We derive second order perturbation expansions of volume-normalized sums of the first $n$ Dirichlet eigenvalues for near-circular and near-spherical domains. The disk (resp. ball) is only a critical shape when $n$ is the last index of an eigenvalue cluster, and in dimension two and three our expansions yield a finitely checkable criterion deciding whether the disk or ball is a weak local minimizer or a saddle point. In two dimensions, this allows us to show that there is an infinite number of cases in which the disk is a saddle point for finite eigenvalue sums.
In this work, we consider relative overdetermined problems for the first eigenfunction of the Laplacian for domains in cones, and the related question of minimizing the first eigenvalue among sets of a given fixed measure. By means of a shape derivative analysis, we show that the spherical sector is a critical shape an...
We prove an asymptotically sharp lower bound for the eigenvalues $\lambda_k$ of the Dirichlet Laplacian on bounded open sets in the plane with area $A>0$. The proof employs a trace identity for the commutator between a certain multiplication operator and its adjoint. As a corollary, we obtain $(2/3)\cdot 4\pi k/A\leq \...
We prove a sharp Szeg\"o-Weinberger-type inequality for balanced doubly connected domains in $\mathbb S^2$, $\mathbb R^2$ and $\mathbb H^2$. More precisely, the first positive Neumann eigenvalue is bounded above by that of the corresponding rotationally symmetric annulus, determined by the area of the domain and the ar...
We compute the limit of the eigenvalue of the $k$-Hessian operator on the unit ball in $\mathbb{R}^n$ when $k\rightarrow \infty$, assuming that the ratio $\frac{n}{k}$ remains fixed. When $n<2ke$, we moreover identify the limit of the corresponding eigenfunctions. We also derive monotonicity results and consider when t...
Nicholas McCleerey, Abhinav Pande, Thidas Wanasinghe· 0 citations
For a polynomial of degree $n$ whose zeros lie in the closed unit disk, we determine the largest possible distance from a prescribed critical point of modulus $r$ to the nearest zero. If $n$ is even, the sharp radius is $\sqrt{1-r^2}$; if $n$ is odd, the sharp radius is strictly smaller for $r\in(0,1)$ and depends on $...
We prove sharp residual stability estimates for the critical Poincar\'e-Sobolev equation near sums of one positive hyperbolic solution and one projected Euclidean bubble in dimensions at least four. The estimates are uniform in the relative positions of the centers. The sharp rate is linear except in dimension six, whe...
