Higher order parabolic boundary Harnack inequality in <i>C</i>¹ and <i>C</i>^{<i>k</i>, <i>α</i>} domains

Abstract

&lt;p style='text-indent:20px;'&gt;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 &lt;i&gt;C&lt;/i&gt;&lt;sup&gt;1&lt;/sup&gt; and &lt;i&gt;C&lt;/i&gt;&lt;sup&gt;&lt;i&gt;k&lt;/i&gt;, &lt;i&gt;α&lt;/i&gt;&lt;/sup&gt; domains, providing that the quotient of two solutions vanishing on the boundary of the domain is as smooth as the boundary.&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;As a consequence of our result, we provide a new proof of higher order regularity of the free boundary in the parabolic obstacle problem.&lt;/p&gt;

Regularity of Free Boundaries in Obstacle Problems

“…The estimate (3.14) follows from the previous inequality combined with a covering argument. See, e.g., Corollary 3.5 and Lemma B.2 in [28]. In the case κ ≥ 0, we can argue analogously to derive (3.15) see, e.g., [28,Corollary 3.5].…”

“…See, e.g., Corollary 3.5 and Lemma B.2 in [28]. In the case κ ≥ 0, we can argue analogously to derive (3.15) see, e.g., [28,Corollary 3.5].…”