeneous boundary conditions / 3.2:
Higher regularity of solutions / 3.3:
Parabolic problems with algebraic-differential boundary conditions / 4:
Homogeneous algebraic boundary conditions / 4.1:
Nonhomogeneous algebraic boundary condition / 4.2:
Hyperbolic problems with algebraic boundary conditions / 4.3:
Existence, uniqueness, and long-time behavior of solutions / 5.1:
Hyperbolic problems with algebraic-differential boundary conditions / 5.2:
The Fourier method for abstract differential equations / 6.1:
First order linear equations / 7.1:
Semilinear first order equations / 7.2:
Second order linear equations / 7.3:
Semilinear second order equations / 7.4:
The semigroup approach for abstract differential equations / 8:
Hyperbolic partial differential systems with nonlinear boundary conditions / 8.1:
Nonlinear nonautonomous abstract differential equations / 9:
First order differential and functional equations containing subdifferentials / 9.1:
An application / 9.2:
Implicit nonlinear differential equations / 10:
Existence of solution / 10.1:
Uniqueness of solution / 10.2:
Continuous dependence of solution / 10.3:
Existence of periodic solutions / 10.4:
Index
Introduction
Preliminaries / 1:
Function and distribution spaces / 1.1:
2.

図書

図書
A. Kneschke
出版情報: Leipzig : B.G. Teubner  v. ; 21 cm
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3.

図書

図書
Dan Henry ; with editorial assistance from Jack Hale and Antônio Luiz Pereira
出版情報: New York : Cambridge University Press, 2005  viii, 206 p. ; 23 cm
シリーズ名: London Mathematical Society lecture note series ; 318
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Introduction
Geometrical preliminaries / 1:
Differential calculus of boundary perturbations / 2:
Examples using the implicit function theorem / 3:
Bifurcation problems / 4:
The transversality theorem / 5:
Generic perturbation of the boundary / 6:
Boundary operators for second-order elliptic equations / 7:
The method of rapidly-oscillating solutions / 8:
Introduction
Geometrical preliminaries / 1:
Differential calculus of boundary perturbations / 2:
4.

図書

図書
Guy David
出版情報: Basel : Birkhäuser, c2005  xiv, 581 p. ; 24 cm
シリーズ名: Progress in mathematics ; v. 233
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Foreword
Presentation of the Mumford-Shah functional
Functions in the Sobolev spaces
Regularity properties for quasiminimizers
Limits of almost-minimizers
Pieces of C^1 curves for almost-minimizers
Global Mumford-Shah minimizers in the plane
Applications to almost-minimizers (n = 2)
Quasi- and almost-minimizers in higher dimensions
Boundary regularity
Foreword
Presentation of the Mumford-Shah functional
Functions in the Sobolev spaces
5.

図書

図書
A. Kneschke
出版情報: Berlin : Verlag Technik, 1960, c1957  540 p. ; 21 cm
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目次情報:
Bd. 1. Gewöhnliche Differentialgleichungen
Bd. 1. Gewöhnliche Differentialgleichungen
6.

図書

図書
Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba ; with assistance and contributions by A. Küster
出版情報: Heidelberg ; Berlin : Springer, c2010  xvii, 623 p. ; 25 cm
シリーズ名: Die Grundlehren der mathematischen Wissenschaften ; 340
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Boundary Behaviour of Minimal Surfaces / Part I:
Minimal Surfaces with Free Boundaries / Chapter 1:
Classes of Admissible Functions. Linking Condition / 1.1:
Existence of Minimizers for the Free Boundary Problem / 1.3:
Stationary Minimal Surfaces with Free or Partially Free Boundaries and the Transversality Condition / 1.4:
Necessary Conditions for Stationary Minimal Surfaces / 1.5:
Existence of Stationary Minimal Surfaces in a Simplex / 1.6:
Stationary Minimal Surfaces of Disk-Type in a Sphere / 1.7:
Report on the Existence of Stationary Minimal Surfaces in Convex Bodies / 1.8:
Nonuniqueness of Solutions to a Free Boundary Problem. Families of Solutions / 1.9:
Scholia / 1.10:
The Boundary Behaviour of Minimal Surfaces / Chapter 2:
Potential-Theoretic Preparations / 2.1:
Solutions of Differential Inequalities / 2.2:
The Boundary Regularity of Minimal Surfaces Bounded by Jordan Arcs / 2.3:
The Boundary Behaviour of Minimal Surfaces at Their Free Boundary: A Survey of the Results and an Outline of Their Proofs / 2.4:
Hölder Continuity for Minima / 2.5:
Hölder Continuity for Stationary Surfaces / 2.6:
Higher Regularity in Case of Support Surfaces with Empty Boundaries. Analytic Continuation Across a Free Boundary / 2.7:
A Different Approach to Boundary Regularity / 2.9:
Asymptotic Expansion of Minimal Surfaces at Boundary Branch Points and Geometric Consequences / 2.10:
The Gauss-Bonnet Formula for Branched Minimal Surfaces / 2.11:
Singular Boundary Points of Minimal Surfaces / 2.12:
The Method of Hartman and Wintner, and Asymptotic Expansions at Boundary Branch Points / 3.1:
A Gradient Estimate at Singularities Corresponding to Corners of the Boundary / 3.2:
Minimal Surfaces with Piecewise Smooth Boundary Curves and their Asymptotic Behaviour at Corners / 3.3:
An Asymptotic Expansion for Solutions of the Partially Free Boundary Problem / 3.4:
References / 3.5:
Hölder Continuity at Intersection Points / 3.5.2:
Geometric Properties of Minimal Surfaces and H-Surfaces / Part II:
Enclosure and Existence Theorems for Minimal Surfaces and H-Surfaces. Isoperimetric Inequalities / Chapter 4:
Applications of the Maximum Principle and Nonexistence of Multiply Connected Minimal Surfaces with Prescribed Boundaries / 4.1:
Touching H-Surfaces and Enclosure Theorems. Further Nonexistence Results / 4.2:
Minimal Submanifolds and Submanifolds of Bounded Mean Curvature. An Optimal Nonexistence Result / 4.3:
An Optimal Nonexistence Result for Minimal Submanifolds of Codimension One / 4.3.1:
Geometric Maximum Principles / 4.4:
The Barrier Principle for Submanifolds of Arbitrary Codimension / 4.4.1:
A Geometric Inclusion Principle for Strong Subsolutions / 4.4.2:
Isoperimetric Inequalities / 4.5:
Estimates for the Length of the Free Trace / 4.6:
Obstacle Problems and Existence Results for Surfaces of Prescribed Mean Curvature / 4.7:
Surfaces of Prescribed Mean Curvature in a Riemannian Manifold / 4.8:
Estimates for Jacobi Fields / 4.8.1:
Riemann Normal Coordinates / 4.8.2:
Enclosure Theorems and Nonexistence / 4.8.3:
The Isoperimetric Problem. Historical Remarks and References to the Literature / 4.9.2:
Experimental Proof of the Isoperimetric Inequality / 4.9.3:
The Plateau Problem for H-Surfaces / 4.9.4:
The Thread Problem / Chapter 5:
Experiments and Examples. Mathematical Formulation of the Simplest Thread Problem / 5.1:
Existence of Solutions to the Thread Problem / 5.2:
Analyticity of the Movable Boundary / 5.3:
Branch Points / 5.4:
The First Five Variations of Dirichlet's Integral, and Forced Jacobi Fields / 6.1:
The Theorem for n + 1 Even and m + 1 Odd / 6.2:
Boundary Branch Points / 6.3:
Bibliography / 6.4:
Index
Boundary Behaviour of Minimal Surfaces / Part I:
Minimal Surfaces with Free Boundaries / Chapter 1:
Classes of Admissible Functions. Linking Condition / 1.1:
7.

図書

図書
Ulrich Dierkes, Stefan Hildebrandt, Anthony J. Tromba
出版情報: Heidelberg ; Berlin : Springer, c2010  xvi, 537 p. ; 25 cm
シリーズ名: Die Grundlehren der mathematischen Wissenschaften ; 341
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Free Boundaries and Bernstein Theorems / Part I:
Minimal Surfaces with Supporting Half-Planes / Chapter 1:
An Experiment / 1.1:
Examples of Minimal Surfaces with Cusps on the Supporting Surface / 1.2:
Setup of the Problem Properties of Stationary Solutions / 1.3:
Classification of the Contact Sets / 1.4:
Nonparametric Representation, Uniqueness, and Symmetry of Solutions / 1.5:
Asymptotic Expansions for Surfaces of Cusp-Types I and III. Minima of Dirichlet's Integral / 1.6:
Asymptotic Expansions for Surfaces of the Tongue/Loop-Type II / 1.7:
Final Results on the Shape of the Trace Absence of Cusps Optimal Boundary Regularity / 1.8:
Proof of the Representation Theorem / 1.9:
Scholia / 1.10:
Embedded Minimal Surfaces with Partially Free Boundaries / Chapter 2:
The Geometric Setup / 2.1:
Inclusion and Monotonicity of the Free Boundary Values / 2.2:
A Modification of the Kneser-Radó Theorem / 2.3:
Properties of the Gauss Map, and Stable Surfaces / 2.4:
Uniqueness of Minimal Surfaces that Lie on One Side of the Supporting Surface / 2.5:
Uniqueness of Freely Stable Minimal Surfaces / 2.6:
Asymptotic Expansions / 2.7:
Edge Creeping / 2.8:
Embedded Minimizers for Nonsmooth Supporting Surfaces / 2.9:
A Bernstein 'Theorem for Minimal Surfaces in a Wedge / 2.10:
Bernstein Theorems and Related Results / 2.11:
Entire and Exterior Minimal Graphs of Controlled Growth / 3.1:
Jörgens's Theorem / 3.1.1:
Asymptotic Behaviour for Solutions of Linear and Quasilinear Equations, Moser's Bernstein Theorem / 3.1.2:
The Interior Gradient Estimate and Consequences / 3.1.3:
First and Second Variation Formulae / 3.2:
First and Second Variation of the Area Integral / 3.2.1:
First and Second Variation Formulae for Singular Minimal Surfaces / 3.2.2:
Some Geometric Identities / 3.3:
Covariant Derivatives of Tensor Fields / 3.3.1:
Simons's Identity and Jacobi's Field Equation / 3.3.2:
Nonexistence of Stable Cones and Integral Curvature Estimates Further Bernstein Theorems / 3.4:
Stability of Minimal Cones / 3.4.1:
Nonexistence of Stable Cones / 3.4.2:
Integral Curvature Estimates for Minimal and ?-Minimal Hypersurfaces. Further Bernstein Theorems / 3.4.3:
Monotonicity and Mean Value Formulae Michael-Simon Inequalities / 3.5:
Pointwise Curvature Estimates / 3.6:
References to the Literature on Bernstein's Theorem and Curvature Estimates for n = 2 / 3.7:
Bernstein Theorems and Curvature Estimates for n ? 3 dimensions / 3.7.2:
Bernstein Theorems in Higher Codimensions / 3.7.3:
Sobolev Inequalities / 3.7.4:
Global Analysis of Minimal Surfaces / Part II:
The General Problem of Plateau: Another Approach / Chapter 4:
The General Problem of Plateau Formulation and Examples / 4.1:
A Geometric Approach to Teichmüller Theory of Oriented Surfaces / 4.2:
Symmetric Riemann Surfaces and Their Teichmüller Spaces / 4.3:
The Mumford Compactness Theorem / 4.4:
The Variational Problem / 4.5:
The Index Theorems for Minimal Surfaces of Zero and Higher Genus / 4.6:
Introduction / 5.1:
The Statement of the Index Theorem of Genus Zero / 5.2:
Stratification of Harmonic Surfaces by Singularity Type / 5.3:
Stratification of Harmonic Surfaces with Regular Boundaries by Singularity Type / 5.4:
The Index Theorem for Classical Minimal Surfaces / 5.5:
The Forced Jacobi Fields / 5.6:
Some Theorems on the Linear Algebra of Fredholm Maps / 5.7:
The Index Theorem for Higher Genus Minimal Surfaces Statement and Preliminaries / 5.8:
Review of Some Basic Results in Riemann Surface Theory / 5.10:
Vector Bundles over Teichmüller Space / 5.11:
Some Results on Maximal Ideals in Sobolev Algebras of Holomorphic Functions / 5.12:
Minimal Surfaces as Zeros of a Vector Field, and the Conformality Operators / 5.13:
The Corank of the Partial Conformality Operators / 5.14:
The Corank of the Complete Conformality Operators / 5.15:
Manifolds of Harmonic Surfaces of Prescribed Branching Type / 5.16:
The Proof of the Index Theorem / 5.17:
Euler Characteristic and Morse Theory for Minimal Surfaces / 5.18:
Fredholm Vector Fields / 6.1:
The Gradient Vector Field Associated to Plateau's Problem / 6.2:
The Sard-Brown Theorem for Functionals / 6.3:
The Morse Lemma / 6.5:
Historical Remarks and References to the Literature / 6.6:
On the Generic Nondegeneracy of Closed Minimal Surfaces in Riemannian Manifolds and Morse Theory / 6.8.2:
Bibliography
Index
Free Boundaries and Bernstein Theorems / Part I:
Minimal Surfaces with Supporting Half-Planes / Chapter 1:
An Experiment / 1.1:
8.

図書

図書
James Ward Brown, Ruel V. Churchill
出版情報: New York ; Singapore : McGraw-Hill, 2012  xvi, 400 p. ; 24 cm
シリーズ名: Churchill-Brown series
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Preface
Fourier Series / 1:
Piecewise Continuous Functions
Fourier Cosine Series
Examples
Fourier Sine Series
Adaptations to Other Intervals
Convergence of Fourier Series / 2:
One-Sided Derivatives
A Property of Fourier Coefficients
Two Lemmas
A Fourier Theorem
A Related Fourier Theorem
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
Partial Differential Equations of Physics / 3:
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
The Fourier Method / 4:
Linear Operators
Principle of Superposition
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
Boundary Value Problems / 5:
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
Fourier Integrals and Applications / 6:
The Fourier Integral Formula
Dirichlet's Integral
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
Orthonormal Sets / 7:
Inner Products and Orthonormal Sets
Generalized Fourier Series
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
Sturm-Liouville Problems and Applications / 8:
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
Bessel Functions and Applications / 9:
The Gamma Function
Bessel Functions Jn(x)
Solutions When v = 0,1,2
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of Jn(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Fourier-Bessel Series
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
Legendre Polynomials and Applications / 10:
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions Pn(cos ¿)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
Verification of Solutions and Uniqueness / 11:
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index
Preface
Fourier Series / 1:
Piecewise Continuous Functions
9.

図書

図書
Н.И. Мусхелишвили
出版情報: Москва : "Наука", 1968  511 p. ; 27 cm
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10.

図書

図書
Ivar Stakgold
出版情報: New York : Wiley, c1998  xvii, 689 p. ; 25 cm
シリーズ名: Pure and applied mathematics
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