English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper we study the existence, the optimal regularity of solutions,and the regularity of the free boundary near the so-called \emph{regularpoints} in a thin obstacle problem that arises as the local extension of theobstacle problem for the fractional heat operator $(\partial_t - \Delta_x)^s$for $s \in (0,1)$. Our regularity estimates are completely local in nature.This aspect is of crucial importance in our forthcoming work on the blowupanalysis of the free boundary, including the study of the singular set. Ourapproach is based on first establishing the boundedness of the time-derivativeof the solution. This allows reduction to an elliptic problem at every fixedtime level. Using several results from the elliptic theory, including theepiperimetric inequality, we establish the optimal regularity of solutions aswell as $H^{1+\gamma,\frac{1+\gamma}{2}}$ regularity of the free boundary nearsuch regular points.

The regular free boundary in the thin obstacle problem for degenerate parabolic equations