Regularity for general functionals with double phase

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.