$L^p$-solvability of boundary value problems for the Laplacian in locally flat unbounded domains

We establish the solvability of the $L^p$-Dirichlet and $L^{p^{\mathrm{\prime }}}$-Neumann problems for the Laplacian for $p\in (\frac{n}{n-1}-\epsilon ,\frac{2n}{n-1}]$ for some $\epsilon >0$ in $2$-sided chord-arc domains with unbounded boundary that is sufficiently flat at large scales and outward unit normal vector whose oscillation fails to be small only at finitely many dyadic boundary balls.