Isoperimetric inequalities for the logarithmic potential operator

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 ∞ . 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.

[1]  N. S. Landkof Foundations of Modern Potential Theory , 1972 .

[2]  Generalized isoperimetric inequalities , 1973 .

[3]  Marion Kee,et al.  Analysis , 2004, Machine Translation.

[4]  J. Troutman The logarithmic eigenvalues of plane sets , 1969 .

[5]  Helmut Linde,et al.  Isoperimetric inequalities for eigenvalues of the Laplacian and the Schrödinger operator , 2012 .

[6]  V. S. Vladimirov,et al.  Equations of mathematical physics , 1972 .

[7]  On Schatten norms of convolution-type integral operators , 2016 .

[8]  C. Bandle Isoperimetric inequalities and applications , 1980 .

[9]  George Polya,et al.  On the characteristic frequencies of a symmetric membrane , 1955 .

[10]  Antoine Henrot,et al.  Extremum Problems for Eigenvalues of Elliptic Operators , 2006 .

[11]  M. Birman,et al.  ESTIMATES OF SINGULAR NUMBERS OF INTEGRAL OPERATORS , 1977 .

[12]  Michael Ruzhansky,et al.  Schatten classes on compact manifolds: Kernel conditions☆ , 2014, 1403.6158.

[13]  G. Pólya,et al.  Isoperimetric inequalities in mathematical physics , 1951 .

[14]  Elliott H. Lieb,et al.  A General Rearrangement Inequality for Multiple Integrals , .

[15]  Naoki Saito,et al.  Data Analysis and Representation on a General Domain using Eigenfunctions of Laplacian , 2008 .

[16]  H. Schubert,et al.  O. D. Kellogg, Foundations of Potential Theory. (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 31). X + 384 S. m. 30 Fig. Berlin/Heidelberg/New York 1967. Springer‐Verlag. Preis geb. DM 32,– , 1969 .

[17]  Spectral estimates of Cauchy's transform inL2(Ω) , 1992 .

[18]  Kurt Bryan,et al.  Elementary Inversion of the Laplace Transform , 1999 .

[19]  Yoshihisa Miyanishi,et al.  Eigenvalues and Eigenfunctions of Double Layer Potentials , 2015, 1501.03627.

[20]  M. Kreĭn,et al.  Introduction to the theory of linear nonselfadjoint operators , 1969 .

[21]  The asymptotic behavior of the singular values of the convolution operators with kernels whose Fourier transforms are rational functions , 2012 .

[22]  Michael Loss,et al.  Competing symmetries, the logarithmic HLS inequality and Onofri's inequality onsn , 1992 .

[23]  T. Kal’menov,et al.  To spectral problems for the volume potential , 2009 .

[24]  M. Kac Integration in function spaces and some of its applications , 1980 .

[25]  M. Kac On Some Connections between Probability Theory and Differential and Integral Equations , 1951 .

[26]  E. Lieb,et al.  Analysis, Second edition , 2001 .