旗形区域上的次椭圆边值问题
Sub-elliptic boundary value problems in flag domains
摘要：
$\mathbb{R}^{3}$中的标志域是形式为$\{(x,y,t) : y < A(x)\}$的$\mathbb{R}^{3}$的子集，其中$A \colon \mathbb{R} \to \mathbb{R}$是Lipschitz函数。我们在旗域$Ω\subset \mathbb{R}^{3}$中求解了次椭圆Kohn-Laplacian算子$\bigtriangleup^{\flat} = X^{2} + Y^{2}$的Dirichlet和Neumann问题，并给出了$L^{2}$ -边值.当边值满足一阶$L^{2}$ -Sobolev正则性时，我们还得到了狄利克雷问题解的正则性的改进.我们的解决方案是作为亚椭圆的单层和双层位势，这是最好的积分算子的第一海森堡群。我们开发的理论，这些运营商的标志域，和他们的边界。
英文摘要：
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.
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