One of the simplest and the most important results in elliptic theory is the maximum principle. It provides sharp estimates for the solutions to elliptic PDEs in $L^infty$ in terms of the corresponding norm of the boundary data. It holds on arbitrary domains for all (real) second order divergence form elliptic operators $- div A nabla$. The well-posedness of boundary problems in $L^p$, $p In the present talk we shall discuss the well-posedness in $L^p$ for elliptic PDEs associated to matrices $A$ of real (possibly non-symmetric) coefficients independent on the transversal direction to the boundary. In combination with our earlier perturbation theorems, this result shows that the Dirichlet and Regularity boundary value problems are well-posed in some $L^p$, $1

This is joint work with S. Hofmann, C. Kenig, and J. Pipher.