论文标题
标志域中的超端边界价值问题
Sub-elliptic boundary value problems in flag domains
论文作者
论文摘要
$ \ Mathbb {r}^{3} $中的一个标志域是$ \ Mathbb {r}^{3} $的子集的子集的表格$ \ {（x，y，t）的$我们解决了dirichlet和neumann的问题，用于下elliptic kohn-laplacian $ \ bigtriangleup^{\ flat} = x^{2} + y^{2} $ in Flag domains $ω\ subset \ subset \ subset \ subset \ subset \ subbb {r} {r}^{r}^{3} $，带有$ l^{2} $ - 如果边界值具有第一阶$ l^{2} $ - sobolev规则性，我们还可以获得改进的解决方案的规律性。我们的解决方案是作为亚椭圆形单层和双层电势获得的，最好将其视为第一个海森堡组的积分运算符。我们在国旗域及其边界上发展了这些操作员的理论。
A flag domain in $\mathbb{R}^{3}$ is a subset of $\mathbb{R}^{3}$ of the form $\{(x,y,t) : y < A(x)\}$, where $A \colon \mathbb{R} \to \mathbb{R}$ is a Lipschitz function. We solve the Dirichlet and Neumann problems for the sub-elliptic Kohn-Laplacian $\bigtriangleup^{\flat} = X^{2} + Y^{2}$ in flag domains $Ω\subset \mathbb{R}^{3}$, with $L^{2}$-boundary values. We also obtain improved regularity for solutions to the Dirichlet problem if the boundary values have first order $L^{2}$-Sobolev regularity. Our solutions are obtained as sub-elliptic single and double layer potentials, which are best viewed as integral operators on the first Heisenberg group. We develop the theory of these operators on flag domains, and their boundaries.
