Neumann Boundary Value Problems on p-Adic Analytic Manifolds

Inspired by graph theory, Neumann Boundary Value Problems on p-adic analytic manifold are formulated. Existence and uniqueness of weak solutions is proven in the ultracontractive case. For this task, a p-adic Gauss-Green identity and an extended eigenvalue formula for wavelets with small supports on the manifold are proven. As an application, the closeness to a zero of an analytic function on the manifold is shown to be detectable via a Neumann Boundary Value Problem.