VERY WEAK SOLUTIONS OF THE STATIONARY STOKES EQUATIONS ON EXTERIOR DOMAINS

초록

We study the nonhomogeneous Dirichlet problem for the stationary Stokes equations on exterior smooth domains Omega in R-n, n >= 3. Our main result is the existence and uniqueness of very weak solutions in the Lorentz space L-p,L-q(Omega)(n), where (p, q) satisfies either (p, q) = (n/ (n - 2), infinity) or n/(n - 2) < p < infinity, 1 <= q <= infinity. This is deduced by a duality argument from our new solvability results on strong solutions in homogeneous Sobolev-Lorentz spaces. Homogeneous Sobolev-Lorentz spaces are also studied in quite details: particularly, we establish basic interpolation and density results, which are not only essential to our results for the Stokes equations but also themselves of independent interest.

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