论文标题
Lipschitz域中Laplacian的Dirichlet问题。抽象的
The Dirichlet problem for the Laplacian in Lipschitz domain. Abstract
论文作者
论文摘要
本文的主要目的是解决有关与拉普拉斯和双拉普拉斯运营商有关的边界价值问题的一些问题，该问题是在经典的$ h^s $ sobolev空间框架内设置在r^n的有限lipschitz域上的。这些问题并不是什么新鲜事物，自80年代以来，许多作者都使用各种技术在这个方向上完成了很多工作。如果对于常规域，几乎所有内容都阐明了，那么Lipschitz的情况并非如此 $ s = k + 1/2 $，带有$ k $ integer。众所周知，这个框架是微妙的。即使在这些情况下，许多结果也已经确立，但有时不令人满意。仍然提出几个问题。通过这项工作，我们的主要目标是一方面通过使用不需要过于复杂的计算的技术来对理论进行一些改进。我们还试图获得解决方案的最大规律性，并且据我们所能获得的结果。
The main purpose of this paper is to address some questions concerning boundary value problems related to the Laplacian and bi-Laplacian operators, set in the framework of classical $H^s$ Sobolev spaces on a bounded Lipschitz domain of R^N. These questions are not new and a lot of work has been done in this direction by many authors using various techniques since the 80's. If for regular domains almost every thing is elucidated, it is not the case for Lipschitz ones and for $s$ of the form $s = k + 1/2$, with $k$ integer. It is well known that this framework is delicate. Even in these cases many results are well established but sometimes not satisfactory. Several questions remain posed. Our main goal through this work is on one hand to give some improvements to the theory and on another one by using techniques which do not require too intricate calculations. We also tried to obtain maximal regularity for the solutions and as far as we can optimality of the results.
