2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/228873This paper contains two results on the $L^p$ regularity problem on Lipschitz domains. For second order elliptic systems and $1<p<\infty$, we prove that the solvability of the $L^p$ regularity problem is equivalent to that of the $L^{p^\prime}$ Dirichlet problem. For higher order elliptic equations and systems, we show that if $p>2$, the solvability of the $L^p$ regularity problem is equivalent to a weak reverse Hölder condition with exponent $p$.To appear in Trans. of AMS. Updated to fix formatting issuesAnalysis of PDEs35J55; 35J40text