Interior C^{1,α} regularity for infinity-harmonic functions
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
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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MANUSCRIPT 01
Uniform Interior C^{1,α} Estimates for Infinity-Harmonic Functions
We prove a uniform interior C^{1,\alpha_d} estimate for bounded infinity-harmonic functions in every dimension d ≥ 3. For some \alpha_d∈(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 B_(1). The…
Full source abstract
We prove a uniform interior C^{1,\alpha_d} estimate for bounded infinity-harmonic functions in every dimension d ≥ 3. For some \alpha_d∈(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 B_(1). The exponent is not explicit, and no endpoint or boundary regularity claim is made. In particular, infinity-harmonic functions are locally C^(1) on every open subset of ℝ^(d), for d ≥ 3.
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