C1 regularity of infinity-harmonic functions in dimension three and higher

Theorem 1.1: for each $d\ge3$ there are $\alpha_d\in(0,1/3]$ and $C_d$ such that every bounded infinity-harmonic $u$ on $B_1\subset\mathbb R^d$ satisfies $\|\nabla u\|_{L^\infty(B_{1/2})}+[\nabla u]_{C^{0,\alpha_d}(B_{1/2})}\le C_d\,\mathrm{osc}_{B_1}u$; hence infinity-harmonic functions are $C^1_{loc}$ on every open set in $d\ge3$. The proof is by compactness, using the Evans-Savin anisotropic blow-up and a new Liouville theorem for the limiting equation. The exponent $\alpha_d$ is not explicit, and no claim is made about the optimal exponent $1/3$, boundary regularity or the planar endpoint.

Theorem 1.1: for each $d\ge3$ there are $\alpha_d\in(0,1/3]$ and $C_d$ such that every bounded infinity-harmonic $u$ on $B_1\subset\mathbb R^d$ satisfies $\|\nabla u\|_{L^\infty(B_{1/2})}+[\nabla u]_{C^{0,\alpha_d}(B_{1/2})}\le C_d\,\mathrm{osc}_{B_1}u$; hence infinity-harmonic functions are $C^1_{loc}$ on every open set in $d\ge3$. The proof is by compactness, using the Evans-Savin anisotropic blow-up and a new Liouville theorem for the limiting equation. The exponent $\alpha_d$ is not explicit, and no claim is made about the optimal exponent $1/3$, boundary regularity or the planar endpoint.