Reflection principles and kernels in \(\mathbb R^n_+\) for the biharmonic and Stokes operators. Solutions in a large class of weighted Sobolev spaces. (English) Zbl 1193.35030

Amrouche, Chérif; Raudin, Yves

Adv. Differ. Equ. 15, No. 3-4, 201-230 (2010).

This paper deals with a class of Stokes systems in the half-space. The authors are interested in a kernel of an operator associated to this problem and symmetrically with the compatibility condition for the data. For this reason, an important part of this paper is devoted to the study of the reflection principles for the biharmonic and Stokes operators. After giving the weak formulation of the problem with the aim of getting the kernels in some distribution spaces, the authors state the main result of this paper. This deals with strong solutions and symmetrically very weak solutions. A central role in the proofs is played by adequate duality arguments.

Reviewer: Teodora-Liliana Rădulescu (Craiova)

MSC:

35J30	Higher-order elliptic equations
35D30	Weak solutions to PDEs
35J50	Variational methods for elliptic systems
35Q30	Navier-Stokes equations
76D07	Stokes and related (Oseen, etc.) flows
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids

Keywords:

Stokes operator; reflection principle; weighted Sobolev space