  We prove sharp regularity results for a general class of functionals of the
type $$ w \mapsto \int F(x, w, Dw) \, dx\;, $$ featuring non-standard growth
conditions and non-uniform ellipticity properties. The model case is given by
the double phase integral $$ w \mapsto \int b(x,w)(|Dw|^p+a(x)|Dw|^q) \,
dx\;,\quad 1 <p < q\,, \quad a(x)\geq 0\;, $$ with $0<\nu \leq b(\cdot)\leq L
$. This changes its ellipticity rate according to the geometry of the level set
$\{a(x)=0\}$ of the modulating coefficient $a(\cdot)$. We also present new
methods and proofs, that are suitable to build regularity theorems for larger
classes of non-autonomous functionals. Finally, we disclose some new
interpolation type effects that, as we conjecture, should draw a general
phenomenon in the setting of non-uniformly elliptic problems. Such effects
naturally connect with the Lavrentiev phenomenon.
