Interior \(C^{1,\alpha}\) regularity for infinity-harmonic functions

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Proves uniform interior \(C^{1,\alpha_d}\) regularity for bounded infinity-harmonic functions in every dimension d ≥ 3, for a positive exponent depending only on dimension. The gradient’s supremum norm and Hölder seminorm on the half unit ball are bounded by a dimension-dependent constant times the oscillation on the unit ball. The exponent is not explicit.

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What the papers state

Theorem 1.1 (Uniform interior estimate)

In Uniform Interior \(C^{1,\alpha}\) Estimates for Infinity-Harmonic Functions, 4 October 2026

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For every integer d\ge3 there are constants \alpha_d\in(0,1/3] and C_d<\infty such that every bounded infinity-harmonic function u\in C(B_1), where B_1\subset\R^d, belongs to C^{1,\alpha_d}(B_{1/2}) and satisfies

\|\nabla u\|_{L^\infty(B_{1/2})} +[\nabla u]_{C^{0,\alpha_d}(B_{1/2})} \le C_d\osc_{B_1}u.

Corollary 1.2 (Interior C^1 regularity)

In Uniform Interior \(C^{1,\alpha}\) Estimates for Infinity-Harmonic Functions, 4 October 2026

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Let d\ge3, let \Omega\subseteq\R^d be a nonempty connected open set, and let u\in C(\Omega) be infinity-harmonic. Then u\in C^1_{\mathrm{loc}}(\Omega).

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Uniform Interior \(C^{1,\alpha}\) Estimates for Infinity-Harmonic Functions

4 October 2026, current edition.

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We prove a uniform interior \(C^{1,\alpha_d}\) estimate for bounded infinity-harmonic functions in every dimension d ≥ 3. For some \(\alpha_d\in(0,1/3]\) depending only on the dimension, the gradient’s supremum norm and α_d-Hölder seminorm on B_1/2 are bounded by a dimension-dependent constant times the oscillation on B₁. The exponent is not explicit, and no endpoint or boundary regularity claim is made. In particular, infinity-harmonic functions are locally C¹ on every open subset of ℝ^d, for d ≥ 3.

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15 references in Uniform Interior C¹,ᵅ Estimates for Infinity-Harmonic Functions

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