A Fatou theorem for the equation $u_t=\Delta (u-1)_+$

Abstract:

In one space dimension and for a given function $u_I (x) \in C_0 ^ {\infty }$ (say such that $u_I (x) > 1$ in some interval), the equation $u_t = \Delta (u-1)_+$ can be thought of as describing the energy per unit volume in a Stefan-type problem where the latent heat of the phase change is given by $1-u_I (x)$. Given a solution $0 \leq u \in L^1 _{\mathrm {loc}} (\mathbb {R} ^n \times (0,T))$ to this equation, we prove that for a.e. $x_0 \in \mathbb {R} ^n$, there exists $\lim _{(x,t) \in \Gamma _{\beta } ^k (x_0),\; (x,t) \to x_0} (u(x,t) -1)_+ =( f(x_0)-1)_+,$ where $f =\partial \mu / \partial |\;|$ is the Radon-Nikodym derivative of the initial trace $\mu$ with respect to Lebesgue measure and $\Gamma _{\beta } ^k (x_0) = \{ (x,t): |x-x_0| < \beta \sqrt t,\; 0<t<k \}$ are the parabolic “non-tangential" approach regions. Since only $(u-1)_+$ is continuous, while $u$ is usually not, $\lim _{(x,t) \in \Gamma _{\beta } ^k (x_0),\; (x,t) \to x_0} u(x,t) = f(x_0)$ does not hold in general.