Fine boundary continuity for degenerate double-phase diffusion

Abstract

We study 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 Hölder continuity of solutions. In particular, the double-phase elicits new questions on the definition of an adapted capacity.

33 pages, 2 figures, appendix at the end of the paper