  We consider the free-boundary motion of two perfect incompressible fluids
with different densities $\rho_+$ and $\rho_-$, separated by a surface of
discontinuity along which the pressure experiences a jump proportional to the
mean curvature by a factor $\epsilon^2$. Assuming the Raileigh-Taylor sign
condition and $\rho_- \leq \epsilon^{3/2}$ we prove energy estimates uniform in
$\rho_-$ and $\epsilon$. As a consequence we obtain convergence of solutions of
the interface problem to solutions of the free-boundary Euler equations in
vacuum without surface tension as $\epsilon$ and $\rho_-$ tend to zero.

Analysis of PDEs