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1.

図書
図書
1. Spectral representations for Schrödinger operators with long-range potentials (: Berlin ; : New York)
Yoshimi Saitō
出版情報: Berlin ; New York : Springer-Verlag, 1979  148 p. ; 25 cm
シリーズ名: Lecture notes in mathematics ; 727
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Preface to the Third Edition
Acknowledgments
Skin, Scalp, and Nail / 1：
Neck / 2：
Breast / 3：
Abdominal Wall and Hernias / 4：
Diaphragm / 5：
Esophagus / 6：
Stomach / 7：
Duodenum / 8：
Pancreas / 9：
Small Intestine / 10：
Appendix / 11：
Colon and Anorectum / 12：
Liver / 13：
Extrahepatic Biliary Tract / 14：
Spleen / 15：
Adrenal Glands / 16：
Vascular System / 17：
Uterus, Tubes, and Ovaries / 18：
Carpal Tunnel / 19：
Microsurgical Procedures / 20：
Index
Preface to the Third Edition
Acknowledgments
Skin, Scalp, and Nail / 1：
2.

図書
図書
2. Approximation of elliptic boundary-value problems
Jean-Pierre Aubin
出版情報: New York : Wiley-Interscience, c1972  xvii, 360 p. ; 24 cm
シリーズ名: Pure and applied mathematics ; v. 26
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Introduction
Aim and Scope / 1：
Neumann Problems / 2：
Introduction of Internal Approximations / 3：
Properties of Internal Approximations / 4：
Stability, Optimal Stability, and Regularity of the Convergence / 5：
The Case of Operators Mapping a Hilbert Space onto Its Dual / 6：
Finite-Element Approximations of Sobolev Spaces / 7：
Approximation of Nonhomogeneous Neumann Problems / 8：
Approximations of Nonhomogeneous Dirichlet Problems / 9：
A Posteriori Error Estimates / 10：
External and Partial Approximations / 11：
General Outline / 12：
Approximation of Solutions of Neumann Problems for Second-Order Linear Differential Equations
Weak Solutions of Neumann Problems for Second-Order Linear Differential Operators
The Neumann Boundary-Value Problem / 1-1：
Definition of Distributions / 1-2：
Weak Derivatives of a Distribution / 1-3：
Variational Formulation of the Problem / 1-4：
Weak Solutions of the Neumann Boundary-Value Problem / 1-5：
Sobolev Spaces / 1-6：
The Lax-Milgram Theorem / 1-7：
Approximation of an Abstract Variational Problem
The Galerkin Approximation of a Separable Hilbert Space / 2-1：
Approximation of a Hilbert Space / 2-2：
Internal Approximation of a Variational Equation / 2-3：
Existence, Uniqueness, and Convergence Properties / 2-4：
Estimates of Global Error / 2-5：
What Kind of Approximations Should Be Chosen? / 2-6：
Examples of Approximations of Sobolev Spaces
Piecewise-Linear Approximations of the Sobolev Space H[superscript 1] (I) / 3-1：
Estimates of Error Functions of Piecewise-Linear Approximations / 3-2：
Examples of Approximate Equations
Construction of a Finite-Difference Scheme / 4-1：
A Simpler Finite-Difference Scheme / 4-2：
Approximations of Hilbert Spaces
Hilbert Spaces and Their Duals
Dual of a Hilbert Space and Canonical Isometry
Example: Finite-Dimensional Hilbert Spaces
Hahn-Banach Theorem
Dual of a Dense Subspace
Imbedding of a Space into Its Dual
Example: Imbedding of Spaces of Functions into Spaces of Distributions
Dual of Closed Subspaces and Factor Spaces
Applications to Error Estimates / 1-8：
Dual of a Product / 1-9：
Dual of Domains of Operators / 1-10：
Examples: Dual of Sobolev Spaces H[subscript 0 superscript m](I) / 1-11：
Properties of Bounded Sets of Operators; Uniform Boundedness / 1-12：
Banach Theorem / 1-13：
Dual of Sobolev Spaces H[superscript m](I) / 1-14：
The Riesz-Fredholm Alternative / 1-15：
V-Elliptic and Coercive Operators / 1-16：
Quasi-Optimal Approximations
Stability Functions
Duality Relations between Error and Stability Functions
Estimates of the Stability Functions
Quasi-Optimal Approximations; Estimate of the Error Function
Truncation Errors and Error Functions
Optimal Approximations
Eigenvalues and Eigenvectors of Symmetric Compact Operators
Optimal Galerkin Approximations
Convergence and Optimality Properties / 3-3：
Spaces H[subscript Theta] / 3-4：
Optimal Restrictions and Prolongations; Applications
Optimal Restrictions and Prolongations
Dual Approximations
Construction of Optimal Prolongations and Restrictions / 4-3：
Miscellaneous Remarks / 4-4：
Characterization of Error and Stability Functions / 4-5：
Spaces of Order [Theta] / 4-6：
Approximation of Operators
Internal Approximations
Construction of an Internal Approximate Equation
The Case of Finite-Dimensional Discrete Spaces
The Case of Operators from V onto V[prime]
Stability of Internal Approximations of Operators
Convergence and Error Estimates
Approximation of a Sum of an Isomorphism and a Compact Operator
Approximation of Coercive and V-Elliptic Operators
Optimal and Quasi-Optimal Stability
Regularity of the Convergence and Estimates of Error in Terms of n-Width
Stability and Convergence in Smaller Spaces
Stability and Convergence in Larger Spaces
Approximation of the Value of a Functional at a Solution
Discrete Convergence, Consistency, and Optimal Approximation of Linear Operators
Discrete Convergence and Consistency
Optimal Approximation of Operators and Internal Approximations
Estimates of Error and Discrete Errors
Finite-Element Approximation of Functions of One Variable
Approximation of Functions of L[superscript 2] by Step Functions and by Convolution
The Space L[superscript 2] and the Discrete Space L[subscript h superscript 2]]
The Prolongations P[subscript h superscript 0]
The Restrictions r[subscript h]
The Theorem of Convergence
Convolution of Functions and Measures
Approximation by Convolution
Piecewise-Polynomial Approximations of Sobolev Spaces H[superscript m]
Finite-Difference Operators
Construction of Approximations of the Space H[superscript m]
Convergence Theorem
Explicit Form of Functions [Pi subscript m]
Properties of the Prolongations p[subscript h superscript m]
Optimal Properties of Prolongations p[subscript h superscript m] / 2-7：
Finite-Element Approximations of Sobolev Spaces H[superscript m]
Finite-Element Approximations
The Criterion of m-Convergence
Characterization of Convergent Finite-Element Approximations
Stability Properties of Finite-Element Approximations
Finite-Element Approximation of Functions of Several Variables
Approximations of the Sobolev Spaces H[superscript m](R[superscript n])
Notations
(2m + 1)[superscript n]-Level Piecewise-Polynomial Approximations
[2(2m)[superscript n] - (2m - 1)[superscript n]]-Level Piecewise-Polynomial Approximations
Approximations of the Sobolev Spaces H[superscript m]([Omega])
Sobolev Spaces H[superscript m]([Omega])
Finite-Element Approximations of H[superscript m]([Omega])
Quasi-Optimal Finite-Element Approximations of H[superscript m]([Omega])
Piecewise-Polynomial Approximations of H[superscript m]([Omega])
Approximation of the Sobolev Spaces H[subscript 0 superscript m]([Omega])
Sobolev Spaces H[subscript 0 superscript m]([Omega])
Finite-Element Approximations of H[subscript 0 superscript m]([Omega])
Convergent Finite-Element Approximations of H[subscript 0 superscript m]([Omega])
Boundary-Value Problems and the Trace Theorem
Some Variational Boundary-Value Problems for the Laplacian
The Laplacian
Characterization of Sobolev Spaces H[subscript 0 superscript 1]([Omega])
The Green Formula
The Dirichlet Problem for the Laplacian
The Neumann Problem for the Laplacian
A Mixed Problem for the Laplacian
An Oblique Problem for the Laplacian
Existence and Uniqueness of the Solutions
Variational Boundary-Value Problems and Their Adjoints
Spaces V, H and Operator [gamma]
Formal Operator [Lambda] Associated with a(u, v)
Abstract Neumann and Dirichlet Problems Associated with a(u, v)
Mixed Type Boundary-Value Problems Associated with a(u, v)
Existence and Uniqueness of the Solutions of Boundary-Value Problems
Formal Adjoint of an Operator and Green's Formula
Theorems of Regularity / 2-8：
The Trace Theorem and Properties of Sobolev Spaces
Statement of the Trace Theorem
Change of Coordinates
Sobolev Spaces H[superscript s](R[superscript n]) for Real Numbers s
Sobolev Spaces H[superscript s]([Gamma] and H[superscript s]([Omega])
Trace Operators and Operators of Extension: Theorems of Density / 3-5：
Properties of the Spaces H[superscript m](R[subscript + superscript n]) / 3-6：
Proof of the Trace Theorem / 3-7：
Sobolev Inequalities and the Trace Theorem in Space H[superscript s]([Omega]) / 3-8：
Theorem of Compactness / 3-9：
Examples of Boundary-Value Problems
Boundary-Value Problems for Second-Order Differential Operators
Second-Order Linear Differential Operators
Elliptic Second-Order Partial Differential Operators
The Dirichlet Problem
The Neumann Problem
Mixed Problems
Oblique Problems
Interface Problems
The Regularity Theorem
Theorems of Isomorphism
Value of the Solution at a Point of the Boundary
Problems with Elliptic Differential Boundary Conditions
Boundary-Value Problems for Differential Operators of Higher Order
Linear Differential Operators of Order 2k
Regularity and Theorems of Isomorphism
Other Boundary-Value Problems
Boundary Value Problems for [Delta][superscript 2] + [lambda]
Approximation of Neumann-Type Problems
Theorems of Convergence and Error Estimates
Internal Approximation of a Neumann-type Problem
Convergence and Estimates of Error in Larger Spaces
Approximation of Neumann Problems for Elliptic Operators of Order 2k
Approximation of Neumann Problems for Elliptic Differential Operators
Convergence Properties of Finite Element Approximations of Neumann Problems
The (2m + 1)[superscript n]-Level Approximations of the Neumann Problem
The [2(2m)[superscript n] - (2m - 1)[superscript n]]-Level Approximations of the Neumann Problem
Approximations of the Spaces H[superscript k]([Omega], [Lambda] and H([Omega], [Lambda]
Approximation of Other Neumann-Type Problems
Approximation of the Value of the Solution at a Point of the Boundary
Approximation of Oblique Boundary-Value Problems
Approximation of a Problem with Elliptic Boundary Conditions
Approximation of Interface Problems
Approximation of the Neumann Problem for [Delta][superscript 2] + [gamma]
Perturbed Approximations and Least-Squares Approximations
Perturbed Approximations
Internal Approximation of a Variational Boundary-Value Problem
Perturbed Approximation of a Variational Boundary-Value Problem
Convergence in the Initial Space
Estimates of Error
Convergence in Smaller Spaces
Convergence in Larger Spaces
Perturbed Approximations of Boundary-Value Problems
Perturbed Approximations by Finite-Element Approximations
Error Estimates and Regularity of the Convergence
The 3[superscript n]-level Perturbed Approximation of the Dirichlet Problem
Least-Squares Approximations
Least-Squares Approximation Schemes
Error Estimates (I)
Error Estimates (II)
Least-Squares Approximations of Dirichlet Problems
Conjugate Problems and A Posteriori Error Estimates
Conjugate Problems of Boundary-Value Problems
First Example of a Conjugate Problem
Second Example of a Conjugate Problem
Construction of Conjugate Problems
Applications to the Approximation of Dirichlet Problems
Approximation of the Dirichlet Problem (I)
Approximation of the Dirichlet Problem (II)
The Case of Second-Order Differential Operators
Finite-Element Approximations of the Spaces H[superscript k]([Omega], D*)
Spaces H[superscript k]([Omega], D*)
Approximations of the Space H[superscript k]([Omega], D*)
Approximation of the Second Example of a Conjugate Problem
Approximation of the Conjugate Dirichlet Problem
Properties of the Discrete Conjugate Problem
External Approximations; Stability, Convergence, and Error Estimates
Definition of External Approximations
Example: Partial Approximations of a Finite Intersection of Spaces
Stability and Convergence of External Approximations of Operators
Estimates of Error and Regularity of the Convergence
Properties of the External Error Functions
External and Partial Approximations of Variational Equations
Partial Approximation of a Split Variational Equation
External Approximation of Variational Equations
Partial Approximation of Neumann Problems
Perturbed Partial Approximation of Boundary-Value Problems
Partial Approximations of Sobolev Spaces
Spaces H([Omega], D[subscript i])
Partial Approximations of the Sobolev Space H[superscript 1]([Omega])
Estimates of Truncation Errors and External Error Functions
Partial Approximations of the Sobolev Spaces H[superscript m]([Omega]) and H[subscript 0 superscript m]([Omega])
Partial Approximation of Boundary-Value Problems
Partial Approximation of Second-Order Linear Operators
Partial Approximation of the Neumann Problem
Perturbed Partial Approximation of Mixed Boundary-Value Problems
Estimates of Error in the Interior
Partial Approximations of Higher-Order Differential Operators
Comments
References
Index
Introduction
Aim and Scope / 1：
Neumann Problems / 2：
3.

図書
図書
3. Methoden der Potentialtheorie für elliptische Differentialgleichungen beliebiger Ordnung
von Bert-Wolfgang Schulze und Günther Wildenhain
出版情報: Basel [etc.] : Birkhäuser, 1977  xv, 408 p. ; 25 cm
シリーズ名: Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften ; . Mathematische Reihe ; Bd. 60
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4.

図書
図書
4. Constructive methods for elliptic equations (: Germany ; : U.S.)
Robert P. Gilbert
出版情報: Berlin ; New York : Springer-Verlag, 1974  vi, 397 p ; 24 cm
シリーズ名: Lecture notes in mathematics ; 365
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5.

図書
図書
5. Elliptic systems in the plane
W.L. Wendland
出版情報: London ; San Francisco : Pitman, c1979  xi, 404 p. ; 24 cm
シリーズ名: Monographs and studies in mathematics ; 3
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6.

図書
図書
6. Partial differential equations of elliptic type. 2d., rev. ed
Carlo Miranda ; translated by Zane C. Motteler
出版情報: Berlin ; New York : Springer-Verlag, 1970  xii, 370 p. ; 24 cm
シリーズ名: Ergebnisse der Mathematik und ihrer Grenzgebiete ; Bd. 2
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7.

図書
図書
7. Probability methods for approximations in stochastic control and for elliptic equations
Harold J. Kushner
出版情報: New York : Academic Press, 1977  xvii, 243 p. ; 23 cm
シリーズ名: Mathematics in science and engineering : a series of monographs and textbooks ; v. 129
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8.

図書
図書
8. Elliptic partial differential equations of second order
David Gilbarg, Neil S. Trudinger
出版情報: Berlin ; New York : Springer-Verlag, 1977 c1957  x, 401 p. ; 25 cm
シリーズ名: Die Grundlehren der mathematischen Wissenschaften ; 224
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目次情報: 続きを見る
Introduction / Chapter 1：
Linear Equations / Part I：
Laplace''s Equation / Chapter 2：
The Mean Value Inequalities / 2.1：
Maximum and Minimum Principle / 2.2：
The Harnack Inequality / 2.3：
Green''s Representation / 2.4：
The Poisson Integral / 2.5：
Convergence Theorems / 2.6：
Interior Estimates of Derivatives / 2.7：
The Dirichlet Problem; the Method of Subharmonic Functions / 2.8：
CapacityProblems / 2.9：
The Classical Maximum Principle / Chapter 3：
The Weak Maximum Principle / 3.1：
The Strong Maximum Principle / 3.2：
Apriori Bounds / 3.3：
Gradient Estimates for Poisson''s Equation / 3.4：
A Harnack Inequality / 3.5：
Operators in Divergence FormNotesProblems / 3.6：
Poisson''s Equation and Newtonian Potential / Chapter 4：
H+ lder Continuity / 4.1：
The Dirichlet Problem for Poisson''s Equation / 4.2：
H+ lder Estimates for the Second Derivatives / 4.3：
Estimates at the Boundary / 4.4：
H+ lder Estimates for the First DerivativesNotes Problems / 4.5：
Banach and Hilbert Spaces / Chapter 5：
The Contraction Mapping / 5.1：
The Method of Cintinuity / 5.2：
The Fredholm Alternative / 5.3：
Dual Spaces and Adjoints / 5.4：
Hilbert Spaces / 5.5：
The Projection Theorem / 5.6：
The Riesz Representation Theorem / 5.7：
The Lax-Milgram Theorem / 5.8：
The Fredholm Alternative in Hilbert Spaces / 5.9：
Weak CompactnessNotesProblems / 5.10：
Classical Solutions; the Schauder Approach / Chapter 6：
The Schauder Interior Estimates / 6.1：
Boundary and Global Estimates / 6.2：
The Dirichlet Problem / 6.3：
Interior and Boundary Regularity / 6.4：
An Alternative Approach / 6.5：
Non-Uniformly Elliptic Equations / 6.6：
Other Boundary Conditions; the Obliue Derivative Problem / 6.7：
Appendix 1: Interpolation Inequalities / 6.8：
Appendix 2: Extension LemmasNotesProblems / 6.9：
Sobolev Spaces / Chapter 7：
L^p spaces / 7.1：
Regularization and Approximation by Smooth Functions / 7.2：
Weak Derivatives / 7.3：
The Chain Rule / 7.4：
The W^(k,p) Spaces / 7.5：
Density Theorems / 7.6：
Imbedding Theorems / 7.7：
Potential Estimates and Imbedding Theorems / 7.8：
The Morrey and John-Nirenberg Estimes / 7.9：
Compactness Results / 7.10：
Difference Quotients / 7.11：
Extension and InterpolationNotesProblems / 7.12：
Generalized Solutions and Regularity / Chapter 8：
Solvability of the Dirichlet Problem / 8.1：
Diferentiability of Weak Solutions / 8.3：
Global Regularity / 8.4：
Global Boundedness of Weak Solutions / 8.5：
Local Properties of Weak Solutions / 8.6：
Local Estimates at the Boundary / 8.7：
H+ lder Estimates for the First Derivatives / 8.11：
The Eigenvalue ProblemNotesProblems / 8.12：
Strong Solutions / Chapter 9：
Maximum Princiles for Strong Solutions / 9.1：
L^p Estimates: Preliminary Analysis / 9.2：
The Marcinkiewicz Interpolation Theorem / 9.3：
The Calderon-Zygmund Inequality / 9.4：
L^p Estimates / 9.5：
A Local Maximum Principle / 9.6：
H+ lder and Harnack Estimates / 9.8：
Local Estimates at the BoundaryNotesProblems / 9.9：
Quasilinear Equations / Part II：
Maximum and Comparison Principles / Chapter 10：
The Comparison Principle / 10.1：
Maximum Principles / 10.2：
A Counterexample / 10.3：
Comparison Principles for Divergence Form Operators / 10.4：
Maximum Principles for Divergence Form Operators Notes Problems / 10.5：
Topological Fixed Point Theorems and Their Application / Chapter 11：
The Schauder Fixes Point Theorem / 11.1：
The Leray-Schauder Theorem: a Special Case / 11.2：
An Application / 11.3：
The Leray-Schauder Fixed Point Theorem / 11.4：
Variational ProblemsNotes / 11.5：
Equations in Two Variables / Chapter 12：
Quasiconformal Mappings / 12.1：
h+ lder Gradient Estimates for Linear Equations / 12.2：
The Dirichlet Problem for Uniformly Elliptic Equations / 12.3：
Non-Uniformly Elliptic EquationsNotesProblems / 12.4：
H+ lder Estimates for the Gradient / Chapter 13：
Equations of Divergence Form / 13.1：
Equations of General Form; the Interior Estimate / 13.2：
Equations of General Form; the Boundary Estimate / 13.4：
Application to the Dirichlet ProblemNotes / 13.5：
Boundary Gradient Estimates / Chapter 14：
General Domains / 14.1：
Convex Domains / 14.2：
Boundary Curvature Conditions / 14.3：
Non-Existence Results / 14.4：
Continuity Estimates / 14.5：
Appendix: Boundary Curvature and the Distance FunctionNotesProblems / 14.6：
Introduction / Chapter 1：
Linear Equations / Part I：
Laplace''s Equation / Chapter 2：
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