Carnot群上凸函数的比较原理

Comparison Principles for Convex Functions on the Carnot Group

Carnot群上凸函数的单调性质对研究完全非线性次椭圆方程的正则性理论起关键作用.通过在Carnot群上引入(Hr)-凸函数类,利用辅助函数方法并结合基于群结构的散度定理,建立了关于(H2)-凸函数的比较原理.此外,作为该结论的应用,得到了高维Heisenberg群上关于凸函数的比较原理.这些结果有望为进一步研究Carnot群上凸函数的性质和完全非线性方程的正则性提供理论基础.

The monotonicity properties of convex functions on the Carnot group are important in studying the regularity of fully nonlinear subelliptic equations. Firstly, the （H）r-convex function class was introduced on the Carnot group. Then, the comparison principle of the （H）_ convex functions was established by constructing auxiliary functions and using divergence theorem based on the group structure. Moreover, as an application of the result, the comparison principle of the convex functions on the higher-dimension Heisenberg group was obtained. These results are expected to provide some theoretical basis for the further study of the properties of convex functions and of the regularity of fully nonlinear equations on the Carnot group.