 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,is solvable exactly whenand any two solutions differ by an element of $\ker L$. Moreover $\dim \ker L=\dim \ker L^{\ast }$.