Limiting smoothness of the solution of a nonstationary problem with one or two obstacles (Q1101885)

The regularity of the solution of a nonstationary problem with an obstacle for various forms of parabolic operators is thoroughly investigated. Under the condition of sufficient smoothness of the data of the problem, one proves that the solution \(W_ q^{2,1}(Q)\) belongs to the Sobolev space \(Q=\Omega \times (0,T)\), \(\Omega \in {\mathbb{R}}^ n\), \(q<+\infty\). In the present paper one establishes that the limiting possible smoothness of the solution of a nonstationary problem with one or two obstacles is the boundedness of the second derivatives of the solution with respect to the spatial variables and of the first derivatives with respect to t. One assumes that the operator is linear and the functions defining the obstacles have the minimal possible smoothness.