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

Abstract

This paper is devoted to the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called regular points in a thin obstacle problem that arises as the local extension of the obstacle problem for the fractional heat operator ( ∂ t − Δ x ) s (\partial _t - \Delta _x)^s for s ∈ ( 0 , 1 ) s \in (0,1) . The regularity estimates are completely local in nature. This aspect is of crucial importance in our forthcoming work on the blowup analysis of the free boundary, including the study of the singular set. The approach is based on first establishing the boundedness of the time-derivative of the solution. This allows reduction to an elliptic problem at every fixed time level. By using several results from the elliptic theory, including the epiperimetric inequality, the optimal regularity is established for solutions as well as the H 1 + γ , 1 + γ 2 H^{1+\gamma ,\frac {1+\gamma }{2}} regularity of the free boundary near such regular points.