What does this research mean for the field?

The viscosity solution to the Dirichlet problem for the inhomogeneous normalized infinity Laplacian equation in a bounded convex domain exhibits 1/2-concavity and C^1-regularity when the right-hand side function has a constant sign. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

This research aims to establish the regularity of viscosity solutions to the Dirichlet problem related to the normalized infinity Laplacian equation.

June 1, 2026Open Access

A C^1 -regularity result to the inhomogeneous normalized infinity Laplacian equation

Key Points

This research aims to establish the regularity of viscosity solutions to the Dirichlet problem related to the normalized infinity Laplacian equation.
Demonstrated the 1/2-concavity of the viscosity solution using the convex envelope method of Alvarez-Lasry-Lions.
Established C1-regularity based on upper estimates of semiconcave functions at singular points.
Proved 1/2-concavity of the viscosity solution for the given Drichlet problem.
Showed C1-regularity for solutions with bounds defined on a convex domain.

Abstract

In this paper, we investigate the regularity for the viscosity solution to the Dirichlet problem -ΔN∞u = f (x) in Ω, u = 0 on ∂Ω, , where Ω is a bounded convex domain and ƒ (x) ∈ C (Ω). For 0 ;lt f₈₍₅ = inf_Ω ƒ ≤ ƒ ≤ sup_Ω ƒ = fₒₔ₏ ;lt +∞, we ƒirst prove the 1/2-concavity of the viscosity solution by the convex envelope method of Alvarez-Lasry-Lions, and then establish the C^{1 -regularity based on the upper estimate of semiconcave functions at the singular point. The similar result holds for-∞: lt f₈₍₅ ≤ ƒ ≤ fₛup < 0.

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