*Preprint*

**Inserted:** 13 feb 2018

**Last Updated:** 13 feb 2018

**Year:** 2018

**Abstract:**

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.

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