= Solution
The <Fredholm alternative for an elliptic Dirichlet problem> says that either the homogeneous adjoint problem has only the zero solution, in which case $Lu=f$ has a unique solution for every admissible $f$, or the homogeneous kernels are nontrivial and finite-dimensional. In the latter case,
$$
Lu=f
$$
is solvable exactly when
$$
\langle f,v\rangle=0
\qquad\text{for every }v\in\ker L^*,
$$
and any two solutions differ by an element of $\ker L$. Moreover $\dim\ker L=\dim\ker L^*$.
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