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.