  We establish the $L^2$-solvability of Dirichlet, Neumann and regularity
problems for divergence-form heat (or diffusion) equations with
H\"older-continuous diffusion coefficients, on bounded Lipschitz domains in
$\mathbb{R}^n$. This is achieved through the demonstration of invertibility of
the relevant layer-potentials which is in turn based on Fredholm theory and a
new systematic approach which yields suitable parabolic Rellich-type estimates.
