  In this paper, we consider a new class of multi phase operators with variable
exponents, which reflects the inhomogeneous characteristics of hardness changes
when multiple different materials are combined together. We at first deal with
the corresponding functional spaces, namely the Musielak-Orlicz Sobolev spaces,
hence we investigate their regularity properties and the extension of the
classical Sobolev embedding results to the new context. Then, we focus on the
regularity properties of our operators, and prove that these operators are
bounded, continuous, strictly monotone, coercive and satisfy the
(S_+)-property. Further, we discuss suitable problems driven by such operators.
In particular, we deal with Dirichlet problems in which the nonlinearity is
gradient dependent. Under very general assumptions, we establish the existence
of a nontrivial solution for such problems. Also, we give additional conditions
on the nonlinearity which guarantee the uniqueness of solution. Lastly, we
produce some local regularity results (namely, Caccioppoli-type inequality,
Sobolev-Poincar\'e-type inequalities and higher integrability) for minimizers
of the integral functionals corresponding to the operators.
