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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.

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You'll need some additional global conditions regarding your domain and its boundary. For example, is the boundary connected? What about $\Gamma_N \cap \Gamma_D$? As to your angle condition - I would then start by looking at special solutions in corner domains, e.g. $u_\alpha(x,y) = \Re (x+iy)^{\alpha}$ in domains like $\Omega_\gamma = \{(x,y) | x \ge 0, \, 0 \le y \le \gamma x\}$ to understand in what sense one can expect maximal regularity to hold. For example, for which $\alpha, \gamma$ is $u_\alpha \in H^{3/2}_{loc}(\Omega_\gamma)$? –  Hans Engler Nov 26 '11 at 18:01

2 Answers 2

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.

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If we consider the simplified problem $$-\Delta u = f\quad\mbox{in}\quad\Omega,$$ $$u|{\Gamma_D}=0,$$ $$\partial_nu|_{\Gamma_N}=h,$$ Then the weak formulation is to find $u\in H^1_E(\Omega)=\{v\in H^1(\Omega):v|_{\Gamma_D}=0\}$ such that $$(\nabla u,\nabla w)=(f,w)+(h,w)_{\Gamma_N}$$ for all $w\in H^1_E(\Omega)$. By the integrability assumptions on $f,w$, and the trace theorem on $H^1$, the right hand side in a continuous linear functional on $H^1_E(\Omega)$, and the left hand side is a coercive bounded bilinear form. So the Lax Milgram theorem gives us existence and uniqueness in $H^1_E(\Omega)$.

I wanted to approach the more general question that you posed by subtracting some extension $\tilde g$ of the Dirichlet boundary data from $u$, and solve your problem for $v=u-\tilde g$, which would reduce to the problem I mentioned, except for I was unsure how this extension would interact with the Neumann condition, perhaps there are such extensions $\tilde g\in H^1_E(\Omega)$ such that $\partial_n \tilde g|_{\Gamma_N}\in L^2$, in which case the above applies.

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