If I want to model the surface of any random crystal, is it safe to assume that it is the graph of a Lipschitz function. Is there a precise result from physicists? How wrong would it be if I assume that it is the graph of a function which is twice differentiable a.e?
2026-03-28 01:48:19.1774662499
Regularity of the surface of a crystal
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Take the simplest (to me) crystal to imagine, a cubic structure. This is just like the three dimensional lattice of integer points. If the faces are along one of the planes, the surface is nice and smooth-a plane even. If the faces are not along the planes, you have lots of sawteeth and the surface is far from Lipschitz if you look close enough.
Most other crystals will have corners everywhere at the atomic scale, so they are not Lipschitz. If you are only interested in scales larger than the atom, you can probably get away with it.