Consider the following PDE on $\Omega\subset \mathbb{R}^n$ for $n\geq 2:$ \begin{align} \Delta u - x\cdot \nabla u &= f(x),\text{ in } \Omega\\ u&=0 \text{ on }\partial \Omega \end{align}

Are there any explicit expressions for a kernel $K$ such that, $$u(x)=\int_{\Omega} K(x,y)f(y)dy?$$

Consider the following PDE on $\Omega\subset \mathbb{R}^n$ for $n\geq 2:$ \begin{align} \Delta u - x\cdot \nabla u &= f(x),\text{ in } \Omega\\ u&=0 \text{ on }\partial \Omega \end{align}

Are there any explicit expressions for a kernel $K$ such that, $$u(x)=\int_{\Omega} K(x,y)f(y)dy$$ when $\Omega=\mathbb{R}^n$ or $\Omega=B(0,1)$?