A generalization of the fundamental theorem of spherical harmonic theory (Q2519275)

In the very interesting note under review the author generalizes the fundamental theorem of spherical harmonic theory stating that a spherical harmonic of degree \(N,\) which is the trace of a polynomial of degree \(N\) on the unit sphere, is also the trace of a harmonic polynomial of the same degree on the sphere. Precisely, consider a self-adjoint elliptic operator of order \(2\ell\) with constant coefficients \[ Lu=\sum_{| \alpha| ,| \beta| =\ell} a_{\alpha\beta}D^\alpha D^\beta u,\quad a_{\alpha\beta}=a_{\beta\alpha}, \] and let \(\Omega\subset\mathbb R^n\) be the interior of an ellipsoid \(\sigma.\) Given an arbitrary polynomial \[ P=P_N(x)=\sum_{| \alpha| \leq N} c_\alpha x^\alpha,\;x\in\mathbb R^n, \] consider the Dirichlet problem \[ \text{LU}=0,\quad \left.{{\partial^m U}\over{\partial\nu^m}}\right| _\sigma= \left.{{\partial^m P}\over{\partial\nu^m}}\right| _\sigma,\quad m=0,1,\ldots,\ell-1, \tag \(*\) \] where \(\nu\) is the outer normal to \(\sigma.\) The following theorem is proved: Theorem. For any polynomial \(P = P_N\) of degree \(N,\) the solution of the boundary-value problem \((*)\), which is clearly unique, is also a polynomial of degree \(N.\)