Let $$AF(x): = \sum_{i,j} a_{ij}(x) \partial_i \partial_j F(x) + \sum_j b_j(x) \partial_jF(x)$$ be an elliptic differential operator.
We have proven that for some $\lambda>0$ $$\lambda I - A:D(A) \to C_K$$ that is, $\lambda I - A$ maps $D(A)$ onto the set $C_K= \{f: K \to \Bbb{R} , f \text{ is continuous }\}$ ($K$ is a bounded closed region of $\Bbb{R}^n$ with sufficiently smooth boundary $\Gamma$)
I am trying now to prove that:
If the coefficients of $A$ are sufficiently smooth then for an everywhere dense set of functions $f$ ($f$ sufficiently smooth) there exists at least one solution $F$ such that $\lambda F - A F = f$ which vanishes on the boundary.