Partial differential equations / by Lipman Bers, Fritz John, Martin Schechter ; with special lectures by Lars Garding and the late A. N. Milgram.
Series Lectures in applied mathematics ; v. 3Editor: New York : Interscience, c1964Descripción: xiii, 343 p. ; 24 cmOtra clasificación: 35-06Contents Part I. Hyperbolic and Parabolic Equations, by Fritz John 1. Equations of Hyperbolic and Parabolic Types [1] 2. The Wave Operator [4] 2.1. The one-dimensional wave equation [4] 2.2. The initial value problem for the wave equation in three-space [10] 2.3. Analysis of the solution [13] 2.4. The method of descent [16] 2.5. The inhomogeneous wave equation [17] 2.6. The Cauchy problem for general initial surfaces [18] 2.7. Energy integrals and a priori estimates [24] 2.8. The general linear equation with the wave operator as principal part [31] 2.9. Mixed problems [35] 3. Cauchy’s Problem, Characteristic Surfaces, and Propagation of Discontinuities [38] 3.1. Notation [38] 3.2. Relations between partial derivatives on a surface [40] 3.3. Free surfaces. Characteristic matrix [43] 3.4. Cauchy’s problem. The uniqueness theorem of Holmgren [45] 3.5. Propagation of discontinuities [53] 4. Linear Hyperbolic Differential Equations [62] 4.1. Solution of the homogeneous equation with constant coefficients by Fourier transform [64] 4.2. Extension to hyperbolic systems of homogeneous equations with constant coefficients [69] 4.3. Method of decomposition into plane waves [71] 4.4. A. priori estimates [74] 4.5. The general linear strictly hyperbolic equation with constant principal part [78] 4.6. First-order systems with constant principal part [82] 4.7. Symmetric hyperbolic systems with variable coefficients [87] 5. A Parabolic Equation: The Equation of Heat Conduction [94] 5.1. Parabolic equations in general [94] 5.2. The heat equation. Maximum principle [96] 5.3. Solution of the initial value problem [98] 5.4. Smoothness of solutions [101] 5.5. The boundary initial value problem for a rectangle [104] 6. Approximation of Solutions of Partial Differential Equations by the Method of Finite Differences [108] 6.1. Solution of parabolic equations [109] 6.2. Stability of difference schemes for other types of equations [115] Bibliography [123] Part II. Elliptic Equations, by Lipman Bers and Martin Schechter [131] 1. Elliptic Equations and Their Solutions [133] 1.1. Introduction [131] 1.2. Linear elliptic equations [134] 1.3. Smoothness of solutions [135] 1.4. Unique continuation [139] 1.5. Boundary conditions [141] Appendix I. Elliptic versus Strongly Elliptic [143] Appendix II. “Weak Equals Strong” [144] 2. The Maximum Principle [150] 2.1. Second-order equations [150] 2.2. Statement and proof of the maximum principle [150] 2.3. Applications to the Dirichlet problem [152] 2.4. Applications to the generalized Neumann problem [154] 2.5. Solution of the Dirichlet problem by finite differences [155] 2.6. Solution of the difference equation by iterations [158] 2.7. A maximum principle for gradients [160] 2.8. Carleman’s unique continuation theorem [162] 3. Hilbert Space Approach. Periodic Solutions [164] 3.1. Periodic solutions [164] 3.2. The Hilbert spaces Ht [165] 3.3. Structure of the spaces Ht [167] 3.4. Basic inequalities [170] 3.5. Differentiability theorem [174] 3.6. Solution of the equation Lu = f [175] Appendix I. The Projection Theorem [177] Appendix II. The Fredholm-Riesz-Schauder Theory [183] 4. Hilbert Space Approach. Dirichlet Problem [190] 4.1. Introduction [190] 4.2. Interior regularity [190] 4.3. The spaces and Ht and H0t [192] 4.4. Some lemmas in H0t [193] 4.5. The generalized Dirichlet problem [196] 4.6. Existence of weak solutions [198] 4.7. Regularity at the boundary [200] 4.8. Inequalities in a half-cube [202] Appendix. Analyticity of Solutions [207] 5. Potential Theoretical Approach [211] 5.1. Fundamental solutions. Parametrix [211] 5.2. Some function spaces [216] 5.3. Fundamental inequalities [220] 5.4. Local existence theorem [228] 5.5. Interior Schauder type estimates [231] 5.6. Estimates up to the boundary [235] 5.7. Applications to the Dirichlet problem [237] 5.8. Smoothness of strong solutions [240] Appendix I. Proofs of the Fundamental Inequalities [242] Appendix II. Proofs of the Interpolation Lemmas [250] 6. Function Theoretical Approach [254] 6.1. Complex notation [255] 6.2. Beltrami equation [257] 6.3. A representation theorem [259] 6.4. Consequences of the representation theorem [261] 6.5. Two boundary value problems [263] Appendix. Properties of the Beltrami Equation. Privaloff’s Theorem [267] 7. Quasi-Linear Equations [282] 7.1. Boundary value problems [282] 7.2. Methods of solution [284] 7.3. Examples [286] Bibliography [291] Supplement I. Eigenvalue Expansions, by Lars GArding [301] Supplement II. Parabolic Equations, by A. N. Milgram [327] Index [341]
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Proceedings of the Summer Seminar, Boulder, Colorado, 1957.
"Third volume of the Proceedings of the Summer Seminar on Applied Mathematics, sponsored by the American Mathematical Society and held at the University of Colorado over the four weeks beginning June 23, 1957."
Contents --
Part I. Hyperbolic and Parabolic Equations, by Fritz John --
1. Equations of Hyperbolic and Parabolic Types [1] --
2. The Wave Operator [4] --
2.1. The one-dimensional wave equation [4] --
2.2. The initial value problem for the wave equation in three-space [10] --
2.3. Analysis of the solution [13] --
2.4. The method of descent [16] --
2.5. The inhomogeneous wave equation [17] --
2.6. The Cauchy problem for general initial surfaces [18] --
2.7. Energy integrals and a priori estimates [24] --
2.8. The general linear equation with the wave operator as principal part [31] --
2.9. Mixed problems [35] --
3. Cauchy’s Problem, Characteristic Surfaces, and Propagation of Discontinuities [38] --
3.1. Notation [38] --
3.2. Relations between partial derivatives on a surface [40] --
3.3. Free surfaces. Characteristic matrix [43] --
3.4. Cauchy’s problem. The uniqueness theorem of Holmgren [45] --
3.5. Propagation of discontinuities [53] --
4. Linear Hyperbolic Differential Equations [62] --
4.1. Solution of the homogeneous equation with constant coefficients by Fourier transform [64] --
4.2. Extension to hyperbolic systems of homogeneous equations with constant coefficients [69] --
4.3. Method of decomposition into plane waves [71] --
4.4. A. priori estimates [74] --
4.5. The general linear strictly hyperbolic equation with constant principal part [78] --
4.6. First-order systems with constant principal part [82] --
4.7. Symmetric hyperbolic systems with variable coefficients [87] --
5. A Parabolic Equation: The Equation of Heat Conduction [94] --
5.1. Parabolic equations in general [94] --
5.2. The heat equation. Maximum principle [96] --
5.3. Solution of the initial value problem [98] --
5.4. Smoothness of solutions [101] --
5.5. The boundary initial value problem for a rectangle [104] --
6. Approximation of Solutions of Partial Differential Equations by the Method of Finite Differences [108] --
6.1. Solution of parabolic equations [109] --
6.2. Stability of difference schemes for other types of equations [115] --
Bibliography [123] --
Part II. Elliptic Equations, by Lipman Bers and Martin Schechter [131] --
1. Elliptic Equations and Their Solutions [133] --
1.1. Introduction [131] --
1.2. Linear elliptic equations [134] --
1.3. Smoothness of solutions [135] --
1.4. Unique continuation [139] --
1.5. Boundary conditions [141] --
Appendix I. Elliptic versus Strongly Elliptic [143] --
Appendix II. “Weak Equals Strong” [144] --
2. The Maximum Principle [150] --
2.1. Second-order equations [150] --
2.2. Statement and proof of the maximum principle [150] --
2.3. Applications to the Dirichlet problem [152] --
2.4. Applications to the generalized Neumann problem [154] --
2.5. Solution of the Dirichlet problem by finite differences [155] --
2.6. Solution of the difference equation by iterations [158] --
2.7. A maximum principle for gradients [160] --
2.8. Carleman’s unique continuation theorem [162] --
3. Hilbert Space Approach. Periodic Solutions [164] --
3.1. Periodic solutions [164] --
3.2. The Hilbert spaces Ht [165] --
3.3. Structure of the spaces Ht [167] --
3.4. Basic inequalities [170] --
3.5. Differentiability theorem [174] --
3.6. Solution of the equation Lu = f [175] --
Appendix I. The Projection Theorem [177] --
Appendix II. The Fredholm-Riesz-Schauder Theory [183] --
4. Hilbert Space Approach. Dirichlet Problem [190] --
4.1. Introduction [190] --
4.2. Interior regularity [190] --
4.3. The spaces and Ht and H0t [192] --
4.4. Some lemmas in H0t [193] --
4.5. The generalized Dirichlet problem [196] --
4.6. Existence of weak solutions [198] --
4.7. Regularity at the boundary [200] --
4.8. Inequalities in a half-cube [202] --
Appendix. Analyticity of Solutions [207] --
5. Potential Theoretical Approach [211] --
5.1. Fundamental solutions. Parametrix [211] --
5.2. Some function spaces [216] --
5.3. Fundamental inequalities [220] --
5.4. Local existence theorem [228] --
5.5. Interior Schauder type estimates [231] --
5.6. Estimates up to the boundary [235] --
5.7. Applications to the Dirichlet problem [237] --
5.8. Smoothness of strong solutions [240] --
Appendix I. Proofs of the Fundamental Inequalities [242] --
Appendix II. Proofs of the Interpolation Lemmas [250] --
6. Function Theoretical Approach [254] --
6.1. Complex notation [255] --
6.2. Beltrami equation [257] --
6.3. A representation theorem [259] --
6.4. Consequences of the representation theorem [261] --
6.5. Two boundary value problems [263] --
Appendix. Properties of the Beltrami Equation. Privaloff’s Theorem [267] --
7. Quasi-Linear Equations [282] --
7.1. Boundary value problems [282] --
7.2. Methods of solution [284] --
7.3. Examples [286] --
Bibliography [291] --
Supplement I. Eigenvalue Expansions, by Lars GArding [301] --
Supplement II. Parabolic Equations, by A. N. Milgram [327] --
Index [341] --
MR, 29 #346
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