# Extremals versus minima for variational problems

A geodesic on a Riemannian manifold is generally not the shortest among nearby curves with the same endoints. But it can always be divided into parts each the shortest among nearby curves between its endpoints.

How much does that generalize? Is there some natural description of Lagrangians with the following property: for each solution $f$ to the Euler-Lagrange equation on a given domain, that domain can be covered by subdomains such that each restriction of $f$ to one of those subdomains actually minimizes (or maximizes) the Lagrangian integral on that subdomain?

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I don't think this "local minimality" is really general. I'm thinking of the variational problem that defines Reeb orbits for a given contact form: it takes the form $\gamma \mapsto \int_\gamma \alpha$, where $\alpha$ is the contact $1$-form and so pieces of $\gamma$ don't seem to be maxima nor minima. For $1$-dimensional problems at least convexity/ellipticity of the integrand seems to play a role. Note: I'm writing this off the top of my head so please check. –  alvarezpaiva Apr 10 '12 at 18:54
On the other hand, I think if the energy functional satisfies an appropriate ellipticity or convexity condition, then any critical point is "locally minimal". I think the same argument used for geodesics can be used more generally. –  Deane Yang Apr 10 '12 at 19:26
I think one also needs to be careful about the regularity of the class of solutions. For instance if $\Sigma$ is a branched minimal surface in $\Real^3$ then $\Sigma$ locally minimizes area away from the branch set but cannot minimize area in any neighborhood of a branch point. –  Rbega Apr 10 '12 at 21:36
Yes, I agree about the need to assume sufficient regularity for the solution. –  Deane Yang Apr 10 '12 at 22:00

For example, it is known that the Lagrangian for $k$-dimensional area in a Riemannian manifold has this property for all $k$. (The case $k=1$ is the case of geodesics.) The technique that works in this case is the technique of calibrations, developed to a very high degree in the fundamental paper Calibrated Geometries by Harvey and Lawson.