Consider a set of isotropic, multivariate Gaussian densities with different centers $\mu_i\in \mathbb{R}^d $, $i\in\{1, \ldots, K\}$, which are denoted $\phi_i(x)$. They all have the same variance parameter $\sigma^2$. One can assume that for any pair $(i,j)$, $\Vert\mu_i- \mu_j \Vert^2$ is bounded below by the same positive constant $C$, but $C$ cannot be too large compared to $\sigma^2$. We also consider a vector of positive weights $\pi$, i.e., $\sum_{i=1}^K\pi_i =1.$ Also we give ourselves a set of second order polynomials $P_i(x)$.
Now consider a set of (shifted-rescaled) Ornstein-Uhlenbeck operators $L_i, i\le K$, $L_i u(x) := \Delta u(x) - \sigma_i^{-2} (x-x_i) \cdot \nabla u(x)$, for arbitrary $x_i$'s and $\sigma_i$'s, and construct the PDE $$ \sum_{i=1}^K\pi_i \phi_i(x) L_i u(x) = \sum_{i=1}^K\pi_i \phi_i(x) P_i(x). $$
What can be said about the solution(s) of this problem? I'll call any such solution $u^\star$. It is clear that each equation $L_i u_i(x)=P_i(x)$ admits explicit polynomial solutions $u_i^\star$. What is not clear to me is how the mixture "forces to stich the $u_i^\star$'s together".
In particular, I would appreciate to get bounds relating $\Vert \nabla u^\star \Vert$ with $\Vert\nabla u_i^\star\Vert$ and the other parameters involved. I tried to browse the literature but it did not yield much. I am not really fluent in PDE's; my apologies for inaccuracies in the question statement.