Improved estimates for the sharp interface limit of the stochastic Cahn-Hilliard equation with space-time white noise

We study the sharp interface limit of the stochastic Cahn-Hilliard equation with cubic double-well potential and additive space-time white noise epsilon sigma W(sic), where epsilon > 0 is an interfacial width parameter. We prove that, for a sufficiently large scaling constant sigma > 0, the stochastic Cahn-Hilliard equation converges to the deterministic Mullins-Sekerka/Hele-Shaw problem for epsilon -> 0. The convergence is shown in suitable fractional Sobolev norms as well as in the L-p-norm for p is an element of (2, 4] in spatial dimension d = 2, 3. This generalizes the existing result for the space-time white noise to dimension d = 3 and improves the existing results for smooth noise, which were so far limited to p is an element of (2, 2d +8 /d+2] in spatial dimension d = 2, 3. As a byproduct of the analysis of the stochastic problem with space-time white noise, we identify minimal regularity requirements on the noise which allow convergence to the sharp interface limit in the H1-norm and also provide improved convergence estimates for the sharp interface limit of the deterministic problem.