  This paper provides new theoretical connections between multi-time
Hamilton-Jacobi partial differential equations and variational image
decomposition models in imaging sciences. We show that the minimal values of
these optimization problems are governed by multi-time Hamilton-Jacobi partial
differential equations. The minimizers of these optimization problems can be
represented using the momentum in the corresponding Hamilton-Jacobi partial
differential equation. Moreover, variational behaviors of both the minimizers
and the momentum are investigated as the regularization parameters approach
zero. In addition, we provide a new perspective from convex analysis to prove
the uniqueness of convex solutions to Hamilton-Jacobi equations. Finally we
consider image decomposition models that do not have unique minimizers and we
propose a regularization approach to perform the analysis using multi-time
Hamilton-Jacobi partial differential equations.
