In this work we establish log type stability estimates for the inverse potential and conductivity problems with partial Dirichlet-to-Neumann map, where the Dirichlet data is homogeneous on the inaccessible part. This result, to some extent, improves our former result on the partial data problem [HW06] in which log-log type estimates were derived.

Consider a parabolic N × N -system of order m on R n with top-order coefficients aα ∈ VMO ∩ L ∞ . Let 1 < p, q < ∞ and let ω be a Muckenhoupt weight. It is proved that systems of this kind possess a unique solution u satisfyingwhere Au = |α| m aαD α u and J = [0, ∞). In particular, choosing ω = 1, the realization of A in L p (R n ) N has maximal L p − L q regularity.

Abstract. In this paper we describe two recent approaches for the L p -theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle.

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