What are the loosest possible conditions for linear second order PDEs to be well-posed: $$c + \sum_{i}b_i\partial_iu + \sum_{i,j}a_{ij}\partial_{ij}u = 0,$$ $$ u(\Gamma) = g_1,\quad \partial_n u(\Gamma) = g_2,$$
where $\Gamma$ is the boundary of the considered domain $\Omega$. I want something looser than requiring $u$ to be $C^2$, and $c,b,a$ to be in $C$.
