= 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: $L=x^2D^2$ 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 $\boxed{N=\{0\}\iff L:X\to Y\text{ is invertible}}$; 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.
Back to article page
