Vol 224, No 3 (2017)

Local regularity up to the flat part of the boundary is proved for certain classes of distributional solutions that are L_∞L^3,q with q finite. The corresponding result for the interior case was recently proved by Wang and Zhang, see also Phuc’s paper. For local regularity up to the flat part of the boundary, q = 3 was established by G. A. Seregin. Our result can be viewed as an extension of it to L^3,q with q finite. New scale-invariant bounds, refined pressure decay estimates near the boundary and development of a convenient new ϵ-regularity criterion, are central themes in providing this extension.

The method proposed by T. I. Zelenjak is applied to the mean curvature flow in the plane. A new type of monotonicity formula for star-shaped curves is obtained.

Variational representations of the constant c_Ω in the inf-sup condition for the Stokes problem in a bounded Lipschitz domain in ℝ^d, d ≥ 2, are deduced. For any pair of admissible functions, the respective variational functional provides an upper bound of c_Ω and the exact infimum of it is equal to c_Ω. Minimization of the functionals over suitable finite dimensional subspaces generates monotonically decreasing sequences of numbers converging to c_Ω and, therefore, they can be used for numerical evaluation of the constant.

The paper is concerned with estimates of the Hölder norms of solutions of model parabolic initialboundary value problems in a half-space. The proof is based on O. A. Ladyzhenskaya’s theorem on the Fourier multipliers in anisotropic Hölder spaces and on K. K. Golovkin’s theorem on equivalent norms in these spaces.