Maximal regularity for local minimizers of non-autonomous functionals

Establishment of regularity theory for partial differential equations and
variational problems with (p, q)-growth condition has been an open issue since the 1980s.
In this talk we introduce recent results of C^{1,α} -regularity for some α ∈ (0, 1) and C^α -
regularity for any α ∈ (0, 1) of local minimizers of the functional v\mapsto\int_\Omega φ(x, |Dv|) dx, where φ satisfies a (p, q)-growth condition and sharp regularity conditions. In addition,
we introduce regularity results for bounded or Hölder continuous minimizers of the above
functional, or weak solutions of PDEs with a generalized Orlicz growth condition.
This is a joint work with Peter Hästö at University of Turku.