Parabolic $L^p$ Dirichlet Boundary Value Problem and VMO-type time-varying domains

Abstract: We prove the solvability of the parabolic $L^p$ Dirichlet boundary value problem for $1 < p \leq \infty$ for a PDE of the form $u_t = \mbox{div} (A \nabla u) + B \cdot \nabla u$ on time-varying domains where the coefficients $A= [a_{ij}(X, t)]$ and $B=[b_i]$ satisfy a certain natural small Carleson condition. This result brings the state of affairs in the parabolic setting up to the elliptic standard. Furthermore, we establish that if the coefficients of the operator $A,\,B$ satisfy a vanishing Carleson condition and the time-varying domain is of VMO type then the parabolic $L^p$ Dirichlet boundary value problem is solvable for all $1 < p \leq \infty$. This result is related to results in papers by Maz\'ya, Mitrea and Shaposhnikova, and Hofmann, Mitrea and Taylor where the fact that boundary of domain has normal in VMO or near VMO implies invertibility of certain boundary operators in $L^p$ for all $1 < p \leq \infty$ which then (using the method of layer potentials) implies solvability of the $L^p$ boundary value problem in the same range for certain elliptic PDEs. Our result does not use the method of layer potentials, since the coefficients we consider are too rough to use this technique but remarkably we recover $L^p$ solvability in the full range of $p$'s as the two papers mentioned above.