Uniqueness for the \(n\)-dimensional half space Dirichlet problem
In \(\mathbb{R}^n\) we prove uniqueness for the Dirichlet problem in the half-space \(x_n> 0\), with continuous data, under the growth condition \(u= o(|x|\text{sec}^\gamma \theta)\) as \(|x|\to \infty\) \((x_n= |x|\cos \theta\), \(\gamma\in \mathbb{R})\). Under the natural integral condition for convergence of the Poisson integral with Dirichlet data, the Poisson integral will satisfy this growth condition with \(\gamma= n-1\). A Phragmén-Lindelöf principle is established under this same growth condition. We also consider the Dirichlet problem with data of higher order growth, including polynomial growth. In this case, if \(u= o(|x|^{N+1} \text{ sec}^\gamma \theta)\) \((\gamma\in \mathbb{R}\), \(N\geq 1)\), we prove that solutions are unique up to the addition of a harmonic polynomial of degree \(N\) that vanishes when \(x_n =0\).
scientific article; zbMATH DE number 950227