Let $\Omega \subset \mathbb R^n$ be a subdomain with Lipschitz boundary, i.e. locally any part of the boundary looks like the graph of a Lipschitz continuous function, after some affine coordinate transformation.
Suppose we are given a "partition" $\Gamma_D$ and $\Gamma_N$ of the boundary, s.t. these sets are submanifolds of $\mathbb R^n$ with Lipschitz boundary by themselves, and their intersection has measure zero.
Let us be given $g \in L^2(\Gamma_D)$ and $h \in L^2(\Gamma_N)$ and some function $f \in L^2(\Omega)$. We want to solve Poisson's equation with mixed boundary conditions
$\operatorname{div}\operatorname{grad} u = f$ over $\Omega$
$u_{|\Gamma_D} = g$ over $\Gamma_D$
$\operatorname{grad} u_{|\Gamma_N} \cdot n = h$ over $\Gamma_N$
It is standard to prove well-posedness of these problems if either $\Gamma_D$ or $\Gamma_N$ is the empty set. I have not found a rigorous proof of well-posedness for general mixed boundary conditions in the standard books like, say, Gilbarg-Trudinger. On the other hand, certain papers suggest the boundary parts are required to meet at an angle that is not 180° in the case of Lipschitz boundaries, so the boundary is necessarily non-smooth.
These influences appear confusing to me. I do not know how to learn more about this. Could please give a reference where to learn more about the Poisson problem with mixed boundary conditions?
EDIT:
In order to motivate why this is interesting and why it confuses me, I would like to point to the paper Ott, Brown: The mixed problem for the Laplacian in Lipschitz domains and R.M. Brown. The mixed problem for Laplace’s equation in a class of Lipschitz domains. On the other hand, in numerical analysis lectures that I attend, this question is usually swept under the rug and one deals freely with mixed boundary conditions. So either I don't know the well-posedness results for simplicial domains, or the numerical examples all belong to the well-posed case.
Best Answer
Conditions for the unique solvability in H1(Ω) are given in Theorem 4 of the paper on discrete maximum principles by Karatson and Korotov in Numer. Math. 99 (2005), 669-698. You need to specialize this result (stated for nonlinear second-order elliptic equations with mixed boundary conditions) to your particular situation.