Regularity for general functionals with double phase

We prove sharp regularity results for a general class of functionals of the type w (sic) integral 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 (sic) integral b(x, w)(vertical bar Dw vertical bar(p) + a(x)vertical bar Dw vertical bar(q))dx, 1 < p < q, a(x) >= 0, with 0 < v <= b(center dot) = L. This changes its ellipticity rate according to the geometry of the level set {a(x) = 0} of the modulating coefficient a(center dot). 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.