Assume that $L$ is an elliptic operator of the form $$L(u) = \sum_{i,j} a_{ij} \partial_i \partial_j u + \sum_i b_i \partial_i u$$ where $a_{i,j}, b_i$ are continuous functions on a bounded domain $\Omega\in \mathbb R^3$.
We assume $L$ is elliptic of uniform module $\lambda$, that is $(a_{ij})$ is symmetric for all $x$ and $\sum_{i,j} a_{ij}\xi_i\xi_j > \lambda \sum_i x_i^2$ for every $\xi\in \mathbb R^3$. Assume furthermore that the boundary of $\Omega$ is a smooth orientable surface (with possibly several connected components $S_k$).
Under these conditions, is there a theorem that ensure the existence of $u\in C^2(\bar\Omega)$ such that $L u = 0$ in $\Omega$, $u = C_k$ on $S_k$, with $C_k$ constant (Dirichlet conditions).
Same question if only $\sum_{i,j} a_{ij} \partial_i\partial_j u $ is specified on $S_k$ (Neuman conditions).
Same question if one or the other conditions above is specified on the $S_k$ (mixed conditions).
Please, provide references if possible.