Higher order parabolic boundary Harnack inequality in C1 and Ck, α domains

<p style='text-indent:20px;'>We study the boundary behaviour of solutions to second order parabolic linear equations in moving domains. Our main result is a higher order boundary Harnack inequality in <i>C</i>¹ and <i>C</i>^{<i>k</i>, <i>α</i>} domains, providing that the quotient of two solutions vanishing on the boundary of the domain is as smooth as the boundary.</p><p style='text-indent:20px;'>As a consequence of our result, we provide a new proof of higher order regularity of the free boundary in the parabolic obstacle problem.</p>

ON THE BOUNDARY BEHAVIOUR OF FRIDMAN INVARIANTS

We prove that the Fridman invariant defined using the Carathéodory pseudodistance does not always go to 1 near strongly Levi pseudoconvex boundary points and it always goes to 0 near nonpseudoconvex boundary points. We also discuss whether Fridman invariants can be extended continuously to some boundary points of domains constructed by deleting compact subsets from other domains.