A Note on Best Sobolev and Relative Iso-perimetric Constants and Neumann Problems in Exterior Domains
首发时间:2010-02-17
Abstract:In this note we review some results in the literature on the best Sobolev and relative iso-perimetric constants. The conjecture of the best constant in the Sobolev inequality in the exterior of a bounded domain with smooth boundary of nonnegative mean curvature was posed by the authors in 1998. This problem is closely related to the problem of the best constant in a relative iso-perimetric inequality on the exterior domains by the work of Talenti. The conjecture was motivated by the problem of existence of the least energy solutions of a semilinear Neumann problem in the exterior domains. Very recently the problem of the best constant in the relative iso-perimetric inequality was solved by Choe, Ghomi and Ritore for the exterior of a convex body. As a consequence it confirms the conjecture of the best constant in the Sobolev inequality in such exterior domains. Then we use this conclusion to discuss the non-existence of the least energy solutions for the semilinear Neumann problem in the exterior of a convex body. We also examine the existence or non-existence of the Dirichlet, Neumann and mixed problems in the ring-type domains.
keywords: best Sobolev constant relative iso-perimetric inequality semilinear Neumann problem exterior domains
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关于最佳索伯列夫常数和相对等周常数及外区域纽曼问题的注记
摘要:在这篇短文中我们考察了文献中关于最佳索伯列夫常数和最佳相对等周常数的一些工作。具有非负平均曲率的光滑边界的有界区域的外部的索伯列夫不等式的最佳常数猜想是作者在1998年提出的。由于Talenti的工作,这个问题与外区域上的相对等周不等式的最佳常数问题密切相关。这个猜想来自于外区域上的半线性纽曼问题的最小能量解的存在性问题。最近,对于凸体的外部,相对等周不等式中的最佳常数问题由Choe, Ghomi,Ritore所解决。作为推论,凸体外部的索伯列夫不等式的最佳常数猜想亦获得解决。我们利用这个结论讨论了凸体外部的半线性纽曼问题的最小能量解的非存在性问题。我们也讨论了环状区域上的狄里克雷问题、纽曼问题与混合问题的解的存在性与非存在性。
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No.4020751589112663****
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