## Fine Regularity of Solutions of Elliptic Partial Differential EquationsThe primary objective of this book is to give a comprehensive exposition of results surrounding the work of the authors concerning boundary regularity of weak solutions of second-order elliptic quasilinear equations in divergence form. The structure of these equations allows coefficients in certain $L^{p}$ spaces, and thus it is known from classical results that weak solutions are locally Holder continuous in the interior. Here it is shown that weak solutions are continuous at the boundary if and only if a Wiener-type condition is satisfied. This condition reduces to the celebrated Wiener criterion in the case of harmonic functions. The work that accompanies this analysis includes the fine analysis of Sobolev spaces and a development of the associated nonlinear potential theory. The term fine refers to a topology of $\mathbf R^{n}$ which is induced by the Wiener condition. The book also contains a complete development of regularity of solutions of variational inequalities, including the double obstacle problem, where the obstacles are allowed to be discontinuous. The regularity of the solution is given in terms involving the Wiener-type condition and the fine topology. The case of differential operators with a differentiable structure and $\mathcal C^{1,\alpha}$ obstacles is also developed. The book concludes with a chapter devoted to the existence theory, thus providing the reader with a complete treatment of the subject ranging from regularity of weak solutions to the existence of weak solutions. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

1 | |

Chapter 2 Potential Theory | 63 |

Chapter 3 Quasilinear Equations | 161 |

Chapter 4 Fine Regularity Theory | 185 |

Chapter 5 Variational Inequalities Regularity | 233 |

Chapter 6 Existence Theory | 253 |

273 | |

283 | |

289 | |

### Other editions - View all

Fine Regularity of Solutions of Elliptic Partial Differential Equations Jan Maly,Jan Malı,William P. Ziemer No preview available - 1997 |

### Common terms and phrases

ac-aco assume B(aco ball boundary bounded capacitary extremal CC Q compact constant convex Corollary defined denote differentiable Dirichlet problem dominated convergence theorem estimate exists finite follows function f function on Q Harnack inequality Hence Hölder continuous Hölder's inequality integral Laplace equation Lebesgue point Lemma Let f Let u e liminf u(x lower semicontinuous LP(Q LP(R n-Green domain nonnegative obtain p-finely continuous p-finely open set p-polar set p-quasi p-quasicontinuous representative p-thin pointwise potential theory proof of Theorem proved Radon measure regularity result Riesz potentials satisfies sequence Sobolev functions Sobolev spaces structure subsection superharmonic functions supersolution of 3.1 Suppose test function u e W*(Q Vu(x weak derivative weak solution weak subsolution weakly Wiener condition Wiener criterion Young's inequality