2023/02/26 by Simon Bortz, Bruno Poggi, Bortz, Simon +5
Computer Science · Mathematics · #31B15 #31E05 #35J15 #35J25 (secondary) #42B37 (primary) #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2302.13294
openalex publication_date 2023/02/26 · openalex created_date 2023/03/03 · openalex updated_date 2026/07/28
Suppose that Ω⊂\mathbb Rn+1, n≥1, is a uniform domain with n-Ahlfors regular boundary and L is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in Ω. We show that the corresponding elliptic measure ωL is quantitatively absolutely continuous with respect to surface measure of ∂Ω in the sense that ωL ∈ A_∞(σ) if and only if any bounded solution u to Lu = 0 in Ω is ε-approximable for any ε ∈ (0,1). By ε-approximability of u we mean that there exists a function Φ= Φε such that ‖u-Φ‖L^∞(Ω) ≤ ε‖u‖L^∞(Ω) and the measure \widetildeμΦ with d\widetildeμ = |∇ Φ(Y)| dY is a Carleson measure with L^∞ control over the Carleson norm. As a consequence of this approximability result, we show that boundary BMO functions with compact support can have Varopoulos-type extensions even in some sets with unrectifiable boundaries, that is, smooth extensions that converge non-tangentially back to the original data and that satisfy L1-type Carleson measure estimates with BMO control over the Carleson norm. Our result complements the recent work of Hofmann and the third named author who showed the existence of these types of extensions in the presence of a quantitative rectifiability hypothesis.