(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let C be the class of C^{1}functions on interval [0,1] satisfying u(0)=0=u(1).

Consider the functional F(u)=

1

∫[(u')^{2}+ 3u^{4}+ cosh(u) + (x^{3}-x)u] dx

0

(note: u is afunctionof x.)

Analyse the functional F term by term. Decide for each term whether it is convex or strictly convex on C.

2. Relevant equations

(Strict) Convexity of functionals.

3. The attempt at a solution

Definition: A functional F is "convex" if for all u,v in C, 0<a<1, we have

F((1-a)u+av) ≤ (1-a)F(u) + aF(v).

F is "strictly convex" if for all u,v in C such that u≠v, and for all 0<a<1, we have

F((1-a)u+av) < (1-a)F(u) + aF(v).

1) I think by linearity of integrals, we can show that the last term is convex, but NOT strictly convex. Am I correct?

2) Each of the first two terms is strictly convex. Am I correct?

[I believe I have a proof for 1) and 2) using strict convexities of f(x)= x]^{2}and g(x)= x^{4}, but there is no answer at the back of the textbook for this problem, and so I'm not sure if I'm correct so far. It would be much appreciated if someone can confirm my answer, or point out if I'm wrong.

3) I'm really stuck for the third term, G(u)=

1

∫cosh(u)dx

0

How can I prove that this is convex/strictly convex? I really don't have much idea on this part of the problem...

Hopefully someone can explain how to prove this. Thanks a million!

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# Homework Help: Convexity & Strict Convexity of Functionals (function of a function)

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