Sufficient conditions for functional extrema

Question

In the calculus of variations, we can develop a sufficient condition for a functional $J: S \to \mathbb{R}$, $$J(y) = \int_a^b f(x,y,y') \, dx$$ to have a local maximum or minimum, where $S \subseteq C^2[a,b]$ and the boundary conditions are arbitrary. The easiest approach, common to all the textbooks I've seen, is to use the Euler--Lagrange equation, the strengthened Legendre condition, and the notion of conjugate points in relation to the Jacobi accessory equation. (Please comment if what I'm referencing is unclear.)

The catch is that the condition pertains to $weak$ local extrema: that is, local extrema with respect to the norm $$||y||_1 := \sup_{x \in [a,b]} |y(x)| + \sup_{x \in [a,b]} |y'(x)|,$$ in contrast to $strong$ extrema with respect to the norm $$||y||_0 := \sup_{x \in [a,b]} |y(x)|.$$ So I have a couple of queries. To what extent is it a deficiency in practical settings that we are constrained to deal with these weak extrema if we want to apply this standard sufficient condition? And what can we do to find sufficient conditions for strong extrema, or extrema with respect to other norms generally?

Answer 1

I don't consider it a deficiency to work with the norm(s) with respect to which the functional is continuous. Functionals involving $y'$ are nowhere continuous in the $C^0$ norm, so perhaps it's not a big deal that we can't easily control their extrema with that norm. Personally, I prefer work with Sobolev spaces, choosing whatever norm will make the functional at least lower semicontinuous. I haven't had a reason to worry about strong vs weak extrema.

For the second question: Chapter 6 of Calculus of Variations by Gelfand and Fomin is mostly devoted to proving a sufficient condition for strong extremum. It is stated as Theorem 1 on page 148. It's also stated in Encyclopedia of Math but you'll have a better chance of deciphering it after reading the book.