In this talk, we will consider the elliptic and parabolic double-phase problems. For the elliptic problem, we discuss the Lipschitz truncation method of the double-phase function space. It features a new aspect in the double-phase function space and provides the absence of Lavrentiev's phenomenon and the regularity property of a very weak solution. The intrinsic geometry for the parabolic double-phase problem is based on the observation in the Lipschitz truncation method. For the parabolic problem, we will discuss the construction of the two phases and the regularity results.
This talk is based on the research papers Baasandorj, Byun and Kim, arXiv and Kim, Kinnunen and Moring, Arch. Ration. Mech. Anal. to appear.