  We present a novel method for solving a class of time-inconsistent optimal
stopping problems by reducing them to a family of standard stochastic optimal
control problems. In particular, we convert an optimal stopping problem with a
non-linear function of the expected stopping time in the objective into
optimization over an auxiliary value function for a standard stochastic control
problem with an additional state variable. This approach differs from the
previous literature which primarily employs Lagrange multiplier methods or
relies on exact solutions. In contrast, we characterize the auxiliary value
function as the unique viscosity solution of a non-linear elliptic PDE which
satisfies certain growth constraints and investigate basic regularity
properties. We demonstrate the construction of an optimal stopping time under
additional regularity assumptions on the auxiliary value function. Finally, we
discuss extensions to more general dynamics and time-inconsistencies, as well
as potential connections to degenerate Monge-Ampere equations.
