Differentiability at lateral boundary for fully nonlinear parabolic equations (Q2358714)

The authors study the regularity of first derivatives of viscosity solutions at the lateral boundary for fully nonlinear uniformly parabolic equations, under Reifenberg Dini conditions. For a domain \(\Omega\) in \(x\) and \(t\), one considers fully nonlinear uniformly parabolic equations of the form \[ \begin{aligned} u_t - F(D^2 u, x, t) = g & \text{ in } \Omega\\ u=\phi(x,t) & \text{ on } \partial_p\Omega, \end{aligned} \] where \(\partial_p \Omega\) is the parabolic boundary of \(\Omega\) and it is assumed that \(F(0,x,t)=0\). The authors focus on optimal boundary conditions that lead to differentiability at \(\partial_p\Omega\), and on understanding the dependence on the modulus of continuity of \(g, \phi\) and \(\partial_p\Omega\). With these goals in mind, estimates on the modulus of continuity of viscosity solutions at the lateral boundary satisfying Reifenberg \(C^{1,\text{dini}}\) conditions are proved. Inspired by [\textit{F. Ma} and \textit{L. Wang}, J. Differ. Equations 252, No. 2, 988--1002 (2012; Zbl 1235.35101)] and [\textit{L. Wang}, Commun. Pure Appl. Math. 45, No. 2, 141--178 (1992; Zbl 0774.35042)], an oscillation argument, combined with the maximum principle, the Harnack inequality and a decay estimate, leads to the main results.