Suppose $\Omega\subseteq\mathbb R^2$ is a compact, connected planar region with a smooth connected boundary $\partial\Omega$. Take $\lambda_0,\lambda_1,\lambda_2,\ldots$ to be the Laplacian spectrum with Dirichlet eigenfunctions $\psi_i$:$$\Delta\psi_i=\lambda_i\psi_i,$$ where we take the sign/ordering convention that $0=\lambda_0<\lambda_1\leq\lambda_2\leq\lambda_3\leq\cdots.$ The Rayleigh-Faber-Krahn inequality gives an isoperimetric result: $\lambda_1$ is minimized when $\partial\Omega$ is a circle.

I can use $\lambda_1$ as a "measure" of isoperimetry: The closer $\lambda_1$ is to that of a disk, the more compact $\Omega$ is in some sense. But $\lambda_1$ is only a single scalar measurement of isoperimetry, and of course there are many ways a shape could deviate from a circle.

Are there other isoperimetric inequalities involving more eigenvalues of $\Delta$? For instance, can I find ones that are more sensitive to small perturbations of the boundary?

For instance, if $\mathcal H_t:=e^{-t\Delta}$ is the heat kernel, does its trace $\mathrm{tr}(\mathcal H_t)$ satisfy any sort of isoperimetric inequality for all $t\geq0$? As $t\rightarrow\infty$, the $\lambda_1$ term decays relatively slowly and we can recover some version of the Rayleigh-Faber-Krahn inequality. But perhaps something can be said about positive but finite $t$? It would be useful for our applications to find a $t$-parameterized family of isoperimetric inequalities.


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