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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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