2023/09/14 by Alberto Enciso, Enciso, Alberto, Antonio J. Fernández +5 · 4 citations
Mathematics · #35B32 #35N25 #35Q31 #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2309.07977
openalex publication_date 2023/09/14 · openalex created_date 2023/09/19 · openalex updated_date 2026/08/01
If on a smooth bounded domain Ω⊂ℝ2 there is a nonconstant Neumann eigenfunction u that is locally constant on the boundary, must Ω be a disk or an annulus? This question can be understood as a weaker analog of the well known Schiffer conjecture, in that the function u is allowed to take a different constant value on each connected component of ∂ Ω yet many of the known rigidity properties of the original problem are essentially preserved. Our main result provides a negative answer by constructing a family of nontrivial doubly connected domains Ω with the above property. As a consequence, a certain linear combination of the indicator functions of the domains Ω and of the bounded component of the complement ℝ2\backslashΩ fails to have the Pompeiu property. Furthermore, our construction implies the existence of continuous, compactly supported stationary weak solutions to the 2D incompressible Euler equations which are not locally radial.