In this dissertation, we study free boundary problems that describe equilibrium configurations of electromechanical systems consisting of a conducting elastic sheet deflected by an external charge distribution. Such systems are non-local in nature---the electrostatic pressure experienced by any individual point on the sheet depends on the entire deflection profile (as a result of the requirement that the deflected sheet must remain an equipotential). The magnitude of the electrostatic pressure varies quadratically with the magnitude of the local electric field. Similar non-local free boundary problems arise in two-layer fluid systems forced by withdrawal flows, but the normal viscous stress experienced by the fluid-fluid interface instead varies linearly with the local velocity gradients.;The analysis presented focuses on two configurations in particular: the electromechanical system described above, forced by a point charge, and an artificially modified version of the same electromechanical system in which the induced electrostatic pressure varies linearly with the local electric field and the forcing is provided by an electric dipole. This second model is constructed as a crude approximation of the two-layer fluid flow forced by a point sink, and is primarily used to explore the influence of the forcing exponent on the bifurcation structure and solution types of the associated system.;Our main contribution is the development of new techniques for the analysis and efficient numerical computation of large-deflection profiles for the true electromechanical system. The induced charge on such profiles accumulates near the interface tip, so that the geometry there is primarily determined by a balance between elastic and electrostatic forces. Away from the tip, the electrostatic pressure is low and the interface relaxes under the influences of gravity and elasticity only. Such interfaces exhibit features on widely disparate length scales. We exploit this separation of the interface into two regions dominated by different force balances to create a separate representation of each region (in appropriately rescaled coordinates), and then match the two representations together while ensuring that the relationship between local induced stress and global interface geometry is respected. This is achieved by combining tools and results from complex analysis and the method of matched asymptotic expansions.
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