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This post is supposed to collect some Theorems and techniques which can be used to analyse variational problems by (a) finding a related variational problem s.t. their optimal values are the same or that their minimizers / maximizers are the same or (b) characterizing the minimizers / maximizers of the problem.

I recognize that the question is quite broad, so let me specify what I am looking for; I'm looking for results that satisfy at least one of the following criteria: Given some variational problem $$(*)\hspace{1cm}\inf_{x\in X} / \sup_{x\in X} F(x),$$ where $X$ is some function space (e.g. $\mathcal C^k(\mathbb R^d)$ or $L_p(\mathbb R^d)$) and $F$ is the function to be minimized (e.g. an energy functional or lagrangian), I would like to find

  1. Another variational problem whose minimal / maximal value is the same as the one of $(*)$.
  2. Another variational problem whose minimizer / maximizer is the same as the one of $(*)$ or at least has a simple relationship to the one of $(*)$
  3. A characterization of the minimizers / maximizers of $(*)$.

To give a few examples, here are a few Theorems and applications thereof which go in the direction of something i'm looking for:

Fenchel-Rockafeller Duality

Let $X$ be a normed space. We define the convex conjugate of a function $\Theta:X\rightarrow\bar{\mathbb R}$ as $$\Theta^{*}(x^{*}):=\sup_{x\in X} \bigg\lbrace\langle x^{*},x\rangle-\Theta(x)\bigg\rbrace, \hspace{1cm}x^*\in X^*$$ Then the Fenchel-Rockafeller Theorem gives a duality between minimization problems over $X$ and maximization problems over $X^*$:

Fenchel-Rockafeller:
Let X be a normed space and let $\Theta,\Sigma:X \rightarrow (−\infty,\infty]$ be convex and lower semicontinuous. If there exists $x_0 \in \text{Dom}(\Theta)\cap \text{Dom}(\Sigma)$ such that $\Theta$ is continuous in $x_0$, then $$\inf_{x\in X}\bigg\{\Theta(x)+\Sigma(x)\bigg\}=\sup_{x^*\in X^*}\bigg\{-\Theta^*(-x^*)-\Sigma^*(x^*)\bigg\}$$

This is an equivalence that guarantees equality of the optimal values, while it does not say anything about the relationship between the minimizers / maximizers of the two problems (to my knowledge). Therefore it fits in category 1.

As an example, this can be applied to show the Kantorovich-Rubinstein duality of optimal transport, i.e. the equivalence between a minimization problem over transport plans (measures) and a maximization problem over pairs of bounded Lipschitz functions. For details, see "Lectures on Optimal Transport" Chapter 3.2.

Euler-Lagrange Equation

If we consider the variational problem $$(\circ)\hspace{1cm}\inf_{f\in \mathcal C^2(\Omega)}\int \mathcal F(x,f(x),Df(x))dx$$ for $\Omega\subset\mathbb R^d$ open, the Euler-Lagrange Equation characterizes minimizers of $(\circ)$, assuming some smoothness of the boundary of $\Omega$ and differentiability of $\mathcal F(x,u,v)\in\mathcal C^2$. More precisely, if $f\in\mathcal C^2(\Omega)$ is a weak local mimimizer among $\mathcal C^1$ functions with the same boundary value, its Euler operator satisfies $$L_{\mathcal F}(f):=D_u\mathcal F(x,f(x),Df(x))-\text{div}\big(D_v\mathcal F(x,f(x),Df(x))\big)=0$$ Together with e.g. convexity of $\mathcal F$, this can help us characterize global minimizers by associating them with a differential equation. This would fall under category 3.
For details, see e.g. chapter 4 of these lecture notes

While I can't cite a good example for category 2 off the top of my head, I hope these two examples clarified what I am looking for. This question is very open-ended, so any answer is appreciated!

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If I'm not mistaken, Fenchel-Rockafeller can provide a result of type 2. I've heard these called "complimentary slackness conditions." I'll give an example in the Dirichlet problem.

Notice that for any $u \in H^1_0(\Omega),f \in L^2(\Omega)$, for a open Lipschitz domain $\Omega \subset \mathbb{R}^n$, we have the following for any $\sigma: \Omega \to \mathbb{R}^n$ that is $L^2(\Omega)$ and $\mathrm{div} \sigma \in L^2(\Omega)$,

$$ \int_{\Omega} \frac{1}{2} | \nabla u |^2 - f u dx \geq \int_{\Omega} \nabla u \cdot \sigma - \frac{1}{2} | \sigma |^2 - f udx = \int_{\Omega} (\mathrm{div} \sigma - f) u - \frac{1}{2} |\sigma|^2 dx $$

By integrating the pointwise Cauchy inequality $|\nabla u|^2/2 + |\sigma|^2/2 \geq \nabla u \cdot \sigma$.

We may then infimize the LHS over $u$, and take the supremum on the RHS over all $\sigma$ with $\mathrm{div} \sigma = f$ and the inequality holds for the supremum and infimum.

Now we may ask when equality holds. Notice the original pointwise inequality is an equality if $\nabla u = \sigma$. Then we have an example of a $u$ and $\sigma$ so that the inequality is actually an equality, and thus the aforementioned infimum and supremum are equal.

But this gives us an easy way to turn the minimizer on the LHS into a maximizer on the RHS! Just write down $\sigma = \nabla u$. We may go the other way with a similar manipulation as the one above.

An pseudo-example of 1. you might be interested is in the theory of relaxation. I refer you to the book of Ekeland and Temam for more info, I am not familiar enough to give as simple an example as above, but the idea is clear: an infimum is not achieved, but you extend the domain of your functional and and extend the functional over the new domain in a 'sensible' way, and you can have a minimizer. It's not exactly duality, but may be interesting to you nonetheless.

A pseudo-example of 3 can be found in A-free measures and Applications by Guido de Philippis. In it, he proves results along the following lines: A nonlinear quantity associated to a minimizing sequence of a variational problem satisfies a constant coefficient-PDE (say its divergence is zero). Then using compactness of the minimizing sequence may imply something about convergence of the desired quantity, which is often a measure. You can derive information about the stress-energy tensor for general variational problems, for instance.

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