Concavity properties of solutions of elliptic equations under conformal deformations
G Khan, S Saha, M Tuerkoen - arXiv preprint arXiv:2403.03200, 2024 - arxiv.org
We study the Dirichlet problem for the weighted Schr\" odinger operator\[-\Delta u+
Vu=\lambda\rho u,\] where $\rho $ is a positive weighting function and $ V $ is a potential …
Vu=\lambda\rho u,\] where $\rho $ is a positive weighting function and $ V $ is a potential …
Modulus of concavity and fundamental gap estimates on surfaces
The fundamental gap of a domain is the difference between the first two eigenvalues of the
Laplace operator. In a series of recent and celebrated works, it was shown that for convex …
Laplace operator. In a series of recent and celebrated works, it was shown that for convex …
Negative curvature constricts the fundamental gap of convex domains
Abstract We consider the Laplace–Beltrami operator with Dirichlet boundary conditions on
convex domains in a Riemannian manifold (M n, g) and prove that the product of the …
convex domains in a Riemannian manifold (M n, g) and prove that the product of the …
On the critical points of semi-stable solutions on convex domains of Riemannian surfaces
M Grossi, L Provenzano - Mathematische Annalen, 2024 - Springer
In this paper we consider semilinear equations-Δ u= f (u) with Dirichlet boundary conditions
on certain convex domains of the two dimensional model spaces of constant curvature. We …
on certain convex domains of the two dimensional model spaces of constant curvature. We …
Concavity for elliptic and parabolic equations in complex projective space
S Aryan, MB Law - arXiv preprint arXiv:2403.16783, 2024 - arxiv.org
We establish a concavity principle for solutions to elliptic and parabolic equations on
complex projective space, generalizing the results of Langford and Scheuer. To our …
complex projective space, generalizing the results of Langford and Scheuer. To our …
Spectral Gap Estimates on Conformally Flat Manifolds
G Khan, M Tuerkoen - arXiv preprint arXiv:2404.15645, 2024 - arxiv.org
The fundamental gap is the difference between the first two Dirichlet eigenvalues of a Schr\"
odinger operator (and the Laplacian, in particular). For horoconvex domains in hyperbolic …
odinger operator (and the Laplacian, in particular). For horoconvex domains in hyperbolic …
A Priori Log-Concavity Estimates for Dirichlet Eigenfunctions
G Khan, S Saha, M Tuerkoen - arXiv preprint arXiv:2501.03504, 2025 - arxiv.org
In this paper, we establish a priori log-concavity estimates for the first Dirichlet eigenfunction
of convex domains of a Riemannian manifold. Specifically, we focus on cases where the …
of convex domains of a Riemannian manifold. Specifically, we focus on cases where the …