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The nonlinear elliptic equation is investigated. It is supposed that u fulfils a mixed boundary value condition and that Ω ⊂ IRn (n ≥ 3) has a piecewise smooth boundary. Ws,2 — regularity (s < 3/2) of u and Lp — properties of the first and the second derivatives of u are proven.

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The degenerate parabolic equation u t = ∆ |u| m−1 u , m>0 is considered in a cylinder Ω × (0, T ) under homogeneous Dirichlet boundary values. The regularity of weak solutions for the fast diffusion equation (m < 1) and the porous medium equation (m > 1) are investigated. Regularity of u and |u| m−1 u in weighted Sobolev spaces and in fractional order Nikolskii spaces are proved.  2005 Elsevier Inc. All rights reserved.