$L^2$ solvability and representation by caloric layer potentials in time-varying domains


We consider boundary value problems for the heat equation in time-varying graph domains of the form $\Omega = \{(x_0,x,t) \in \mathbb{R} \in \mathbb{R}^{n-1} \times \mathbb{R}\colon x_0 > A(x,t)\}$, obtaining solvability of the Dirichlet and Neumann problems when the data lie in $L^2(\partial \Omega)$. We also prove optimal regularity estimates for solutions to the Dirichlet problem when the data lie in a parabolic Sobolev space of functions having a tangential (spatial) gradient, and one half of a time derivative in $L^2(\partial \Omega)$. Furthermore, we obtain representations of our solutions as caloric layer potentials. We prove these results for functions $(A(x,t)$ satisfying a minimal regularity condition which is essentially sharp from the point of view of the related singular integral theory. We construct counterexamples which show our results are in the nature of “best possible.”




Steve Hofmann

John Lewis