Dirichlet problem for mean curvature equation

Question

Let $\Omega$ be a uniformly convex domain in $\mathbb{R}^n$ with smooth boundary $\partial \Omega$. For any $\varphi \in C^{\infty} (\partial \Omega)$, is there a stictly convex smooth solution $u$ to the Dirichlet problem of prescribed mean curvature equation \begin{aligned} H(x) =&\; c, &\mbox{in}\; \Omega\\ u =&\; \varphi, &\mbox{on}\; \partial \Omega \end{aligned} where $H(x)$ is the mean curvature of the graph $(x, u(x))$ and $c> 0$ is a constant. Is there anyone that remind me any references where this result is proved?

Answer 1

In $\mathbb{R}^3$, the Plateau problem for constant mean curvature surfaces was solved in

Hildebrandt, S., On the Plateau problem for surfaces of constant mean curvature, Commun. Pure Appl. Math. 23, 97-114 (1970). ZBL0181.38703.

Now, this version does not match directly with what you asked:

• Compared to what you asked about, this is only about $n = 2$
• You are only looking at the graphical case, whereas Hildebrandt's result does not make the assumption that the resulting surface is obtained as a graph over a hyperplane.

But potentially tracing forwards and backwards from this paper on MathSciNet will get you something more akin to the precise statement you are asking about.

Answer 2

If we ignore "convexity" everywhere in your question, Gillbarg & Trudinger Theorem 16.11 gves:

For $\Omega$ a bounded $C^2$ domain in $\mathbb{R}^n$, the Dirichlet problem $H(\textrm{graph}(u)) = c$, $u|_{\partial\Omega} = \varphi$ is solvable (for all $\varphi$) if and only if the boundary mean curvature of $\partial\Omega$ satisfies $H_{\partial\Omega} \geq \frac{n}{n-1}|c|$.

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I think that for $\Omega= B_1$ and $\varphi =0$, no solution exists (convex or not) for $|c|$ large as can be seen by analyzing rotationally symmetric CMC surfaces.