PDEs and functionals
I’m suddenly interested in the connections between PDEs and functionals. Specifically, if I have a PDE and associated BCs/ICs on a given domain, how can I find a functional (and on what space) such that the minimizer of this functional is a solution to the PDE, and in what sense is it a solution? Conversely, what PDE (if any), does the minimizer of a given functional satisfy?
I’m not sure under what conditions these questions even make sense: e.g. why would a given functional have a unique minimizer? But, at least the latter is an important question to me, because it would allow me to convert optimization problems– in particular, I’m thinking of variational image processing– into PDEs, which, thanks to this term, I have confidence in my ability to attack numerically.
Here’s an example of the flavor of what I’d like to do. In the appendix of Variational Image Segmentation using Boundary Functionals, the authors show (sort of– the argument is neat, but not very rigorous) that the minimizer of the functional
satisfies the PDE
on 
, 
on 
.In the image processing context, the minimizer of 
/ the solution of the PDE represents a balance between fidelity to the original image 
and being slowly changing.
Possibly relevant posts:
- Eikonal equation (11/14/2006)
- Calculus of Variations (9/23/2004)
- Exactness of differential forms (7/28/2007)