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,<graphic></graphic>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,<graphic></graphic>with 0<ν≤b(·)≤L<inline-graphic></inline-graphic>. This changes its ellipticity rate according to the geometry of the level set {a(x)=0}<inline-graphic></inline-graphic> of the modulating coefficient a(·)<inline-graphic></inline-graphic>. 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. [ABSTRACT FROM AUTHOR]