Pointwise Estimates for the Heat Equation. Application to the Free Boundary of the Obstacle Problem with Dini Coefficients

摘要

We study the pointwise regularity of solutions to parabolic equations. As a first result, we prove that if the modulus of mean oscillation of Δu ? u_t at the origin is Dini (in L~p average), then the origin is a Lebesgue point of continuity (still in L~p average) for D~2u and ?_tu. We extend this pointwise regularity result to the parabolic obstacle problem with Dini right-hand side. In particular, we prove that the solution to the obstacle problem has, at regular points of the free boundary, a Taylor expansion up to order two in space and one in time (in the L~p average). Moreover, we get a quantitative estimate of the error in this Taylor expansion. Our method is based on decay estimates obtained by contradiction, using blow-up arguments and Liouville-type theorems. As a by-product of our approach, we deduce that the regular points of the free boundary are locally contained in a C~1 hypersurface for the parabolic distance √x2 + |t|.