## Partial Differential EquationsThis 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. |

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### Contents

What Are Partial Differential Equations? | 1 |

The Maximum Principle | 31 |

Methods Based on the Maximum | 51 |

Parabolic Methods The Heat | 77 |

The Wave Equation and Its Connections with the Laplace | 113 |

Through the Darboux Equation | 117 |

The Heat Equation Semigroups and Brownian Motion | 127 |

The Dirichlet Principle Variational Methods for the Solu | 157 |

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Algebraic already apply approximation assume assumptions ball Banach space boundary values bounded called chapter choose claim coefficients condition consider const constant contained continuous converges convex Corollary corresponding course defined Definition depend derivatives difference differential equation Dirichlet problem eigenvalues elliptic estimate example exists extend formula function given harmonic harmonic function heat equation hence holds implies inequality initial values integral interior Laplace equation Lemma limit linear maximum principle method minimizer namely obtain operator particular PDEs positive preceding present problem Proof proof of Theorem prove regularity Remark require respect result right-hand side satisfies semigroup sense sequence smooth Sobolev solves subharmonic suppose term Theorem Theory tion uniformly unique variational wave equation weak solution wish yields