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On the Stokes system and on the biharmonic equation in the half-space: An approach via weighted Sobolev spaces. (English) Zbl 0998.35037
The author of this paper studies the Stokes system, $-\nu \triangle\mathbf u +\nabla p=\mathbf f \quad\text{div } \mathbf u =h \quad \text{in } \mathbb R_+^n$ with an adhesion condition $$\mathbf u$$$$=\mathbf g$$ at $$\Sigma$$ or under a slip condition $$\mathbf u$$$$\cdot \mathbf n$$$$=g$$, $$\mathbf e$$$$_i\cdot$$$$\mathcal T$$$$(\mathbf u$$$$,p)\cdot \mathbf n$$$$=\mathbf z$$$$\cdot\mathbf e$$$$_i$$, $$i\in \{1,2,\dots,n-1\}$$ at $$\Sigma$$. Here $$\mathcal T$$ denotes the stress tensor, $$\mathbf n$$ is the normal vector on $$\Sigma$$ and $$\mathbf e$$$$_i$$ ($$i=1,2,\dots,n-1$$) are the tangential vectors on $$\Sigma$$. The approach applied here rests on the use of a family of weighted Sobolev spaces as a framework for describing the behavior at infinity. A complete class of existence, uniqueness and regularity results for the considered problem is proved. Another problem investigated here is the biharmonic equation $$\triangle^2 u=f$$ with boundary condition $$u=g$$ at $$\Sigma$$, $$\partial u/\partial n=h$$ at $$\Sigma$$. Existence, uniqueness and regularity results for this problem are shown as well. The proofs of the main results are based on the principle of reflection.

##### MSC:
 35Q30 Navier-Stokes equations 35J25 Boundary value problems for second-order elliptic equations 76D07 Stokes and related (Oseen, etc.) flows 31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
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