zbMATH — the first resource for mathematics

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 \[ \int_{\Omega^\prime}H(u)\,d\xi\leq C({\text{osc}}_\Omega u)^2. \]

35B50 Maximum principles in context of PDEs
35H20 Subelliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
PDF BibTeX Cite
Full Text: DOI
[1] Balogh Z. M., Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) pp 847– (2003)
[2] Bieske T., Comm. Partial Differential Equations 27 (3) pp 727– (2002) · Zbl 1090.35063
[3] Danielli D., Comm. Anal. Geom. 11 (2) pp 263– (2003) · Zbl 1077.22007
[4] Gutiérrez C. E., The Monge-Ampère Equation (2001) · Zbl 0989.35052
[5] Gutiérrez C. E., Ann. Scuola Norm. Sup. Pisa Cl. Sci (5) pp 349– (2004)
[6] Jensen R., Arch. Rational Mech. Anal. 123 (1) pp 51– (1993) · Zbl 0789.35008
[7] Lu G., Calc. Var. Partial Differential Equations 19 (1) pp 1– (2003) · Zbl 1072.49019
[8] Reilly R. C., Michigan Math. J. 20 pp 373– (1973)
[9] Stein E. M., Princeton Math. Series, 43, in: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals (1993)
[10] Trudinger N. S., Topol. Methods Nonlinear Anal. 10 pp 225– (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.