Determination of the nature of stationary values in variational calculus

In variational calculus, when we solve the Euler-Lagrange equation $$\frac{d}{dx}L_p(u',u,x)-L_z(u',u,x)$$, where $$L=L(p,z,x)$$, to find stationary inputs of the functional $$I[u]=\int_0^1 L(u',u,x)dx,$$ we also need to determine whether this solution $$u$$ is a maximum, a minimum, or a saddle point. However, the second order variation $$\left.\frac{d^2}{dt^2}\right|_{t=0} I[u+tv]= \int_0^1 v'^2L_{pp}+2vv'L_{pz}+L_{zz}v^2 dx$$ is very hard to work with since we need to prove, for instance, this is positive for all functions $$v$$ to show that $$u$$ is a minimum. I have kept everything in one dimension for simplicity, but of course the above applies to higher dimensions as well.

I can think of two ways to confirm that it is a minimum:

1. If the function $$L$$ satisfy certain convexity assumption, then a minimizer exists, and the minimizer is a solution to the differential equation. Therefore, if the differential equation has a unique solution, it is the minimizer.
2. In some situations, we can write an inequality satisfied by $$I$$, and then consider the condition for the equality to be achieved in the inequality. An example is the equation for geodesic on the sphere $$S^2$$.

However, both methods applies only to a limited range of functions.

What are some other methods to establish that the stationary solution is a minimum/maximum? I am looking for some other ideas.

Perhaps I need some reference on this, since parts of my variational calculus and PDE books that I have read does not focus on this question.

• I think you've summarized the two most common strategies. In general, it can be quite difficult to establish whether a stationary value is an extremal value, whether using one of the two strategies mentioned or some other approach. Usually, ad hoc techniques must be developed for the specific problem being studied. As a start, I suggest reading about past work done for problems related to the one you're particularly interested in studying.. – Deane Yang Oct 29 '20 at 18:41
• Convexity in 'native' variables may not always be so apparent. Change of variables, and expansion of dimension of the underlying space can lead to convexity. See for example the famous dynamic optimal transport : BENAMOU, Jean-David, and Yann BRENIER. "A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem." Numerische Mathematik 84.3 (2000): 375-393. – Piyush Grover Oct 29 '20 at 19:30
• The Weierstrass conditions were designed to check the extremality of a solution of the EL equations. – Igor Khavkine Oct 29 '20 at 21:54
• @IgorKhavkine Could you recommend a book that discusses this? Thank you! – Ma Joad Oct 30 '20 at 17:42
• In my opinion, the first volume of the monograph "Calculus of Variations" by Mariano Giaquinta and Stefan Hildebrandt would be a very nice introduction to this range of problems. – Daniele Tampieri Oct 30 '20 at 20:55