Let $(M,g)$ be a closed, smooth Riemannian manifold of dimension $m \geq 1$. It is not difficult to produce an example of a fourth order formally self-adjoint elliptic differential operator $P: C^4(M) \rightarrow C^0(M)$ that has the property that there exists a sign-changing eigenfunction that is associated with the lowest eigenvalue. Indeed, $\Delta_g^2 + \lambda_2 \Delta_g$ does the job, where $\Delta_g :=div_g \nabla_g$. and where $\lambda_2$ is the second lowest eigenvalue of the operator $-\Delta_g$. The question that remains is how rare are formally self-adjoint elliptic differential operators whose lowest eigenvalue has this property. In this paper, the author proves that elliptic operators of the form $\Delta_g^2 - div_g(T-\lambda_2 g^{ -1})d$, where $T$ is a smooth, symmetric, 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 exists a lot of formally self-adjoint fourth order elliptic operators with the property that there exists a sign-changing eigenfunction that is associated with the lowest eigenvalue of the operator.
Categories: math.AP
