## 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

1 | |

The Maximum Principle | 31 |

Principle | 51 |

Parabolic Methods The Heat | 77 |

and Heat Equations | 113 |

6 | 122 |

7 | 157 |

Sobolev Spaces and L2 Regularity Theory | 193 |

Strong Solutions | 243 |

Method Existence Techniques IV | 255 |

The Moser Iteration Method and the Regularity Theorem | 275 |

Appendix Banach and Hilbert Spaces The LpSpaces | 309 |

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a e Q aij(x assume assumptions B(ao B(yo ball Banach space boundary values bounded domain C# Q CC Q coefficients consider constant converges convex Corollary defined Definition depend differential equation Dirichlet problem domain of class eigenvalues elliptic estimate exists harmonic function heat equation hence Hilbert space Hölder continuous Hölder's inequality implies initial values integral iteration Laplace equation Laplace operator Lemma Let Q C R Let u e Let u e W*(Q linear LP(Q maximum principle mean value minimizer obtain open and bounded PDEs Poisson equation proof of Theorem regularity theory representation formula respect right-hand side satisfies Section semigroup Sobolev embedding theorem Sobolev spaces solves subharmonic subsolution u e W*(Q uniformly unique variational problems wave equation weak derivatives weak solution yields