Asymptotically Hölder gradient regularity for degenerate or singular parabolic normalized $p$-Laplace equations

Abstract

We study bounded viscosity solutions of the degenerate or singular parabolic equation \[ u_t-|Du|^γΔ_p^{\mathrm{N}}u=f \qquad \text{in } Q_1, \] where \[ Δ_p^{\mathrm{N}}u := Δu+(p-2) \left\langle D^2u\frac{Du}{|Du|}, \frac{Du}{|Du|} \right\rangle, \qquad -1<γ<\infty,\quad 1<p<\infty, \] and $f\in C^0(Q_1)\cap L^{\infty }(Q_1)$. Given any prescribed $\alpha \in (0,1)$, we prove that there exists $\epsilon =\epsilon (n,\alpha )>0$ such that \[ |p-2|+|γ|\leq\varepsilon \quad\Longrightarrow\quad u\in C_{\mathrm{loc}}^{1+α,\frac{1+α}{2}}(Q_1). \] In particular, the spatial gradient Hölder exponent can be chosen arbitrarily close to $1$ as $(p,\gamma )\to (2,0)$. This extends the result of Andrade and Santos (Calc. Var. Partial Differential Equations \textbf{61}, Paper No. 196, 2022) for $\gamma =0$ to the joint regime $(p,\gamma )\to (2,0)$ and answers affirmatively the question raised in Remark~1.1 therein.