In this paper the authors prove the existence, optimal regularity of solutions, and regularity of the free boundary near regular points in a thin obstacle problem arising as the local extension of the obstacle problem for the fractional heat operator \((\partial_t-\Delta_x)^s\), for \(s\in (0,1)\).
More precisely, the operator in question is \[\mathcal{L}_aU=|y|^a\partial_tU-\text{div}_X(|y|^a\nabla_XU),\] with \(a\in(-1,1)\). Here \(X=(x,y)\in\mathbb{R}^n\times\mathbb{R}\) and \(t\in \mathbb{R}\). Given a domain \(\mathbb{D}\subset\mathbb{R}^{n+1}\) that is symmetric in \(y\), define \(\mathbb{D}^{\pm}=\mathbb{D}\cap\{\pm y>0\}\) and \(D=\mathbb{D}\cap\{y=0\}\). The thin obstacle is denoted with \(\psi\), defined on \(D\times[t_0,T]\), for \(t_0<T\), and the obstacle problem considered consists of finding a function \(U\) in \(\mathbb{D}^{\pm}\times(t_0,T]\) such that \[\begin{cases} \mathcal{L}_aU=0 \text{ in } \mathbb{D}^{+}\times(t_0,T],\\ \min\{U(x,0,t)-\psi(x,t), -\partial_y^aU(x,0,t) =0 \text{ on } D\times(t_0,T]. \end{cases} \] Here \(\partial_y^aU(x,0,t)=\lim\limits_{y\rightarrow 0+}y^a\partial_yU(x,y,t)\).
Lateral boundary conditions are imposed: \[ \begin{cases} U(X,t_0)=\varphi_0(X) \text{ on } \mathbb{D}^+,\\ U=g \text{ on } (\partial\mathbb{D})^+\times(t_0,T), \end{cases}\] under the compatibility assumptions \(\varphi_0=g(\cdot,t_0)\) on \((\partial\mathbb{D})^+\), \(\varphi_0\ge \psi(\cdot, t_0)\) on \(D\), and \(g\ge \psi\) on \(\partial D\times(t_0,T)\).
A key ingredient in the proofs is the boundedness of the time-derivative of the solution, which allows the authors to reduce the problem to an elliptic thin obstacle problem at every fixed time level. The authors then use results from elliptic theory, among them a truncated Almgren type frequency formula, a Weiss-type monotonicity formula and an epiperimetric inequality, to obtain the optimal regularity of solutions, along with the \(H^{1+\gamma,\frac{1+\gamma}{2}}\) regularity of the free boundary near regular points.