Lowest eigenvalues and higher order elliptic differential operators
David Raske
TL;DR
The paper explores whether the ground state of fourth-order elliptic operators on closed Riemannian manifolds must be positive, showing that many such operators admit a sign-changing eigenfunction at the bottom of the spectrum. It introduces a variational framework for the operator $P_g=\\Delta_g^2 - \\mathrm{div}_g (A d)$ and proves the existence of a ground state by minimizing the Rayleigh-type functional $F(u)=\\frac{\\int_M ((\\Delta_g u)^2 + A(du,du)) \\, dvol_g}{\\int_M u^2 \\, dvol_g}$, yielding the eigenvalue problem $\\Delta_g^2 z - \\mathrm{div}_g (A dz) = \\lambda z$. Under the condition $A = T - \\lambda_2 g^{-1}$ with $T$ negative semidefinite and \\max_x \\lambda_m(x) \\le -\\lambda_2$, the lowest eigenfunction is shown to change sign, establishing that $P_g$ is not ESLEES. These results imply a broad prevalence of non-ESLEES fourth-order operators and connect to geometric analysis questions, including $Q$-curvature prescription on closed manifolds.
Abstract
Let $(M,g)$ be a closed, smooth Riemannian manifold of dimension $m \geq 1$. It is not difficult to produce an example of an elliptic differential operator on $(M,g)$ that has the property that there exists a sign-changing eigenfunction that is associated with the lowest eigenvalue. Indeed, $Δ_g^2 + λ_2 Δ_g$ does the job, where $Δ_g:=div_g \nabla_g$. and where $λ_2$ is the second lowest eigenvalue of the operator $-Δ_g$. The question that remains is how rare are elliptic differential operators whose lowest eigenvalue has this property. In this paper, the author proves that elliptic operators of the form $Δ_g^2 - div_g(T-λ_2 g^{-1}) d$, where $T$ is a negative semi-definite $(2,0)$-tensor field on $M$, and where $g^{-1}$ is the inverse metric tensor, have the property that there exists a sign-changing eigenfunction that is associated with the lowest eigenvalue of the operator. This suggests that there are a lot of fourth-order elliptic operators with the property that there exists a sign-changing eigenfunction that is associated with the lowest eigenvalue of the operator.