- Sponsor
- Richard Laugesen
- Speaker
- Jiya Rose Johnson (Indian Institute of Technology Madras)
- Contact
- Richard Laugesen
- laugesen@illinois.edu
TITLE: Spectral properties of the logarithmic potential operator
ABSTRACT: The classical Faber--Krahn inequality states that, among domains of given area, the ball minimizes the first Dirichlet eigenvalue of the Laplacian. We discuss an analogue for the logarithmic potential operator on $L^2(\Omega)$,
\[\mathcal{L}u(x)=\int_\Omega \log\frac{1}{|x-y|}u(y)\,dy,\]
where the planar domain $\Omega$ is bounded and open. Here the situation is reversed: symmetrization increases the largest eigenvalue. Assuming the diameter of the domain is at most $1$, we establish reverse Faber--Krahn inequalities under polarization and Schwarz (symmetric decreasing) symmetrization.
We also consider domains containing a hole, investigating how the largest eigenvalue varies as an obstacle is translated or rotated within the domain. Finally, we characterize the existence of a negative eigenvalue in terms of the logarithmic capacity (transfinite diameter) of the domain.
This talk is based on joint work with T. V. Anoop.