  By solving the Young Laplace equation of capillary hydrostatics one can
accurately determine equilibrium shapes of droplets on relatively smooth solid
surfaces. The solution, however of the Young Laplace equation becomes tricky
when a droplet is sitting on a geometrically patterned surface and multiple,
and unknown a priori, three phase contact lines have not been accounted for,
since air pockets are trapped beneath the liquid droplet. In this work, we
propose an augmented Young-Laplace equation, in which a unified formulation for
the liquid/vapor and liquid/solid interfaces is adopted, incorporating
microscale interactions. This way, we bypass the implementation of the Young's
contact angle boundary condition at each three phase contact line. We
demonstrate the method's efficiency by computing equilibrium wetting states of
entire droplets sitting on geometrically structured surfaces, and compare the
results with those of the mesoscopic Lattice Boltzmann simulations. The
application of well-established parameter continuation techniques enables the
tracing of stable and unstable equilibrium solutions if necessary for the
determination of energy barriers separating co-existing stable wettign states.
Since energy barriers determine whether a surface facilitates or inhibits
certain wetting transitions, their computation is important for many technical
applications. Our continuum-level analysis can readily be applied to patterned
surfaces with increased and unstructured geometric complexity, having a
significant computational advantage, as compared to the computationally
demanding mesoscopic simulations that are usually employed for the same task.
