2023/07/21 by Dai, Guowei, Zhang, Yong
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2307.11797
In this paper, we prove the existence of k families of smooth unbounded domains Ωs⊂ℝN+1 with N≥1, where Ωs=\(x,t)∈ ℝN× ℝ:\vert x\vertlt;1+scos ((2π)/(T(s))t)+s ws((2π)/(T(s))t)\,such that -Δu=λu in Ω, ∂νu=0, u=const on ∂Ωadmits a bounded sign-changing solution with exactly k+1 nodal domains. These results can be regarded as counterexamples to the Schiffer conjecture on unbounded domain. These results also indicate that there exist non-spherical unbounded regions without Pompeiu property. Our construction shows that the condition "∂Ω is homeomorphic to the unit sphere" is necessary for Williams conjecture to hold. In addition, these conclusions may have potential applications in remote sensing or CT.