Partial Differential Equations (Lectures in Applied Mathematics)
By Lipman Bers, Fritz John, Martin Schechter
• Publisher: Amer Mathematical Society
• Number Of Pages: 343
• Publication Date: 1981-06
• ISBN-10 / ASIN: 0821800493
• ISBN-13 / EAN: 9780821800492
Product Description:
This book consists of two main parts. The first part, "Hyperbolic and Parabolic Equations", written by F. John, contains a well-chosen assortment of material intended to give an understanding of some problems and techniques involving hyperbolic and parabolic equations. The emphasis is on illustrating the subject without attempting to survey it. The point of view is classical, and this serves well in furnishing insight into the subject; it also makes it possible for the lectures to be read by someone familiar with only the fundamentals of real and complex analysis.The second part, "Elliptic Equations", written by L. Bers and M. Schechter, contains a very readable account of the results and methods of the theory of linear elliptic equations, including the maximum principle, Hilbert-space methods, and potential-theoretic methods. It also contains a brief discussion of some quasi-linear elliptic equations. The book is suitable for graduate students and researchers interested in partial differential equations.
Part I . Hyperbolic and Parabolic Equations.
. . . . . . . . . . . . .
1. Equations of Hyperbolic and Parabolic Types . . . . . .
2 . The Wave Operator . . . . . . . . . . . . .
2.1. The one-dimensional wave equation . . . . . . .
2.2. Thc initial value problem for the wave equation in three-space
2.3. Analysis of the solution . . . . . . . . . . .
2.4. The method of descent . . . . . . . . . . .
2.5. The inhomogenous wave equation . . . . . . . .
2.6. The. Cauehy problem for general initial surfaces . . . .
2.7. Energy integrals and a priori estimates . . . . . .
2.8. The general linear equation with the wave operator as principal part .
2.9. Mixed problems . . . . . . . . . . . . .
3 . Cauchy 's Problem. Characteristic Surfaces. and Propagation of
Discontinuities .
3.1. Notation . . . . . . . . . . . . . . .
3.2. Relations between partial derivatives on a surface . . .
3.3. Free surfaces. Characteristic matrix . . . . . . . .
3.4. Cauchy's problem . The uniqueness theorem of Holmgren .
3.5. Propagation of discontinuities . . . . . . . . .
4. Linear Hyperbolic Differential Equations . . . . . . .
4.1. Solution of the homogeneous equation with constant coeffi-
cients by Fourier transform . . . . . . . . . .
4.2. Extension to hyperbolic systems of homogeneous equations
with constant coefficients . . . . . .
4.3. Method of decomposition into plane waves . . . . .
4.4. Afirion'estimates . . . . . . . . . . . . .
4.5. The general linear strictly hyperbolic equation with constant
principal part . . . . . . . . . . .
4.6. First-order system with constant principal part . . . .
4.7. Symmetric hyptrboIic systems with variable coefficients .
5 . A Parabolic Equation: 'I'hc Equation of Heat Conduction . .
5.1. Parabolic equations in gcneral . . . . . . .
5.2. 'I'he heat equation . SIaxirnum principle . . . . .
5.3. Solution of the initial value problem . . . . . .
5.4. Smoothnessofsolutions . . . . 101
5.5. The boundary initial value problem for a rectangle .
6 . Approxinlat ion of Solutions of Partial Differential Equations by
the Xfethod of Finite Diqerences . . . . . . . .
6.1. Solution of parabolic equations . . . . . . . .
6.2. Stability of difference schemes for other types of equations
Bibliography . . . . . . . . . .
Part II . Elliptic Equations.
by LIPMANB ERSa nd MARTINS CHECHTE.R . . . . .
1. Elliptic Equations and Their Solutions . . . . . . . .
1.1. Introdtiction . . . .
1.2. Linear ellipticequations . . . . . . . . .
1.3 . Smoothness ofsolutions . . . . . . . . . .
1.4. Unique con tinuation . .
1.5. Boundary conditions . . . . . . . . . . .
Appendix I . Elliptic velasusS trongly elliptic . . . . . .
Appendix I I . "Weak Equals Strong" . . . . .
2 . The Maximum Principle . . . . . . .
2.1. Second-order equations . . . . . . . . .
2.2. Statement and proof of the maximum principle . .
2.3. Applications to the Dirichlet problem . . . .
2.4. Applications to thc generalized Neumann problem .
2.5. Solution or the Diriclllrt problem by finite difference
2.6. Solution of thc difference equation by iterations .
2.7. . \ maximum principle for gradients . . . . . .
2.8. Carleman's unique continuation theorem . . . .
3 . Hiibert Space Approach . Periodic Solutions .
3.1. Periodic sotutions . . . . . .
3.2. The Hilbcrt spaces 4 . .
3.3. Structure of the spaces, . . . . +
3.4 . Basic ineqttalities . . . . .
3.5. TlifferentiabiIity theorem . . . . . .
3.6. Solutionoftheequation hu = f . . . . .
Appendix I . The Projection Theorem . . . .
Appendix II . The Fredholm-Riesz-Schauder Theory
4 . Hilbert Space Approach . Dirichlet Problem . .
4 .I . Introduction . . . . . .
4.2. lntcrior regularity . . . . . . . . .
4.3. Method of decomposition into plane waves . . . . .
4.4. Some lemmas . . . . . .
4.5. The generalized Dirichlct problem . . .
4.6. Existence of weak solutions . . . . . . .
4.7. Regularity at the boundary . . . .
4.8. InequalitirsinahalCcube . . .
Appendix . Analyticity of Solutions . . . . . .
5 . Potentcial theoritical Approach . . . .
5.1. FundamentaI solutions . Parametrix . .
5.2. The heat equation . . . . .
5.3. Fundamental inequalities . . . .
5.4. Local existence theorem . . . . . . .
5.5. Interior Schauder type estimates . . . . .
5.6. Estimates up to the boundary . . . .
5.7 Applications to the Dirichlet problem .
5.8. Smoothness of strong solutions . . .
.4 ppendix I . Proofs of the Fundamental Inequalities
Appendix 11 . l'roofs of thc Interpolation Lemmas
.
6 . Function Theore tical Approach . . . . . . .
6.1. Complexnotation . . . . . . . . .
6.2. Beltrarmi equation . . . . . . . . .
6.3. A representation theorem . . . . . . .
6.4. Consequences of the representation theorem . . . . .
6.5. Two boundary value problems .
Appendix . Properties of the Beltrami Equation . Privaloff 's Theorem
7 . Quasi-Linear Equations . . . . . . . . .
7 . I . Boundary vatue problems . . . . .
7.2. -Methods of solution . . . . . . . . . . . .
7.3. Examples . . . . . . . . . . .
Bibliography . . . . . .
Supplement I . Eigenvalue Expansions. by LARS GARDING . .
Supplement II . Parbolic quations. . .
Index
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By Lipman Bers, Fritz John, Martin Schechter
• Publisher: Amer Mathematical Society
• Number Of Pages: 343
• Publication Date: 1981-06
• ISBN-10 / ASIN: 0821800493
• ISBN-13 / EAN: 9780821800492
Product Description:
This book consists of two main parts. The first part, "Hyperbolic and Parabolic Equations", written by F. John, contains a well-chosen assortment of material intended to give an understanding of some problems and techniques involving hyperbolic and parabolic equations. The emphasis is on illustrating the subject without attempting to survey it. The point of view is classical, and this serves well in furnishing insight into the subject; it also makes it possible for the lectures to be read by someone familiar with only the fundamentals of real and complex analysis.The second part, "Elliptic Equations", written by L. Bers and M. Schechter, contains a very readable account of the results and methods of the theory of linear elliptic equations, including the maximum principle, Hilbert-space methods, and potential-theoretic methods. It also contains a brief discussion of some quasi-linear elliptic equations. The book is suitable for graduate students and researchers interested in partial differential equations.
Part I . Hyperbolic and Parabolic Equations.
. . . . . . . . . . . . .
1. Equations of Hyperbolic and Parabolic Types . . . . . .
2 . The Wave Operator . . . . . . . . . . . . .
2.1. The one-dimensional wave equation . . . . . . .
2.2. Thc initial value problem for the wave equation in three-space
2.3. Analysis of the solution . . . . . . . . . . .
2.4. The method of descent . . . . . . . . . . .
2.5. The inhomogenous wave equation . . . . . . . .
2.6. The. Cauehy problem for general initial surfaces . . . .
2.7. Energy integrals and a priori estimates . . . . . .
2.8. The general linear equation with the wave operator as principal part .
2.9. Mixed problems . . . . . . . . . . . . .
3 . Cauchy 's Problem. Characteristic Surfaces. and Propagation of
Discontinuities .
3.1. Notation . . . . . . . . . . . . . . .
3.2. Relations between partial derivatives on a surface . . .
3.3. Free surfaces. Characteristic matrix . . . . . . . .
3.4. Cauchy's problem . The uniqueness theorem of Holmgren .
3.5. Propagation of discontinuities . . . . . . . . .
4. Linear Hyperbolic Differential Equations . . . . . . .
4.1. Solution of the homogeneous equation with constant coeffi-
cients by Fourier transform . . . . . . . . . .
4.2. Extension to hyperbolic systems of homogeneous equations
with constant coefficients . . . . . .
4.3. Method of decomposition into plane waves . . . . .
4.4. Afirion'estimates . . . . . . . . . . . . .
4.5. The general linear strictly hyperbolic equation with constant
principal part . . . . . . . . . . .
4.6. First-order system with constant principal part . . . .
4.7. Symmetric hyptrboIic systems with variable coefficients .
5 . A Parabolic Equation: 'I'hc Equation of Heat Conduction . .
5.1. Parabolic equations in gcneral . . . . . . .
5.2. 'I'he heat equation . SIaxirnum principle . . . . .
5.3. Solution of the initial value problem . . . . . .
5.4. Smoothnessofsolutions . . . . 101
5.5. The boundary initial value problem for a rectangle .
6 . Approxinlat ion of Solutions of Partial Differential Equations by
the Xfethod of Finite Diqerences . . . . . . . .
6.1. Solution of parabolic equations . . . . . . . .
6.2. Stability of difference schemes for other types of equations
Bibliography . . . . . . . . . .
Part II . Elliptic Equations.
by LIPMANB ERSa nd MARTINS CHECHTE.R . . . . .
1. Elliptic Equations and Their Solutions . . . . . . . .
1.1. Introdtiction . . . .
1.2. Linear ellipticequations . . . . . . . . .
1.3 . Smoothness ofsolutions . . . . . . . . . .
1.4. Unique con tinuation . .
1.5. Boundary conditions . . . . . . . . . . .
Appendix I . Elliptic velasusS trongly elliptic . . . . . .
Appendix I I . "Weak Equals Strong" . . . . .
2 . The Maximum Principle . . . . . . .
2.1. Second-order equations . . . . . . . . .
2.2. Statement and proof of the maximum principle . .
2.3. Applications to the Dirichlet problem . . . .
2.4. Applications to thc generalized Neumann problem .
2.5. Solution or the Diriclllrt problem by finite difference
2.6. Solution of thc difference equation by iterations .
2.7. . \ maximum principle for gradients . . . . . .
2.8. Carleman's unique continuation theorem . . . .
3 . Hiibert Space Approach . Periodic Solutions .
3.1. Periodic sotutions . . . . . .
3.2. The Hilbcrt spaces 4 . .
3.3. Structure of the spaces, . . . . +
3.4 . Basic ineqttalities . . . . .
3.5. TlifferentiabiIity theorem . . . . . .
3.6. Solutionoftheequation hu = f . . . . .
Appendix I . The Projection Theorem . . . .
Appendix II . The Fredholm-Riesz-Schauder Theory
4 . Hilbert Space Approach . Dirichlet Problem . .
4 .I . Introduction . . . . . .
4.2. lntcrior regularity . . . . . . . . .
4.3. Method of decomposition into plane waves . . . . .
4.4. Some lemmas . . . . . .
4.5. The generalized Dirichlct problem . . .
4.6. Existence of weak solutions . . . . . . .
4.7. Regularity at the boundary . . . .
4.8. InequalitirsinahalCcube . . .
Appendix . Analyticity of Solutions . . . . . .
5 . Potentcial theoritical Approach . . . .
5.1. FundamentaI solutions . Parametrix . .
5.2. The heat equation . . . . .
5.3. Fundamental inequalities . . . .
5.4. Local existence theorem . . . . . . .
5.5. Interior Schauder type estimates . . . . .
5.6. Estimates up to the boundary . . . .
5.7 Applications to the Dirichlet problem .
5.8. Smoothness of strong solutions . . .
.4 ppendix I . Proofs of the Fundamental Inequalities
Appendix 11 . l'roofs of thc Interpolation Lemmas
.
6 . Function Theore tical Approach . . . . . . .
6.1. Complexnotation . . . . . . . . .
6.2. Beltrarmi equation . . . . . . . . .
6.3. A representation theorem . . . . . . .
6.4. Consequences of the representation theorem . . . . .
6.5. Two boundary value problems .
Appendix . Properties of the Beltrami Equation . Privaloff 's Theorem
7 . Quasi-Linear Equations . . . . . . . . .
7 . I . Boundary vatue problems . . . . .
7.2. -Methods of solution . . . . . . . . . . . .
7.3. Examples . . . . . . . . . . .
Bibliography . . . . . .
Supplement I . Eigenvalue Expansions. by LARS GARDING . .
Supplement II . Parbolic quations. . .
Index
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