Prescription of the Robin spectrum

2025/03/03 by Xiang He, Zuoqin Wang, He, Xiang +1
Medicine · Biochemistry, Genetics and Molecular Biology · #Medical and Biological Sciences #Hedgehog Signaling Pathway Studies #Congenital limb and hand anomalies

paper · pdf · doi:10.48550/arxiv.2503.01451

Abstract

Let M be a compact connected smooth manifold with smooth boundary, and let ρ be a positive continuous function on the boundary which is served as the Robin parameter. In this paper, we study three problems concerning the prescription of finite Robin spectrum: (1) Prescribing finitely many Robin eigenvalues and the volume. (2) Within a fixed conformal class, prescribing the multiplicities of finitely many Robin eigenvalues. (3) Within a fixed conformal class, prescribing finitely many distinct Robin eigenvalues and the volume. A key step in our method is to solve the corresponding problems for the Dirichlet spectrum. As a consequence, we also obtain analogous results for the Dirichlet case.

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