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This dissertation focuses on developing analytical methods for elliptic partial differential equations with conditions imposed on internal boundaries. Internal boundaries are formed where materials with different properties meet to form interfaces. These interfaces arise in a variety of physical and engineering contexts such as in the evaporation of water droplets, dielectric double-spheres, and soft-material Janus drops. The solutions to problems with interfaces are often singular where the interfaces meet the boundaries or two interfaces meet. This causes difficulties when attempting to solve these problems solely with numerical approaches. In contrast analytical approaches (while limited to relatively simple geometries) lend significantly insight into the nature of the singularities with full resolutions in some cases. Potentially this knowledge can then be used to improve the quality of numerical solutions for more generic situations.

We will focus here on four important elliptic PDE problems: Laplace, Poisson, biharmonic and Stokes flow. First we introduces our main analytic result known as the Parity Split Method (PSM) developed in the context of the Laplace and the Poisson equation. The method is then applied to the problem of a thermally driven evaporative liquid bridge in a long V-shaped channel. The problem involves solving acoupled temperature-concentration system of the Laplace equation. Complex analysis based analytic solutions to the concentration equation are also developed along the way. Finally, we extend the PSM for biharmonic equation and addresses several numerical issues in regards to solving for the fluid flow around a soft-material Janus drop