Extrapolation of solvability of the regularity and the Poisson regularity problems in rough domains

2023/06/09 by Gallegos, Josep M., Mourgoglou, Mihalis, Tolsa, Xavier
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2306.06185

Abstract

Let Ω⊂ \mathbb Rⁿ⁺¹, n≥2, be an open set satisfying the corkscrew condition with n-Ahlfors regular boundary ∂Ω, but without any connectivity assumption. We study the connection between solvability of the regularity problem for divergence form elliptic operators with boundary data in the Hajłasz-Sobolev space M^1,1(∂Ω) and the weak-\mathcal A_∞ property of the associated elliptic measure. In particular, we show that solvability of the regularity problem in M^1,1(∂Ω) is equivalent to the solvability of the regularity problem in M^1,p(∂Ω) for some p>1. We also prove analogous extrapolation results for the Poisson regularity problem defined on tent spaces. Moreover, under the hypothesis that ∂Ω supports a weak (1,1)-Poincaré inequality, we show that the solvability of the regularity problem in the Hajłasz-Sobolev space M^1,1(∂Ω) is equivalent to a stronger solvability in a Hardy-Sobolev space of tangential derivatives.

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