Stability Estimates for Commutativity Properties of the Dirichlet-to-Neumann Operator
Romain Speciel
TL;DR
The paper addresses when the Dirichlet-to-Neumann map $\Lambda$ commutes with or is a function of the boundary Laplacian $\Delta_{\partial M}$, focusing on Euclidean domains and manifolds conformal to the ball. It develops stability results via microlocal analysis and Calderón–type methods: in $\mathbb{R}^3$, a small commutator $[\Lambda,\Delta_{\partial\Omega}]$ forces the boundary to be near spherical, using Gohberg's lemma to bound $\nabla\mathrm{II}$, Topping's inequality for diameter control, and De Lellis–Müller stability for nearly umbilical surfaces; in the conformal-ball setting, $q$ is radial iff $\Lambda_q$ is a function of $\Delta_{\mathbb{S}^{n-1}}$, with a logarithmic-stability bound on the radial component and an infinitesimal commutativity result $[\Lambda',\Delta_{\mathbb{S}^{n-1}}]=0$ iff $q'=Pq'$. The work also provides a modern exposition of Gohberg's lemma and discusses perturbative versus nonperturbative questions, contributing to the Calderón problem and geometric inverse problems by linking operator commutativity to geometric symmetry and deriving explicit stability rates.
Abstract
The Laplacian $Δ_{\mathbb{S}^{n-1}}$ on the unit sphere $\mathbb{S}^{n-1}\subset \mathbb{R}^n$ has the property that it can explicitly be expressed in terms of $Λ$, the Dirichlet-to-Neumann map of the unit ball, as $Δ_{\mathbb{S}^{n-1}}=Λ^2+(n-2)Λ$. 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 $[Λ,Δ_{\mathbb{S}^{n-1}}]$ 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.