Double phase problems

The double phase energy
 \int_{\Omega} |Dw|^p+a(x) |Dw|^q dx
is probably the most extreme example of non-homogeneous structure exhibiting different degeneration rates while simultaneously depending on the space variable. Introduced by V. V. Zhikov in order to study homogeneization problems and the occurrence of Lavrentiev phenomenon. Such functionals are particularly relevant in Materials Science since they can be used to describe the behaviour of strongly anisotropic materials whose hardening properties, linked to the gradient growth exponent, change with the point. In particular, a mixture of two different media, with hardening exponents $p$ and $q$ respectively, can be realized according to the geometry dictated by the zero set of the modulating coefficient $a$. The regularity theory for minima of such variational integrals has attracted lots of attention in the last few years. I will discuss the last advances on this subject with strong emphasis on fine properties of solutions to certain problems set in the double phase framework. This talk is based on papers:

I. Chlebicka, C. De Filippis, Removable sets in non-uniformly elliptic problems. Ann. Mat. Pura Appl. https://doi.org/10.1007/s10231-019-00894-1.
C. De Filippis, Regularity for solutions of fully nonlinear elliptic equations with non-homogeneous degeneracy. Submitted. https://arxiv.org/abs/1906.06125
C. De Filippis, G. Mingione, A borderline case of Calderon-Zygmund estimates for non-uniformly elliptic problems. St Petersburg Mathematical Journal, to appear.
C. De Filippis, G. Mingione, Manifold constrained non-uniformly elliptic problems. J. Geom. Analysis, (2019). https://doi.org/10.1007/s12220-019-00275-3
C. De Filippis, J. Oh, Regularity for multi-phase variational problems. Journal of Differential Equations 267, 3, 1631-1670, (2019). https://doi.org/10.1016/j.jde.2019.02.015
C. De Filippis, G. Palatucci, Hoelder regularity for nonlocal double phase equations, J. Differential Equations, 267, 1, 547-586, (2019).