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