On spherical functions connected with a general PDE of the second order in the unit ball

The report is devoted to a connection between the Dirichlet problem in the unit ball for a general PDE of the second order and spherical functions which are zero on null-variety of the PDE-symbol.
Let $L=L(x,D)=\sum_{|\alpha |\le 2}a_\alpha D^\alpha$ be a general linear differential operation with constant coefficients, which can be complex-valued or matrix, and let $\Omega \subset \mathbb R^n$ be a bounded domain with smooth boundary $\partial \Omega$.
Let us consider the Dirichlet problem
$$\begin{equation} \label{443:eq1} Lu=f,\qquad u|_{\partial \Omega }=0 \end{equation}$$
in the Sobolev space $W^2_2(\Omega)$. We extend functions $f$ and $u$ by zero: $\widetilde u=u$ in $\Omega$, $\widetilde u=0$ outside of $\Omega$. Then
$$\begin{equation} \label{443:eq2} L\widetilde u=\widetilde f+L_{1}u\delta_{\partial \Omega }, \end{equation}$$
where $L_{1}u$ is a linear differential expression on $\psi$ and $u_\nu ^{\prime }\langle \delta _{\partial \Omega },\varphi \rangle =\int_{\partial \Omega }\overline{\varphi }\,ds$. Let the domain $\Omega$ be defined by means of the inequality $P(x)>0$ where $P\in \mathbb R[x]$ is a polinomial, $|\nabla P|_{P=0}\neq 0$. We multiply equality $\eqref{443:eq2}$ by $P(x)$ and apply the Fourier transform. We obtain
$$\begin{equation} \label{443:eq3} P(-D_\xi )[L(\xi )F(\widetilde u\mspace{1mu})(\xi )]=g(\xi ) \end{equation}$$
with a known function $g$. Here $L(\xi)$ is the symbol and $L_2(\xi)$ is the major symbol.
Statement. The solvability of the last equation in some classes of entire functions is equivalent to the solvability of problem $\eqref{443:eq1}$.
If the domain is the unit ball, then $P(-D_\xi)=\Delta_\xi$ and if, moreover, the right-hand side $f=0$, then $g=0$ and for the uniqueness problem in problem $\eqref{443:eq1}$ we obtain the equivalent problem of the following form: $(\Delta_\xi+1)[L_2(\xi)v(\xi)]=0$. Now for lowest term $v_m(\xi)$ of the power series for $v$ we have the equation $\Delta_\xi[L_2(\xi)v_m(\xi)]=0$.
The application of this methods gives, in particular, the following results. Let us consider
$$Lu=u_{ x_1x_1}+\dots +u_{ x_kx_k}-a^2(u_{ x_{k+1}x_{k+1}}+\dots +u_{ x_nx_n}).$$

\begin{estatement} Problem $\eqref{443:eq1}$ with $f=0$ has a nontrivial solution in $W_2^2({\Omega})$ if and only if there exist natural numbers $m$, $i$, $j$, $i+j\leqslant m$ such that

• 1) $m-i-j$ even and
  $$P^{(\frac{n-k}{2}+j-1, i+\frac{k}{2}-1)}_{\frac{m-i-j}{2}+1} \biggl(\frac{a^2-1}{a^2+1}\biggr)=0$$
  or
  
• 2) $m+n-k-i+j$ even and
  $$P^{(1-j-\frac{n-k}{2}, i+\frac{k}{2}-1)}_{\frac{m+n-k-i+j}{2}} \biggl(\frac{a^2-1}{a^2+1}\biggr)=0$$
  or
  
• 3) $m+n+i+j$ even and
  $$P^{(1-j-\frac{n-k}{2}, 1-i-\frac{k}{2})}_{\frac{m+n+i+j}{2}-1} \biggl(\frac{a^2-1}{a^2+1}\biggr)=0$$
  or
  
• 4) $m+k+i-j$ even and
  $$P^{(\frac{n-k}{2}+j-1, 1-i-\frac{k}{2})}_{\frac{m+k+i-j}{2}} \biggl(\frac{a^2-1}{a^2+1}\biggr)=0,$$
  

where $P^{(\alpha,\beta)}_N (x)$ is the Jacoby polynomial. \end{estatement}
For the case $n=2$ the result conforms with the well-known result for the string equation.
There is also an application of these results to problems of the interal geometry.