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Let $\Omega\subset \mathbb R^{n+1}$, $n\geq2$, be an open set satisfying thecorkscrew condition with $n$-Ahlfors regular boundary $\partial\Omega$, butwithout any connectivity assumption. We study the connection betweensolvability of the regularity problem for divergence form elliptic operatorswith boundary data in the Haj{\l}asz-Sobolev space $M^{1,1}(\partial\Omega)$and the weak-$\mathcal A_\infty$ property of the associated elliptic measure.In particular, we show that solvability of the regularity problem in$M^{1,1}(\partial\Omega)$ is equivalent to the solvability of the regularityproblem in $M^{1,p}(\partial\Omega)$ for some $p>1$. We also prove analogousextrapolation results for the Poisson regularity problem defined on tentspaces. Moreover, under the hypothesis that $\partial\Omega$ supports a weak$(1,1)$-Poincar\'e inequality, we show that the solvability of the regularityproblem in the Haj{\l}asz-Sobolev space $M^{1,1}(\partial\Omega)$ is equivalentto a stronger solvability in a Hardy-Sobolev space of tangential derivatives.

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