Qualitative-to-quantitative regularity for degenerate equations

Abstract

We establish a principle that converts qualitative regularity into uniform interior estimates for degenerate equations. For the $p$-Laplace equation, $p>2$, we prove that local $C^{1,\gamma}$ regularity of every solution, without any uniform bound on the corresponding seminorms, implies uniform $C^{1,\beta}$ estimates for every $0<\beta<\gamma$. Thus, the optimal qualitative and quantitative gradient H\"older exponents coincide as suprema in every dimension. In the plane, this yields estimates controlled by the $L^p$ norm of the solution for every exponent below the sharp Iwaniec--Manfredi threshold. Applied to Xu's recent qualitative $C^{1,1/3}$ regularity for planar infinity-harmonic functions, we obtain uniform interior $C^{1,\beta}$ estimates for every $0<\beta<1/3$. For the nonlocal superconductivity equation $|Du|(-\Delta)^s u=0$, with $1/2<s<1$, we prove that pointwise differentiability of all solutions with integrable tails implies uniform interior $C^{1,\alpha}$ estimates for bounded solutions, for some $\alpha>0$. The proofs combine compactness for shifted equations with an improvement-of-flatness iteration at variable scales.