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1.

図書

図書
Jürgen Jost
出版情報: New York ; Tokyo : Springer, c2002  xi, 325 p. ; 25 cm
シリーズ名: Graduate texts in mathematics ; 214
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Introduction
The Laplace equation as the prototype of an elliptic partial differential equation of 2nd order
The maximum principle
Existence techniques
methods based on the maximum principle / I:
Parabolic methods / II:
The Head equation
The wave equation and its connections with the Laplace and heat equation
The heat equation, semigroups, and Brownian motion
The Dirichlet principle
Variational methods for the solution of PDE (Existence techniques III)
Sobolev spaces and L2 regularity theory
Strong solutions
The regularity theory of Schauder and the continuity method (Existence techniques IV)
The Moser iteration method and the reqularity theorem of de Giorgi and Nash
Banach and Hilbert spaces
The Lp-spaces
Bibliography
Introduction
The Laplace equation as the prototype of an elliptic partial differential equation of 2nd order
The maximum principle
2.

図書

図書
Giuseppe Da Prato, Jerzy Zabczyk
出版情報: Cambridge : Cambridge University Press, 2002  xvi, 379 p. ; 23 cm
シリーズ名: London Mathematical Society lecture note series ; 293
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Theory in the Space of Continuous Functions / Part I:
Gaussian measures / 1:
Spaces of continuous functions / 2:
Heat equation / 3:
Poisson's equation / 4:
Elliptic equations with variable coefficients / 5:
Ornstein-Uhlenbeck equations / 6:
General parabolic equations / 7:
Parabolic equations in open sets / 8:
Theory in Sobolev Spaces with a Gaussian Measure / Part II:
L2 and Sobolev spaces / 9:
Ornstein-Uhlenbeck semigroups on Lp(H, mu) / 10:
Perturbations of Ornstein-Uhlenbeck semigroups / 11:
Gradient systems / 12:
Applications to Control Theory
Second order Hamilton-Jacobi equations / 13:
Hamilton-Jacobi inclusions / 14:
Theory in the Space of Continuous Functions / Part I:
Gaussian measures / 1:
Spaces of continuous functions / 2:
3.

図書

図書
Andrew Majda
出版情報: New York : Courant Institute of Mathematical Sciences, New York University , Providence, R.I : American Mathematical Society, 2003  ix, 234 p. ; 26 cm
シリーズ名: Courant lecture notes in mathematics ; 9
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Preface
Introduction / Chapter 1.:
Basic Properties of the Equations with Rotation and Stratification / 1.1.:
Two-Dimensional Exact Solutions / 1.2.:
Buoyancy and Stratification / 1.3.:
Jet Flows with Rotation and Stratification / 1.4.:
From Vertical Stratification to Shallow Water / 1.5.:
Some Remarkable Features of Stratified Flow / Chapter 2.:
Energy Principle / 2.1.:
Vorticity in Stratified Fluids and Exact Solutions Motivated by Local Analysis / 2.2.:
Use of Theorem 2.4: Exact Two-Dimensional Solutions / 2.3.:
Nonlinear Plane Waves in Stratified Flow: Internal Gravity Waves / 2.4.:
Exact Solutions with Large-Scale Motion and Nonlinear Plane Waves / 2.5.:
More Details for Theorem 2.7 on Special Exact Solutions for the Boussinesq Equations Including Plane Waves / 2.6.:
Linear and Nonlinear Instability of Stratified Flows with Strong Stratification / Chapter 3.:
Boussinesq Equations and Vorticity Stream Formulation / 3.1.:
Nonlinear Instability of Stratified Flows / 3.2.:
Shear Flows / 3.3.:
Some Background Facts on ODEs / 3.4.:
Rotating Shallow Water Theory / Chapter 4.:
Rotating Shallow Water Equations / 4.1.:
Conservation of Potential Vorticity / 4.2.:
Nonlinear Conservation of Energy / 4.3.:
Linear Theory for the Rotating Shallow Water Equations / 4.4.:
Nondimensional Form of the Rotating Shallow Water Equations / 4.5.:
Derivation of the Quasi-Geostrophic Equations / 4.6.:
The Quasi-Geostrophic Equations as a Singular PDE Limit / 4.7.:
The Model Rotating Shallow Water Equations / 4.8.:
Preliminary Mathematical Considerations / 4.9.:
Regorous Convergence of the Model Rotating Shallow Water Equations to the Quasi-Geostrophic Equations / 4.10.:
Proof of the Convergence Theorem / 4.11.:
Linear and Weakly Nonlinear Theory of Dispersive Waves with Geophysical Examples / Chapter 5.:
Linear Wave Midlatitude Planetary Equations / 5.1.:
Dispersive Waves: General Properties / 5.2.:
Interpretation of Group Velocity / 5.3.:
Distant Propagation from a Localized Source / 5.4.:
WKB Methods for Linear Dispersive Waves / 5.5.:
Beyond Caustics: Eikonal Equation Revisited / 5.6.:
Weakly Nonlinear WKB for Perturbations Around a Constant State / 5.7.:
Nonlinear WKB and the Boussinesq Equations / 5.8.:
Simplified Equations for the Dynamics of Strongly Stratified Flow / Chapter 6.:
Nondimensionalization of the Boussinesq Equations for Stably Stratified Flow / 6.1.:
The Vorticity Stream Formulation and Elementary Properties of the Limit Equations for Strongly Stratified Flow / 6.2.:
Solutions of the Limit Dynamics with Strong Stratification as Models for Laboratory Experiments / 6.3.:
The Stratified Quasi-Geostrophic Equations as a Singular Limit of the Rotating Boussinesq Equations / Chapter 7.:
The Rotating Boussinesq Equations / 7.1.:
The Nondimensional Rotating Boussinesq Equations / 7.3.:
Formal Asymptotic Derivation of the Quasi-Geostrophic Equations as a Distinguished Asymptotic Limit of Small Rossby and Froude Numbers / 7.4.:
Rigorous Convergence of the Rotating Boussinesq Equations to the Quasi-Geostrophic Equations / 7.5.:
Introduction to Averaging over Fast Waves for Geophysical Flows / 7.6.:
Motivation for Fast-Wave Averaging / 8.1.:
A General Framework for Averaging over Fast Waves / 8.3.:
Elementary Analytic Models for Comparing Instabilities at Low Froude Numbers with the Low Froude Number Limit Dynamics / 8.4.:
The Rapidly Rotating Shallow Water Equations with Unbalanced Initial Data in the Quasi-Geostrophic Limit / 8.5.:
The Interaction of Fast Waves and Slow Dynamics in the Rotating Stratified Boussinesq Equations / 8.6.:
Waves and PDEs for the Equatorial Atmosphere and Ocean / Chapter 9.:
Introduction to Equatorial Waves for Rotating Shallow Water / 9.1.:
The Equatorial Primitive Equations / 9.2.:
The Nonlinear Equatorial Long-Wave Equations / 9.3.:
A Simple Model for the Steady Circulation of the Equatorial Atmosphere / 9.4.:
Bibliography
Preface
Introduction / Chapter 1.:
Basic Properties of the Equations with Rotation and Stratification / 1.1.:
4.

図書

図書
V. Lakshmikantham and S. Köksal
出版情報: London : Taylor & Francis, c2003  x, 318 p. ; 26 cm
シリーズ名: Series in mathematical analysis and applications / Edited by Ravi P. Agarwal and Donal O'Regan ; v. 7
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5.

図書

図書
Ovidiu Calin, Der-Chen Chang
出版情報: Boston : Birkhäuser, c2005  xv, 278 p. ; 24 cm
シリーズ名: Applied and numerical harmonic analysis / series editor, John J. Benedetto
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Preface
Introductory Chapter
Laplace Operator on Riemannian Manifolds
Lagrangian Formalism on Riemannian Manifolds
Harmonic Maps from a Lagrangian Viewpoint
Conservation Theorems
Hamiltonian Formalism
Hamilton-Jacobi Theory
Minimal Hypersurfaces
Radially Symmetric Spaces
Fundamental Solutions for Heat Operators with Potentials
Fundamental Solutions for Elliptic Operators
Mechanical Curves
Bibliography
Index
Preface
Introductory Chapter
Laplace Operator on Riemannian Manifolds
6.

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:
7.

図書

図書
Luca Capogna, Carlos E. Kenig, Loredana Lanzani
出版情報: Providence, R.I. : American Mathematical Society, c2005  x, 155 p. ; 26 cm
シリーズ名: University lecture series ; 35
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Motivation and statement of the main results
The relation between potential theory and geometry for planar domains
Preliminary results in potential theory
Reifenberg flat and chord arc domains
Further results on Reifenberg flat and chord arc domains
From the geometry of a domain to its potential theory
From potential theory to the geometry of a domain
Higher codimension and further regularity results
Bibliography
Motivation and statement of the main results
The relation between potential theory and geometry for planar domains
Preliminary results in potential theory
8.

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
9.

図書

図書
L. Ambrosio, N. Dancer ; edited by G. Buttazzo, A. Marino, M.K.V. Murthy
出版情報: Berlin : Springer, c2000  viii, 347 p. ; 24 cm
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10.

図書

図書
Irena Lasiecka, Roberto Triggiani
出版情報: Cambridge ; New York : Cambridge University Press, 2000  xxi, 644 p.
シリーズ名: Encyclopedia of mathematics and its applications / edited by G.-C. Rota ; v. 74 . Control theory for partial differential equations : continuous and approximation theories / Irena Lasiecka, Roberto Triggiani ; v. 1
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