论文标题
MOSCO的Sobolev空间和非平滑域的Sobolev不平等的不平等现象
Mosco convergence of Sobolev spaces and Sobolev inequalities for nonsmooth domains
论文作者
论文摘要
我们发现非常通用的非平滑开放式套件，可确保MOSCO收敛相应的Sobolev空间以及具有均匀常数的Sobolev不等式的有效性。我们结果的一个重要特征是，我们对MOSCO收敛的开放式集合和Sobolev不等式的条件具有相同的性质，因此很容易检查两者何时满足。 我们的分析是通过研究直接声学散射问题相对于散射器的稳定性而激发的，我们还讨论了这一点。 关于MOSCO在维度3或更高的MOSCO收敛，我们的结果扩展了文献中先前已知的所有结果。关于Sobolev的不平等，我们的方法似乎是新的，并且大大简化了声学直接散射问题所需的条件。
We find extremely general classes of nonsmooth open sets which guarantee Mosco convergence for corresponding Sobolev spaces and the validity of Sobolev inequalities with a uniform constant. An important feature of our results is that the conditions we impose on the open sets for Mosco convergence and for the Sobolev inequalities are of the same nature, therefore it is easy to check when both are satisfied. Our analysis is motivated, in particular, by the study of the stability of the direct acoustic scattering problem with respect to the scatterer, which we also discuss. Concerning Mosco convergence in dimension 3 or higher, our result extends all those previously known in the literature. Concerning Sobolev inequalities, our approach seems to be new and considerably simplifies the conditions previously required for the stability of acoustic direct scattering problems.
