ap.analysis of pdes – Regularity of harmonic functions for a degenerate elliptic operator

This is a question on a degenerate elliptic operator.

Let $E$ be a closed unit ball in $mathbb{R}^d$ centered at the origin. For $c>0$ and $f in C^2(E)(:=C^2(mathbb{R}^d)|_E)$, we
define
begin{align*}
mathcal{L}f(x)=(1-|x|^2)sum_{i=1}^dfrac{partial^2 }{partial x_i^2}f(x)-csum_{i=1}^d(x,nabla f(x)),quad x in E.
end{align*}

Here, $|x|$ denotes the Euclidean norm on $mathbb{R}^d$. For an open subset $U subset E$, a harmonic function $h$ on $U$ (with respect to $(mathcal{L},C^2(E))$) should be properly introduced. That is, $h$ belongs to $C^2(U)$ and satisfies $mathcal{L}h=0$ on $U$.

Let $z in partial E$ and $r in (0,1)$, and set $U=E cap B(z,r)$. Here, $B(z,r)$ denotes the open ball in $mathbb{R}^d$ centered at $z$ with radius $r>0$. We consider a harmonic function $h$ on $U$ (with respect to $(mathcal{L},C^2(E))$) which satisfies the Dirichlet boundary condition: $h=0$ on $B(z,r)cap partial E$.

My question.

  • Does such a function exist in the first place?
  • If it exists, can we study the regularity of $nabla h$ on $U$?

I feel that it is difficult to study harmonic functions for degenerated differential operators like $(mathcal{L},C^2(E))$. Please let me know if there is any related preceding research.