= Fredholm alternative for an elliptic Dirichlet problem
{c}
For a uniformly elliptic operator $L$ on a bounded regular domain, exactly one of the following equivalent descriptions applies to its homogeneous <Dirichlet boundary condition>[Dirichlet problem]. If $\ker L^*=0$, every forcing has a unique solution. Otherwise $Lu=f$ is solvable exactly when $f$ is orthogonal to $\ker L^*$, and any two solutions differ by an element of $\ker L$. Moreover $\dim\ker L=\dim\ker L^*$.
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