On the Lp-Poisson Semigroup Associated with Elliptic Systems
On the Lp-Poisson Semigroup Associated with Elliptic Systems
复制标题
与椭圆系统相关的 Lp-Poisson 半群
作者：
J. M. Martell;D. Mitrea;I. Mitrea;M. Mitrea
研究了上半空间上齐次二阶复系数强椭圆系统在Lp上的Poisson半群的无穷小生成，证明了它是Dirichlet-to-Normal映射。在正则性问题可解的函数构成的上半空间的边界上，将其定义域确定为基于lp的1阶Sobolev空间的线性子空间。此外，对于一类包含lam<s:1>系统，以及所有二阶标量椭圆算子，具有常复系数的系统，无穷小发生器的作用被显式地描述为奇异积分算子，其核涉及给定系统规范基本解的一阶导数。此外，还用高阶Sobolev空间和系统的高阶正则性问题描述了所述泊松半群的无穷小发生器的任意幂。最后，我们指出了我们的技术如何适用于处理图Lipschitz域中的高阶系统。
We study the infinitesimal generator of the Poisson semigroup in Lp associated with homogeneous, second-order, strongly elliptic systems with constant complex coefficients in the upper-half space, which is proved to be the Dirichlet-to-Normal mapping in this setting. Also, its domain is identified as the linear subspace of the Lp-based Sobolev space of order one on the boundary of the upper-half space consisting of functions for which the Regularity problem is solvable. Moreover, for a class of systems containing the Lamé system, as well as all second-order, scalar elliptic operators, with constant complex coefficients, the action of the infinitesimal generator is explicitly described in terms of singular integral operators whose kernels involve first-order derivatives of the canonical fundamental solution of the given system. Furthermore, arbitrary powers of the infinitesimal generator of the said Poisson semigroup are also described in terms of higher order Sobolev spaces and a higher order Regularity problem for the system in question. Finally, we indicate how our techniques may be adapted to treat the case of higher order systems in graph Lipschitz domains.