Let Ω⊂ℝ n , n≥1, be a bounded domain and A an elliptic operator given by Au=∑ i,j=1 n ∂ ∂x i (a ij (x)∂u ∂x j ), with a ij ∈L ∞ (Ω), 1≤i, j≤n, a ij (x)ξ i ξ j ≥d|ξ| 2 , ξ∈ℝ n , a.e. x∈Ω. For f∈L p (Ω) the authors study existence results for problems of the form: -Au+β(x,u)∋f in Ω, u=0 on ∂Ω, where β(x,·) a maximal monotone graph. This extends a well-known theory of H. Brezis (see [Problèmes unilatéraux, J. Math. Pure Appl. 51, 1-162 (1972); Noveaux théorèmes de regularité pur les problèmes unilatéraux, Recontres entre physiciens théoriciens et mathématiciens 12 (1971), Strasbourg]) to the case where β depends on x.

Chipot, M; Kis, L (2000). *Existence results for monotone perturbations of the Dirichlet problem.* Advances in Mathematical Sciences and Applications, 10(1):81-101.

## Abstract

Let Ω⊂ℝ n , n≥1, be a bounded domain and A an elliptic operator given by Au=∑ i,j=1 n ∂ ∂x i (a ij (x)∂u ∂x j ), with a ij ∈L ∞ (Ω), 1≤i, j≤n, a ij (x)ξ i ξ j ≥d|ξ| 2 , ξ∈ℝ n , a.e. x∈Ω. For f∈L p (Ω) the authors study existence results for problems of the form: -Au+β(x,u)∋f in Ω, u=0 on ∂Ω, where β(x,·) a maximal monotone graph. This extends a well-known theory of H. Brezis (see [Problèmes unilatéraux, J. Math. Pure Appl. 51, 1-162 (1972); Noveaux théorèmes de regularité pur les problèmes unilatéraux, Recontres entre physiciens théoriciens et mathématiciens 12 (1971), Strasbourg]) to the case where β depends on x.

## 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: | uniqueness; Yosida approximation; estimates; variational inequality; maximal monotone graph |

Language: | English |

Date: | 2000 |

Deposited On: | 12 Jul 2010 12:46 |

Last Modified: | 05 Apr 2016 13:25 |

Publisher: | Gakko Tosho |

ISSN: | 1343-4373 |

Official URL: | http://www1.gifu-u.ac.jp/~aiki/AMSA/vol10.html |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=1769179 |

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