Partial Differential Equations

1. Front Matter

   Pages i-ix

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2. The single first-order equation

   • Fritz John

   Pages 1-30

3. Second-order equations: hyperbolic equations for functions of two independent variables

   • Fritz John

   Pages 31-51

4. Characteristic manifolds and the Cauchy problem

   • Fritz John

   Pages 52-71

5. The Laplace equation

   • Fritz John

   Pages 72-102

6. Hyperbolic equations in higher dimensions

   • Fritz John

   Pages 103-155

7. Higher-order elliptic equations with constant coefficients

   • Fritz John

   Pages 156-165

8. Parabolic equations

   • Fritz John

   Pages 166-189

9. Back Matter

   Pages 191-198

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The book has been completely rewritten for this new edition. While most of the material found in the earlier editions has been retained, though in changed form, there are considerable additions, in which extensive use is made of Fourier transform techniques, Hilbert space, and finite difference methods. A condensed version of the present work was presented in a series of lectures as part of the Tata Institute of Fundamental Research -Indian Insti­ tute of Science Mathematics Programme in Bangalore in 1977. I am indebted to Professor K. G. Ramanathan for the opportunity to participate in this excit­ ing educational venture, and to Professor K. Balagangadharan for his ever ready help and advice and many stimulating discussions. Very special thanks are due to N. Sivaramakrishnan and R. Mythili, who ably and cheerfully prepared notes of my lectures which I was able to use as the nucleus of the present edition. A word about the choice of material. The constraints imposed by a partial differential equation on its solutions (like those imposed by the environment on a living organism) have an infinite variety of con­ sequences, local and global, identities and inequalities. Theories of such equations usually attempt to analyse the structure of individual solutions and of the whole manifold of solutions by testing the compatibility of the differential equation with various types of additional constraints.

• Cauchy problem
• Equations
• Finite
• Fourier transform
• Partielle Differentialgleichung
• constraint
• differential equation
• education
• hyperbolic equation
• maximum principle
• partial differential equation
• solution
• testing
• types
• wave equation
• partial differential equations

Fourth Edition

F. John

Partial Differential Equations

"An excellent second-reading text. Should be accessible to any mathematician. Highly recommended."

—THE MATHEMATICAL GAZETTE

• Courant Institute of Mathematical Sciences, New York University, New York, USA

  Fritz John