标题： 第一特征值和亚椭圆算子的基本间隙的极值性质 显示英文标题

标题： Extremal properties of the first eigenvalue and the fundamental gap of a sub-elliptic operator

摘要： 我们考虑在势能$V$受到$p$-范数约束时，带有狄利克雷边界条件的次椭圆算子的第一特征值和基本间隙的极值问题。 给出了弱解的存在性结果、紧嵌入定理和次椭圆方程的谱理论。 此外，我们提供了相应最优势能函数的具体特性。 显示英文摘要

摘要： We consider the problems of extreming the first eigenvalue and the fundamental gap of a sub-elliptic operator with Dirichlet boundary condition, when the potential $V$ is subjected to a $p$-norm constraint. The existence results for weak solutions, compact embedding theorem and spectral theory for sub-elliptic equation are given. Moreover, we provide the specific characteristics of the corresponding optimal potential function.