Eleutherius Symeonidis
Catholic University of Eichstätt-Ingolstadt, Eichstätt, Germany

The solutions $u:\Omega\rightarrow {\mathbb R}$ ($\Omega$ a domain in ${\mathbb R}^d$, $d\ge 2$) of the equation $$ x_d\cdot\left(\frac{\partial^2 u}{\partial x_1^2}+\ldots +\frac{ \partial^2 u}{\partial x_d^2}\right)+k\cdot\frac{\partial u} {\partial x_d}=0 $$ are called k-modified harmonic functions, where $k\in {\mathbb R}$. In our talk we study solutions that are homogeneous polynomials on ${\mathbb R}^d$ of a fixed (but arbitrary) degree $n$ and determine the dimension of their space in all but a finite number of values of $k$.