TY - JOUR
T1 - Isoperimetric inequalities for the logarithmic potential operator
AU - Ruzhansky, Michael
AU - Suragan, Durvudkhan
N1 - Funding Information:
The authors were supported in parts by the EPSRC grant EP/K039407/1 and by the Leverhulme Trust Grant RPG-2014-02 , as well as by the MESRK grant 5127/GF4 .
Publisher Copyright:
© 2015 The Authors.
PY - 2016/1/1
Y1 - 2016/1/1
N2 - 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, 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 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.
AB - 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, 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 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.
KW - Characteristic numbers
KW - Isoperimetric inequality
KW - Logarithmic potential
KW - Pólya inequality
KW - Rayleigh-Faber-Krahn inequality
KW - Schatten class
UR - http://www.scopus.com/inward/record.url?scp=84946423627&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84946423627&partnerID=8YFLogxK
U2 - 10.1016/j.jmaa.2015.07.041
DO - 10.1016/j.jmaa.2015.07.041
M3 - Article
AN - SCOPUS:84946423627
SN - 0022-247X
VL - 434
SP - 1676
EP - 1689
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 2
ER -