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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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