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We prove that the $L^{p'}$-solvability of the homogeneous Dirichlet problemfor an elliptic operator $L=-\operatorname{div}A\nabla$ with real and merelybounded coefficients is equivalent to the $L^{p'}$-solvability of the PoissonDirichlet problem $Lw=H-\operatorname{div} F$, which is defined in terms of an$L^{p'}$ estimate on the non-tangential maximal function, assuming that$\operatorname{dist}(\cdot, \partial \Omega) H$ and $F$ lie in certain$L^{p'}$-Carleson-type spaces, and that the domain $\Omega\subset\mathbbR^{n+1}$, $n\geq2$, satisfies the corkscrew condition and has $n$-Ahlforsregular boundary. In turn, we use this result to show that, in a bounded domainwith 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 $L^p$-regularityproblem 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 forDahlberg-Kenig-Pipher operators is new even if $\Omega$ is the unit ball,despite the fact that the $L^{p'}$-solvability of the Dirichlet problem forthese operators in Lipschitz domains has been known since 2001.  Further novel applications include i) new local estimates for the Green'sfunction and its gradient in rough domains, ii) a local $T1$-type theorem forthe $L^{p}$-solvability of the ``Poisson-Regularity problem'', itselfequivalent to the $L^{p'}$-solvability of the homogeneous Dirichlet problem, interms of certain gradient estimates for local landscape functions, and iii) new$L^p$ estimates for the eigenfunctions (and their gradients) of symmetricoperators $L$ on bounded rough domains.

Solvability of the Poisson-Dirichlet problem with interior data in $L^{p'}$-Carleson spaces and its applications to the $L^{p}$-regularity problem