- Let $X$ be a vector field on a Riemmanian manfiold $M$.
I recently read that solving: \begin{equation} \operatorname{argmin}_{\phi}\int_M (\nabla \phi - X)^2 d\mu, \end{equation} where $\mu$ is the Riemmanian measure thereon and $\phi$ is a scalar valued function thereon is equivalent to solving the PDE (poisson equation): \begin{equation} L\phi = Div(X), \end{equation} where $L$ is the Laplacian and $Div$ is the divergence operator.
- My question is if I want to solve a penalized optimization problem: \begin{equation} \operatorname{argmin}_{\phi}\int_M (\nabla \phi - X)^2 d\mu + \lambda C(\phi,X), \end{equation} where $\lambda \in [0,\infty)$ and $C$ is a convex penalty function depending on the choice of $X$ and $\phi$, then is this equivalent to solving a certain PDE, which generalizes the poisson equation above?