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Isoperimetric inequalities for the logarithmic potential operator

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Abstract
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 R-2, for all even integers 2 <= p < infinity. 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-Faber-Krahn or Polya 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.
Keywords
Logarithmic potential, Characteristic numbers, Schatten class, Isoperimetric inequality, Rayleigh-Faber-Krahn inequality, Polya inequality, EIGENVALUES, LAPLACIAN

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Chicago
Ruzhansky, Michael, and Durvudkhan Suragan. 2016. “Isoperimetric Inequalities for the Logarithmic Potential Operator.” Journal of Mathematical Analysis and Applications 434 (2): 1676–1689.
APA
Ruzhansky, M., & Suragan, D. (2016). Isoperimetric inequalities for the logarithmic potential operator. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 434(2), 1676–1689.
Vancouver
1.
Ruzhansky M, Suragan D. Isoperimetric inequalities for the logarithmic potential operator. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. 2016;434(2):1676–89.
MLA
Ruzhansky, Michael, and Durvudkhan Suragan. “Isoperimetric Inequalities for the Logarithmic Potential Operator.” JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 434.2 (2016): 1676–1689. Print.
@article{8585360,
  abstract     = {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 R-2, for all even integers 2 {\textlangle}= p {\textlangle} infinity. 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-Faber-Krahn or Polya 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.},
  author       = {Ruzhansky, Michael and Suragan, Durvudkhan},
  issn         = {0022-247X},
  journal      = {JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS},
  language     = {eng},
  number       = {2},
  pages        = {1676--1689},
  title        = {Isoperimetric inequalities for the logarithmic potential operator},
  url          = {http://dx.doi.org/10.1016/j.jmaa.2015.07.041},
  volume       = {434},
  year         = {2016},
}

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