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A type of uniqueness for the Dirichlet problem on a half-space with continuous data. (English) Zbl 0853.31004

Let $T=\left\{\left(X,y\right):X\in {ℝ}^{n-1}$ and $y>0\right\}$, where $n\ge 2$. If $f:{ℝ}^{n-1}\to ℝ$ is continuous and satisfies $\int |f\left(X\right)|\left(1+{|X|\right)}^{-n}dX<+\infty$, then the Poisson integral of $f$ yields a harmonic function $h$ on $T$ with boundary values $f$ on $\partial T\equiv {ℝ}^{n-1}$. It is known that, for functions $f$ which fail to satisfy this integrability condition, the corresponding Dirichlet problem can be solved by suitably modifying the Poisson kernel. Although these solutions are certainly not unique, the paper under review presents a type of uniqueness theorem which answers a question posed by D. Siegel.

To be more precise, suppose that $\int |f\left(X\right)|\left(1+{|X|\right)}^{-n-l}dX<+\infty$ for some $l$ in $ℕ$. Then there exists a harmonic function $h$ on $T$ with boundary values $f$ such that ${r}^{-l-2}𝒟\left(|h|,r\right)\to 0$ as $r\to +\infty$, where $𝒟\left(|h|,r\right)$ denotes the mean value of $y|h\left(x,y\right)|$ with respect to surface area measure on the hemisphere $\left\{\left(X,y\right)\in T:|\left(X,y\right)|=r\right\}$. Further, all other solutions satisfying this growth condition are of the form $h\left(X,y\right)+y{\Pi }\left(X,y\right)$, where ${\Pi }$ is a polynomial in ${ℝ}^{n}$ of degree at most $l-1$ and even with respect to the variable $y$.

##### MSC:
 31B05 Harmonic, subharmonic, superharmonic functions (higher-dimensional) 31B20 Boundary value and inverse problems (higher-dimensional potential theory)