Regularity of weak solutions of parabolic variational inequalities

Abstract:

In this paper, parabolic operators of the form \[ {u_t} - \operatorname {div} A(x, t, u, Du) - B(x, t, u, Du)\] are considered where $A$ and $B$ are Borel measurable and subject to linear growth conditions. Let $\psi : \Omega \to {R^1}$ be a Borel function bounded above (an obstacle) where $\Omega \subset {R^{n + 1}}$. Let $u \in {W^{1,2}}(\Omega )$ be a weak solution of the variational inequality in the following sense: assume that $u \geqslant \psi$ q.e. and \[ \int _\Omega {{u_t}\varphi + A \cdot D\varphi - B\varphi \geqslant 0} \] whenever $\varphi \in W_0^{1,2}(\Omega )$ and $\varphi \geqslant u - \psi$ q.e. Here q.e. means everywhere except for a set of classical parabolic capacity. It is shown that $u$ is continuous even though the obstacle may be discontinuous. A mild condition on $\psi$ which can be expressed in terms of the fine topology is sufficient to ensure the continuity of $u$. A modulus of continuity is obtained for $u$ in terms of the data given for $\psi$.