On the $L^p$-Poisson semigroup associated with elliptic systems

TL;DR

This paper investigates the infinitesimal generator of the Poisson semigroup for elliptic systems in the upper-half space, identifying its domain, explicit action, and extensions to higher order systems and Lipschitz domains.

Contribution

It characterizes the generator as the Dirichlet-to-Normal map, explicitly describes its action via singular integrals, and extends the analysis to higher order systems and Lipschitz domains.

Findings

Generator equals the Dirichlet-to-Normal map.

Explicit description via singular integral operators.

Extension to higher order systems and Lipschitz domains.

Abstract

We study the infinitesimal generator of the Poisson semigroup in $L^p$ 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 $L^p$-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 Lam\'e 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 adapted to treat the case of higher order systems in graph Lipschitz domains.