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

Solvability of the Dirichlet, Neumann and the regularity problems for parabolic equations with H\"older continuous coefficients