Analysis Seminar

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Finding counterexamples to Pompeiu’s and Schiffer’s conjectures

Speaker: Juame de Dios Pont, NYU, Courant

Location: Warren Weaver Hall 1302

Date: Thursday, October 1, 2026, 11 a.m.

Synopsis:

Let $\Omega \subset \mathbb R^2$ be a bounded domain. Pompeiu asked whether a function $f$ is determined by its integrals over all translations and rotations of $\Omega$. Williams showed that nonuniqueness occurs exactly when $\Omega$ supports a nonconstant Neumann eigenfunction of the Laplacian that is constant on $\partial\Omega$. The disk is the clearest example with this property, and Schiffer conjectured that it is the only smooth, connected planar domain with this property.

In this talk, we will construct a sequence of counterexamples to these conjectures using a bifurcation argument inspired by the work of Fall, Minlend, and Weth. A classical consequence of Bourget’s hypothesis on Bessel zeros is that bifurcation methods cannot directly yield such counterexamples. Instead, we will first construct counterexamples to a natural generalization of Schiffer’s conjecture and then transfer them to the original problem. The talk is based on joint work with Gonzalo Cao-Labora.