We prove that the $L^{p'}$-solvability of the homogeneous Dirichlet problem for an elliptic operator $L=-\operatorname{div}A\nabla$ with real and merely bounded coefficients is equivalent to the $L^{p'}$-solvability of the Poisson Dirichlet problem $Lw=H-\operatorname{div} F$, which is defined in terms of an $L^{p'}$ estimate on the non-tangential maximal function, assuming that $\operatorname{dist}(\cdot, \partial Ω) H$ and $F$ lie in certain $L^{p'}$-Carleson-type spaces, and that the domain $Ω\subset\mathbb R^{n+1}$, $n\geq2$, satisfies the corkscrew condition and has $n$-Ahlfors regular boundary. In turn, we use this result to show that, in a bounded domain with uniformly $n$-rectifiable boundary that satisfies the corkscrew condition, $L^{p'}$-solvability of the homogeneous Dirichlet problem for an operator $L=-\operatorname{div} A\nabla$ satisfying the Dahlberg-Kenig-Pipher condition (of arbitrarily large constant) implies solvability of the $L^p$-regularity problem for the adjoint operator $L^*=-\operatorname{div} A^T \nabla$, where $1/p+1/p'=1$ and $A^T$ is the transpose matrix of $A$. This result for Dahlberg-Kenig-Pipher operators is new even if $Ω$ is the unit ball, despite the fact that the $L^{p'}$-solvability of the Dirichlet problem for these operators in Lipschitz domains has been known since 2001. Further novel applications include i) new local estimates for the Green's function and its gradient in rough domains, ii) a local $T1$-type theorem for the $L^{p}$-solvability of the ``Poisson-Regularity problem'', itself equivalent to the $L^{p'}$-solvability of the homogeneous Dirichlet problem, in terms of certain gradient estimates for local landscape functions, and iii) new $L^p$ estimates for the eigenfunctions (and their gradients) of symmetric operators $L$ on bounded rough domains.

69 pages. To appear in the Journal of the European Mathematical Society