A note on Hölder regularity of weak solutions to linear elliptic equations

2024/05/06 by Karthik Adimurthi, Adimurthi, Karthik
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2405.03802

openalex publication_date 2024/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we show that weak solutions of -div \mathbbA(x)∇ u = 0 where \mathbbA(x)= \mathbbA(x)^T and λ|ζ|² ≤ ⟨ \mathbbA(x)ζ,ζ⟩ ≤ Λ|ζ|², and \mathbbA(x) ≡ \mathbbA is a constant matrix are Hölder continuous u ∈ C^α_loc with α≥ \frac12 (-(n-2) + √(n-2)² + \frac4(n-1)λΛ ). This implies that the example constructed by Piccinini - Spagnolo is sharp in the class of constant matrices \mathbbA(x) ≡ \mathbbA. The proof of Hölder regularity does not go through a reduction of oscillation type argument and instead is achieved through a monotonicity formula. In the case of general matrices \mathbbA(x), we obtain the same regularity under some additional hypothesis.

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