Partial Differential Equations

1. Front Matter

   Pages i-xi

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2. Introduction: What Are Partial Differential Equations?

   Pages 1-6

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3. The Laplace Equation as the Prototype of an Elliptic Partial Differential Equation of Second Order

   Pages 7-30

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4. The Maximum Principle

   Pages 31-50

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5. Existence Techniques I: Methods Based on the Maximum Principle

   Pages 51-75

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6. Existence Techniques II: Parabolic Methods. The Heat Equation

   Pages 77-112

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7. The Wave Equation and Its Connections with the Laplace and Heat Equations

   Pages 113-125

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8. The Heat Equation, Semigroups, and Brownian Motion

   Pages 127-156

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9. The Dirichlet Principle. Variational Methods for the Solution of PDEs (Existence Techniques III)

   Pages 157-192

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10. Sobolev Spaces and L² Regularity Theory

    Pages 193-242

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11. Strong Solutions

    Pages 243-254

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12. The Regularity Theory of Schauder and the Continuity Method (Existence Techniques IV)

    Pages 255-274

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13. The Moser Iteration Method and the Regularity Theorem of de Giorgi and Nash

    Pages 275-307

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14. Back Matter

    Pages 309-325

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This textbook is intended for students who wish to obtain an introduction to the theory of partial di?erential equations (PDEs, for short), in particular, those of elliptic type. Thus, it does not o?er a comprehensive overview of the whole ?eld of PDEs, but tries to lead the reader to the most important methods and central results in the case of elliptic PDEs. The guiding qu- tion is how one can ?nd a solution of such a PDE. Such a solution will, of course, depend on given constraints and, in turn, if the constraints are of the appropriate type, be uniquely determined by them. We shall pursue a number of strategies for ?nding a solution of a PDE; they can be informally characterized as follows: (0) Write down an explicit formula for the solution in terms of the given data (constraints). This may seem like the best and most natural approach, but this is possible only in rather particular and special cases. Also, such a formula may be rather complicated, so that it is not very helpful for detecting qualitative properties of a solution. Therefore, mathematical analysis has developed other, more powerful, approaches. (1) Solve a sequence of auxiliary problems that approximate the given one, and show that their solutions converge to a solution of that original pr- lem. Di?erential equations are posed in spaces of functions, and those spaces are of in?nite dimension.

• Maximum
• PDE
• Partial Differential Equations
• Sobolev space
• calculus
• differential equation
• hyperbolic equation
• maximum principle
• partial differential equation
• wave equation

From the reviews:

MATHEMATICAL REVIEWS

"..the composition of this book is somewhat classical. Indeed the author covers the main properties of the elliptic, parabolic and hyperbolic equations. But he always adds some interesting extensions or links between these chapters…this textbook is self-contained…Because of the nice global presentation, I recommend this book to students and young researchers who need the now classical properties of these second-order partial differential equations. Teachers will also find in this textbook the basis of an introductory course on second-order partial differential equations."

"The author covers the main properties of the elliptic parabolic and hyperbolic equations. But he always adds some interesting extensions or links between these chapters. … With an appendix on general results concerning Banach or Hilbert spaces, this textbook is self-contained. … Because of the nice global presentation, I recommend this book to students and young researchers who need the now classical properties of these second-order partial differential equations." (Alain Brillard, Mathematical Reviews, 2003f)

"Jost’s book … focuses mainly on elliptic PDEs and gives a comprehensive overview of the modern theory of solving such equations. … There are extremely helpful summaries and a handful of exercises at the end of each of the 11 chapters; an appendix covers background functional analysis. … Throughout, Jost achieves an impeccable blend of motivation, orientation and analysis … . Beautifully written and superbly well-organised, I strongly recommend this book to anyone seeking a stylish, balanced, up-to-date survey of this central area of mathematics." (Nick Lord, The Mathematical Gazette, Vol. 88 (512), 2004)

• Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Germany

  Jürgen Jost

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