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) is an element of R x R(n-1) x R: x(0) > A(x, t)}, obtaining solvability of the Dirichlet and Neumann problems when the data lie in L(2)(partial derivative 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 derivative 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 that our results are in the nature of ''best possible.''