Gonzalo Cao will talk in the geometry seminar at 3.15pm on 25th May in the Salle des Profs on the 9th floor of building NO as (part of a double bill of geometry talks, with Anna Roig-Sanchez talking at 2pm). Gonzalo’s title is “The Schiffer conjecture on the half-sphere” and his title is below.
Consider a fixed domain in Euclidean space. A natural question, introduced by Dimitrie Pompeiu in 1929, asks whether one can uniquely determine a function f(x) from its integrals over all rigid motions of the domain. While the answer is false for balls, it is conjectured to be true for any other domain. For contractible domains, this is equivalent to the Schiffer conjecture, which asks whether the domain admits any Neumann eigenfunction of the Laplacian that is constant on the boundary. In this formulation, the conjecture remains one of the open problems featured in S. T. Yau’s 1982 list.
A recent line of research explores the Schiffer conjecture in other geometries, constructing counterexamples via local bifurcation techniques (such as the Crandall-Rabinowitz theorem). We extend this research by introducing a new computer-assisted methodology. This approach allows us to establish a wider class of counterexamples, bypassing the limitations of classical techniques that rely on bifurcations arising from an asymptotic regime. In particular, we will construct the first contractible counterexamples to the Schiffer conjecture on the sphere, which also serve as the first examples in a half-sphere. This is particularly notable because it represents the first geometry where the related Serrin problem (zero Dirichlet and constant Neumann conditions) exhibits rigidity—due to the classical moving plane method—whereas our example demonstrates the flexibility of the Schiffer problem.
The talk will also provide an introductory exposition to computer-assisted techniques in the context of PDEs and their practical applications. If time permits, we will discuss further developments regarding the Schiffer problem.
The talk will be based on joint work with Antonio Fernández