On C¹ regularity for degenerate elliptic equations in the plane

2024/06/30 by Lacombe, Thibault, Lamy, Xavier
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.00775

Abstract

We show that Lipschitz solutions u of div G(∇ u)=0 in B₁⊂\mathbb R² are C¹, for strictly monotone vector fields G∈ C⁰(\mathbb R²;\mathbb R²) satisfying a mild ellipticity condition. If G=∇ F for a strictly convex function F, and 0≤ λ(ξ)≤ Λ(ξ) are the two eigenvalues of ∇² F(ξ), our assumption is that the set \lbraceλ=0\rbrace ∩ \lbrace Λ=∞\rbrace, where ellipticity degenerates both from below and from above, is finite. This extends results by De Silva and Savin (Duke Math. J. 151, No. 3, p.487-532, 2010), which assumed either that set empty, or the larger set \lbrace λ=0\rbrace finite. Our main new input is to transfer estimates in \lbrace λ> 0 \rbrace to estimates in \lbrace Λ<∞\rbrace by means of a conjugate equation. When G is not a gradient, the ellipticity assumption needs to be interpreted in a specific way, and we highlight the nontrivial effect of the antisymmetric part of ∇ G.

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