This research investigates one-dimensional two-phase free boundary problems
motivated by the flow of two liquids in models of jets and cavities. Specifically, the
project examines three minimization problems, each considering a functional
involving a two-phase function and its derivative, subject to boundary conditions.
The methodology uses techniques from the calculus of variations, functional analysis,
and Sobolev spaces to parametrize solutions of the Euler–Lagrange equations within
each phase and reduce to standard minimization problems. In the first and simplest
problem, we fully characterize the minimizers of the functional, giving explicit
equations for the solutions. In the second, more complicated problem, we
prove that minimizers exist and provide a method to solve for the explicit
equations of the minimizers. In the third problem, when no minimizers exist, we
provide conditions that guarantee the existence of nonconstant solutions
satisfying the
Euler–Lagrange equation in
.
