The Dirichlet problem when the boundary function is entire (Q1433362)

Let \(X= (x_1,\dots, x_n)\in \mathbb{R}^n\), \(n\geq 2\) and let \(\Omega\) be an ellipsoid centered at the origin of \(\mathbb{R}^n\), that is \[ \Omega= \Biggl\{x\in \mathbb{R}^n\,\Biggl|\, \sum^n_{j=1} \frac{x^2_j}{a^2_j}< 1\Biggr\}, \] where \(a_1,\dots, a_n\) are positive numbers. It is known that, if \(f\) is entire function on \(\mathbb{C}^n\), then there exists a (complex-valued) harmonic function \(h\) on \(\mathbb{R}^n\) such that \(h= f\) on \(\partial\Omega\). There are indications that the growth of \(h\) (as a function on \(\mathbb{C}^n\)) is governed by that of \(f\). This paper gives more detailed results relating the growth of \(h\) to that of \(f\).