Isoperimetric inequalities for the logarithmic potential operator

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• 作者：Michael Ruzhansky^a ; ^{m.ruzhansky@imperial.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; Durvudkhan Suragan^b ; ^a ; ^{d.suragan@imperial.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Logarithmic potential ; Characteristic numbers ; Schatten class ; Isoperimetric inequality ; Rayleigh&ndash ; Faber&ndash ; Krahn inequality ; Pó ; lya inequality
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：434
• 期：2
• 页码：1676-1689
• 全文大小：352 K

文摘

In this paper we prove that the disc is a maximiser of the Schatten p  -norm of the logarithmic potential operator among all domains of a given measure in R²${\mathbb{R}}^2$, for all even integers 2≤p<∞$2\le p<\infty$. We also show that the equilateral triangle has the largest Schatten p-norm among all triangles of a given area. For the logarithmic potential operator on bounded open or triangular domains, we also obtain analogies of the Rayleigh&ndash;Faber&ndash;Krahn or Pólya inequalities, respectively. The logarithmic potential operator can be related to a nonlocal boundary value problem for the Laplacian, so we obtain isoperimetric inequalities for its eigenvalues as well.