I have a family of simply connected domains Lipschitz domains D(t) on the unit n-1 sphere in Euclidean n-space. The domains vary continuously in t. Since the uniform limit of Lipschitz functions is a Lipschitz function, can I apply the Leiniz-Reynolds transport theorem to differentiate
t--> int_{D(t)} f(x,t) dS(x)
with respect to t where f may be assumed as infinitely smooth in all variables and S is the surface measure on the sphere. In other words, does this result in something like standard integration by parts in n-space because the domains relative boundaries in the sphere have well-defined surface measures and an outward normals that is defined(tangent to the sphere) almost everywhere on bd(D(t)) for all t?
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