The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight vertical bar y vertical bar(a) for a is an element of (-1, 1). Such problem arises as the local extension of the obstacle problem for the fractional heat operator (partial derivative(t) - Delta(x))(s) for s is an element of (0, 1). Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation (a = 0).