Regularity for general functionals with double phase

We prove sharp regularity results for a general class of functionals of the type w↦∫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↦∫b(x,w)(|Dw|p+a(x)|Dw|q)dx,1<p<q,a(x)≥0,with 0 < ν≤ b(·) ≤ L. This changes its ellipticity rate according to the geometry of the level set a(x) = 0 of the modulating coefficient a(·). 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.