  We present new second-kind integral-equation formulations of the interior and
exterior Dirichlet problems for Laplace's equation. The operators in these
formulations are both continuous and coercive on general Lipschitz domains in
$\mathbb{R}^d$, $d\geq 2$, in the space $L^2(\Gamma)$, where $\Gamma$ denotes
the boundary of the domain. These properties of continuity and coercivity
immediately imply that (i) the Galerkin method converges when applied to these
formulations; and (ii) the Galerkin matrices are well-conditioned as the
discretisation is refined, without the need for operator preconditioning (and
we prove a corresponding result about the convergence of GMRES). The main
significance of these results is that it was recently proved (see
Chandler-Wilde and Spence, Numer. Math., 150(2):299-271, 2022) that there exist
2- and 3-d Lipschitz domains and 3-d starshaped Lipschitz polyhedra for which
the operators in the standard second-kind integral-equation formulations for
Laplace's equation (involving the double-layer potential and its adjoint)
$\textit{cannot}$ be written as the sum of a coercive operator and a compact
operator in the space $L^2(\Gamma)$. Therefore there exist 2- and 3-d Lipschitz
domains and 3-d starshaped Lipschitz polyhedra for which Galerkin methods in
$L^2(\Gamma)$ do $\textit{not}$ converge when applied to the standard
second-kind formulations, but $\textit{do}$ converge for the new formulations.
