Geometric gradient estimates for solutions to degenerate elliptic equations

A proof of the Cp′-regularity conjecture in the plane

“…This will allow us to frame the C p ′ conjecture into the formalism of the so called geometric tangential analysis, e.g. [7], [2,1] and [17,18,19,20,21,22].…”

Optimal C1,α regularity for degenerate fully nonlinear elliptic equations with Neumann boundary condition

“…Notice that solutions u of (1.1) cannot be more regular than C 1,α . More precisely for 0 < α < 1, the function u(x) = |x| 1+α (as mentioned in [1,17]) satisfies |∇u| γ ∆u = C|x| (1+α)(γ+1)−(γ+2)…”

“…Interior regularity properties of viscosity solutions of (1.1) have been studied since a long time, starting with the seminal paper of C. Imbert and L. Silvestre [17], which contains interior C 1,α estimates for (1.1) when f ∈ L ∞ (B 1 ). Very recently, optimal regularity was proved in [1], with the optimal Hölder's coefficient α = 1 1+γ .…”

“…In this paper we will develop the optimal regularity theory for (1.2). Precisely, we will apply the technique presented in [1] to prove that viscosity solution to (1.2) are C 1,α , where α = min{α − 0 .β , 1 1+γ } (see (4.2)). The main difference between this article and [2] is bases on the following fundamental steps to achieve this property are: ABP type estimate with boundary Neumann condition "when the gradient is large" (see Section 3.1) and C 1,α 0 estimate for F-harmonic functions combined with a certain geometric oscillation estimate which has its roots in the paper [1].…”

Sharp regularity estimates for quasilinear evolution equations

“…Although weak solutions of (1.1) under the compatibility assumptions (C) are known to be locally of the class C 1+α (in the parabolic sense) for some α ∈ (0, 1), the sharp exponent is known only for some specific cases (see [5,6,26,28,33]). This type of quantitative information plays an essential role in the study of blow-up analysis, related geometric and free boundary problems and for proving Liouville type results (see [3,4,12,16,18,40] for some enlightening examples).…”