Material and shape derivatives for solutions to the Dirichlet Laplacian in a half-space are derived by an application of the speed method. The proposed method is general and can be used for shape sensitivity analysis in unbounded domains for the Neumann Laplacian as well as for the elasticity boundary value problems.
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We consider the time-periodic Oseen flow around a rotating body in ℝ³. We prove a priori estimates in $L^{q}$-spaces of weak solutions for the whole space problem under the assumption that the right-hand side has the divergence form. After a time-dependent change of coordinates the problem is reduced to a stationary Oseen equation with the additional term -(ω ∧ x)·∇u + ω ∧ u in the equation of momentum where ω denotes the angular velocity. We prove the existence of generalized weak solutions in $L^{q}$-space using Littlewood-Paley decomposition and maximal operators.
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