  We investigate existence and uniqueness of strong solutions of mean-field
stochastic differential equations with irregular drift coefficients. Our direct
construction of strong solutions is mainly based on a compactness criterion
employing Malliavin Calculus together with some local time calculus.
Furthermore, we establish regularity properties of the solutions such as
Malliavin differentiablility as well as Sobolev differentiability in the
initial condition. Using this properties we formulate an extension of the
Bismut-Elworthy-Li formula to mean-field stochastic differential equations to
get a probabilistic representation of the first order derivative of an
expectation functional with respect to the initial condition.
