Stability Estimates for Commutativity Properties of the Dirichlet-to-Neumann Operator

Abstract

The Laplacian ${\Delta }_{{\mathbb{S}}^{n-1}}$ on the unit sphere ${\mathbb{S}}^{n-1}\subset {\mathbb{R}}^n$ has the property that it can explicitly be expressed in terms of $\Lambda$, the Dirichlet-to-Neumann map of the unit ball, as ${\Delta }_{{\mathbb{S}}^{n-1}}={\Lambda }^2+(n-2)\Lambda$. In this paper, we seek to characterize those manifolds for which such an exact relationship holds, and more generally measure the discrepancy of such a relationship holding in terms of geometric data. To this end, we obtain a stability estimate which shows that, for a smoothly bounded domain in ${\mathbb{R}}^3$, if the commutator $[\Lambda ,{\Delta }_{{\mathbb{S}}^{n-1}}]$ is small then that domain is itself close to a ball. We then study the case of manifolds conformal to the ball, show that a relationship as above implies a radial metric structure, and discuss stability in this setting. Finally, we provide a modern exposition of Gohberg's lemma, a foundational result in microlocal analysis which we employ as a starting step for our reasoning.