The Wiener Criterion for Double-Phase Diffusion

We present a recent study on the boundary behavior of solutions to parabolic double-phase equations, through the celebrated Wiener’s sufficiency criterion. The analysis is conducted for cylindrical domains and the regularity up to the lateral boundary is shown in terms of either its p or q capacity, depending on whether the phase vanishes at the boundary or not. Eventually we obtain a fine boundary estimate that, when considering uniform geometric conditions (as density or fatness), leads us to the boundary Holder continuity of solutions. In particular, the double-phase elicits new questions on the definition of an adapted capacity. This is a joint work in collaboration with Eurica Henriques and Ihor Skrypnik.