Simplicity and isolation of the first eigenvalue associated with some differential operators

Question

Consider the operator $\Delta$ or more generally a second order differential operator $L$, in which the principal part is symmetric and positive definite. It can be proved (see here page 336) that the first eigenvalue "$\lambda_1$" of $L$ is simple, i.e. the space generated by the eigenfunctions associated with $\lambda_1$ is one-dimensional. Moreover, $\lambda_1$ is isolated.

The same is true for the $p$-Laplace operator $\Delta_p$ ($1<p<\infty$) where $\Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$, for this result see here page 20.

We can generalize this result to a more general operator $\Delta_\Phi$, called $\Phi$-Laplacian where $\operatorname{div}(\Phi(|\nabla u|)\nabla u)$ (see here and references therein).

So my questions are: Why so much effort in proving such a result? What is the intuition behind such result? Is there any physical application?

Remark 1: Note that if $\Phi(t)=t^{p-2}$, then we have the $p$-Laplace operator.

Remark 2: In the nonlinear cases, simplicity is to be understood as the property that if $u$ is a eigenfunction associated to $\lambda_1$, then $u=c\varphi_1$, where $\varphi_1$ is an eigenfunction associated with $\lambda_1$.

Thank you

Answer 1

The simplicity of the first eigenvalue governs the behavior of parabolic problems associated with the operator. Lower bounds on the size of the gap between the first and second eigenvalue allow one to make quantitative statements about asymptotic behavior. To see why, expand $u$ into eigenfunctions of $\mathcal L$, say $u=\sum c_j u_j$. Assuming $\mathcal L$ is linear and positive, the gradient flow $U_t=-\mathcal L U$ has solution $$U=\sum c_j e^{-\lambda_j t}u_j\tag1$$ where $\lambda_j$ is the eigenvalue for $u_j$.

As $t\to \infty$, (1) tells us that $u$ is asymptotic to a linear combination of eigenfunctions for the lowest eigenvalue. If we know that $\lambda_1$ is simple, the statement is more precise: $$e^{\lambda_1 t}U\to c_1 u_1\tag2$$ And if we can bounds the spectral gap $\lambda_2-\lambda_1$ from below, then the rate of convergence in (2) can be estimated.

The search for "spectral gap" (exact match in the abstract only) in Physics ArXiv brings up 340 papers. You can sample a few to see what physicists do with this stuff.

I don't know anything about the use of spectral gaps for nonlinear operators, but the paper Towards a Calculus for Non-Linear Spectral Gaps by Mendel and Naor suggests they are useful in theoretical computer science.