 Solution

Interpret strict ellipticity as uniform ellipticity up to the boundary, as required for the global elliptic boundary value problem. Mere pointwise positive definiteness in the interior is insufficient: $L=x^2{D}^2$ on $(0,1)$ has trivial homogeneous Dirichlet kernel but $Lu=1$ has no $C^2$ solution up to $x=0$.

Set $X=\{u\in C^{2,\alpha }(\overline{\Omega }):u{|}_{\partial \Omega }=0\}$ and $Y=C^{0,\alpha }(\overline{\Omega })$. The Fredholm alternative for an elliptic Dirichlet problem says $L:X\to Y$ is a Fredholm operator of index zero. Its null space $N$ is finite dimensional, its range is closed, and the range codimension equals $\dim N$. Hence $\boxed{N=\{0\}⟺L:X\to Y\text{ is invertible}}$; in this case every forcing has a unique solution.

Otherwise there are nonzero homogeneous solutions and finitely many compatibility conditions: $Lu=f$ is solvable exactly when $\ell (f)=0$ for every continuous linear functional on $Y$ vanishing on $L(X)$. When solvable, its solutions are $u_0+N$. This Fredholm solvability condition can be expressed with formal-adjoint null solutions for smoother coefficients. The functional formulation avoids differentiating the merely $C^{0,\alpha }$ coefficients.