Let $\Omega$ be a smooth bounded open set of ${\bf R}^n,A$ a uniformly elliptic second-order operator and $f$ a function belonging to $W^{1,p}(\Omega)\ (p>n)$. For $\varphi,\psi\in H_0^1(\Omega)\cap W^{2,\infty}(\Omega)$, $\varphi\leq\psi$, we prove that any solution $u$ of the problem $(-Au,v-u)\geq(f,v-u)$ for all $v\in K$, $u\in K=${$v\in H_0^1(\Omega)\colon\varphi\leq v\leq\psi$ a.e. in $\Omega$}, belongs to $W^{2,\infty}(\Omega)$. This generalizes the results of H. Brézis and D. Kinderlehrer [Indiana Univ. Math. J. 23 (1973/74), 831--844] in the case of two barriers.

Chipot, M (1979). *Sur la régularité de la solution d‘inéquations variationnelles elliptiques.* Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B, 288(10):A543-A546.

## Abstract

Let $\Omega$ be a smooth bounded open set of ${\bf R}^n,A$ a uniformly elliptic second-order operator and $f$ a function belonging to $W^{1,p}(\Omega)\ (p>n)$. For $\varphi,\psi\in H_0^1(\Omega)\cap W^{2,\infty}(\Omega)$, $\varphi\leq\psi$, we prove that any solution $u$ of the problem $(-Au,v-u)\geq(f,v-u)$ for all $v\in K$, $u\in K=${$v\in H_0^1(\Omega)\colon\varphi\leq v\leq\psi$ a.e. in $\Omega$}, belongs to $W^{2,\infty}(\Omega)$. This generalizes the results of H. Brézis and D. Kinderlehrer [Indiana Univ. Math. J. 23 (1973/74), 831--844] in the case of two barriers.

## Citations

## Additional indexing

Item Type: | Journal Article, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | Elliptic Operator; Generalized Solutions |

Language: | French |

Date: | 1979 |

Deposited On: | 30 Oct 2009 14:27 |

Last Modified: | 05 Apr 2016 13:30 |

Publisher: | Elsevier |

ISSN: | 0151-0509 |

Related URLs: | http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0397.35015 |

## Download

Full text not available from this repository.

TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.

You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.