PDE seminar
Speaker: Harsh Prasad, Universität Bielefeld
Host: Harsha Hutridurga
Title: Nonlocal Ground States Are Superharmonic
Time, day and date: 4:00:00 PM – 5:00:00 PM, Monday, August 10
Venue: Ramanujan Hall
Abstract: he first eigenfunction of the fractional Laplacian is expected to share many of the geometric properties of the classical ground state, yet even its superharmonicity has remained open in general. Previous results were limited to special exponents, low dimensions, or particular classes of domains.
Based on joint work with Florian Grube, I will present a new proof showing that every first Dirichlet eigenfunction $\phi$ satisfies
[
-\Delta\phi \ge \lambda_1^{1/s}\phi
]
for every $s\in(0,1)$. In one dimension, this yields strict log-concavity and resolves a long-standing conjecture of Bañuelos, Kulczycki and Méndez-Hernández.
The proof is remarkably short and is based on a new combination of Bernstein function theory and maximum principle arguments. Besides establishing the superharmonicity estimate, the method suggests a broader framework for comparing local and nonlocal elliptic operators and may extend to more general subordinate processes.
Finally, I will discuss the Brascamp–Lieb type log-concavity questions that originally motivated this line of research. I will present evidence indicating that the nonlocal picture may be considerably more subtle than in the classical setting, including indications that natural Brascamp–Lieb analogues may fail for symmetric $\alpha$-stable processes, and I will conclude with several open problems and possible future directions.
Combinatorics seminar