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Maximum and comparison principles for convex functions on the Heisenberg group. (English) Zbl 1056.35033
The function $$u\in C^2(\Omega)$$ is convex in $$\Omega$$ if the symmetric matrix ${\mathcal H}(u)=\left[\begin{matrix} X^2u &(XYu+YXu)/2\\ (XYu+YXu)/2 & Y^2u\end{matrix}\right]$ is positive semidefinite in $$\Omega$$. The authors defined a Monge-Ampère type operator as follows; $$H(u)=\det{\mathcal H}(u)+12(\partial_tu)^2$$. In this paper, the authors proved the following theorems.
Theorem 1 (Comparison Principle). Let $$u,v\in C^2(\bar\Omega)$$ such that $$u+v$$ is convex in $$\Omega$$ satisfying $$u=v$$ on $$\partial\Omega$$ and $$v<u$$ in $$\Omega$$. Then $\int_\Omega H(u)d\xi\leq \int_\Omega H(v)\,d\xi.$ Theorem 2 (Maximum Principle). Let $$u\in C^2(B_R)$$ be convex, $$u=0$$ on $$\partial B_R$$. If $$u(\xi_0)=\min_{B_R}u$$, then there exists a positive constant $$c$$, depending on $$d(\xi_0, \partial B_R)$$, such that $| u(\xi_0)| ^2\leq c\int_{B_R}H(u)\,d\xi.$ Theorem 3 (Oscillation Estimate). Let $$u\in C^2(\Omega)$$ be convex. For any compact domain $$\Omega^\prime\subset\subset\Omega$$ there exists a positive constant $$C$$ depending on $$\Omega^\prime$$ and $$\Omega$$ and independent of $$u$$ such that $\int_{\Omega^\prime}H(u)\,d\xi\leq C({\text{osc}}_\Omega u)^2.$

##### MSC:
 35B50 Maximum principles in context of PDEs 35H20 Subelliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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