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Nowadays people usually consider PDEs in weak formulations only, so I have a hard time finding statements about the existence of classical solutions of the Poisson equation with mixed Dirichlet-Neumann boundary conditions.

Maybe someone here can help me and point to a book or article where I can find sufficient conditions on the right hand side that guarantee the existence of a $C^2$ solution.

Note that the unique solvability in $H^1(\Omega)$ follows, e.g., from the paper on discrete maximum principles by Karatson and Korotov in Numer. Math. 99 (2005), 669-698. So (a slightly more precise version of) my question boils down to finding conditions such that this solution is actually C^2 in the interior and continuous differentiable at the boundary.

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WWelcome to MSE. I am adding a reference request tag to your question. –  user21436 Mar 3 '12 at 22:08
    
Have you looked at Troianiello, Elliptic Differential Equations and Obstacle Problems? Chapter 3 might have something on Hölder regularity, and if I recall correctly, he treats mixed boundary conditions (although only homogeneous ones). –  Christian Clason Nov 28 '12 at 20:11

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I believe the papers by Gary Lieberman will answer your question: one is Mixed boundary value problem for elliptic and parabolic differential equations of second order. Another one is Optimal holder regularity for mixed boundary value problems. Thanks

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Thanks. These and especially their reference to Azzam and Kreyszig are very useful! –  Arnold Neumaier Dec 2 '12 at 14:11

From my rather primitive knowledge of PDEs, for a well-posed mixed boundary value problem for Poisson equation, I think the effect of the Neumann boundary condition on the regularity of the solution is equivalent to Dirichlet boundary condition of one less differentiability.

Intuitively speaking(most likely I am wrong here), if the uniqueness of the mixed problem can be established(via maximum principle for example), and enough regularity has been assumed for the $u$ to be well-defined on $\Gamma_N$, then it is same that we have $u = H g_N$ on $\Gamma_N$ where $H$ is the Neumann-to-Dirichlet map. If we know all the Dirichlet and Neumann data, the regularity of the solution depends on the regularity of the boundary like the classical theory states.

For reference, in the book Elliptic Partial Differential Equations of Second Order by Gilbarg and Trudinger, they use an entire chapter six to discuss the classical Schauder estimate for Hölder continuous solutions with various boundary conditions, in Section 6.7 of that book, Lemma 6.27 reads the solution to the Neumann problem of Poisson equation is $C^{2,\alpha}(\bar{\Omega})$-regular if Neumann data $g_N$ is $C^{1,\alpha}$, $f$ is $C^{0,\alpha}$, and the domain is $C^{2,\alpha}$. Now back to the regularity estimate for the Dirichlet problem in Theorem 6.19, if Dirichlet data $g_D$ is $C^{2,\alpha}$, other the conditions are the same, we will get the same $C^{2,\alpha}$-regularity for the solution. Hence I am guessing for mixed boundary problem we should have a similar result that a $C^2(\Omega)\cap C^{1,1}(\bar{\Omega})$ solution should require a $C^2$ domain, $C^1$ Neumann boundary data, and $C^2$ Dirichlet boundary data.

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Thanks. I hadn't realize before that the pure Neumann case is in Gilbarg/Trudinger which I generally know quite well. I still hope to get info on the mixed case from somewhere.... –  Arnold Neumaier May 24 '12 at 12:19

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