From Heisenberg uniqueness pairs to properties of the Helmholtz and Laplace equations

TL;DR

This paper proves that solutions to the Helmholtz and Laplace equations are uniquely determined by their values on two intersecting submanifolds, extending the understanding of uniqueness in PDEs.

Contribution

It establishes new uniqueness results for solutions of Helmholtz and Laplace equations based on data on intersecting submanifolds.

Findings

Solutions are uniquely determined by their values on intersecting submanifolds.

Generic position of submanifolds ensures global uniqueness.

Extends classical uniqueness theorems to new geometric configurations.

Abstract

The aim of this paper is to establish uniqueness properties of solutions of the Helmholtz and Laplace equations. In particular, we show that if two solutions of such equations on a domain of R d agree on two intersecting d -- 1-dimensional submanifolds in generic position, then they agree everywhere.