# Special version of Tonelli’s theorem

I am trying to prove this theorem. I have not found anything similar to it on the internet.

Special version of Tonelli’s theorem Assume that the functions $$f(x,u): [a,b] \times \mathbb{R} \to \mathbb{R}$$, $$g(x, \xi): [a,b] \times \mathbb{R} \to \mathbb{R}$$ are continuous, $$f$$ is bounded below, $$g$$ is convex in $$\xi$$ and satisfies

$$\exists r>1,\, \exists C>0\,\, \text{such that}\,\, g(x,\xi) \ge C| \xi|^r,\,\, \forall (x, \xi) \in [a,b] \times \mathbb{R}.$$

Then there exists a minimizer of the functional $$J[u] = \displaystyle\int\limits_a^b \big(f(x,u(x)) + g(x,u'(x))\big) dx$$ in the space $$X= \{ u \in AC([a,b]); u(a)=\alpha, u(b)= \beta \}.$$

Proof. Since $$f$$ is bounded then there is a real number $$m \in \mathbb{R}$$ such that $$m (b-a)\le f(x,u(x)), \quad \forall (x,u(x)) \in [a,b] \times \mathbb{R}$$. From the properties of $$g$$ we get

$$m+ C \int\limits_a^b |u'(x)|^r dx \leq J[u] \Rightarrow m+ C \| u'\|_{L^r[a,b]}^r \leq J[u]\,\,\, \forall u \in X.$$

We can see that $$J[u]$$ is bounded below and from the definition of the infimum there is a minimizing sequence $$\{u_n\}_{n\in \mathbb{N}} \subset X$$ such that

$$\underset{n \to \infty}{\lim} J[u_n] = \inf \{ J[u] | u \in X \}> -\infty \,\, \text{ in } \mathbb{R}.$$

and hence, $$\{ u_n'\}_{n \in \mathbb{N}}$$ is uniformly bounded, i.e. there is $$N>0$$ such that $$\forall n >N$$ we have

$$\| u'_n\|_{L^r[a,b]} \leq \left(\frac{J[u_N] -m}{c} \right)^\frac{1}{r}.$$

Now, since $$\{u_n\}$$ is equicontinuous, and uniformly bounded in $$L^r[a,b]$$, then according to the Arzelà-Ascoli theorem there is a subsequence $$\{ u_{n_k} \}_{k \in \mathbb{N}}$$ and $$\overline{u} \in AC[a,b]$$ such that $$u_{n_k} \to \overline{u}$$ uniformly, and $$u'_{n_k} \to \overline{u}'$$ in the sense of $$L^r[a,b]$$. $$\blacksquare$$

I am not sure if my last argument is right. I want to make it more rigorous. Although I found the general idea of the proof on page 140 in the book of Hansjörg Kielhöfer named (Calculus of Variations An Introduction to the One-Dimensional Theory with Examples and Exercises) I have no idea about completing the proof of the theorem. Could you please help.

$$\renewcommand\bar\overline$$Indeed, it is not obvious why "$$u'_{n_k} \to \overline{u}'$$ in the sense of $$L^r[a,b]$$".

Look at this example: $$[a,b]=[0,2\pi]$$, $$u_n(x)=\dfrac{\sin nx}n$$, $$\bar u=0$$. Then $$u_n\to\bar u$$ uniformly, but $$u_{n_k}'\not\to\bar u'$$ in $$L^r$$ for any increasing sequence $$(n_k)$$ of natural numbers, because $$u_n'(x)=\cos nx$$ and hence $$\|u_n'\|_r^r=c_r:=\int_0^{2\pi}|\cos u|^r\,du>0$$ for all $$n$$.

Note also that your argument does not use the condition that $$g$$ is convex.

(Also, your post seems to have hardly anything to do with the Tonelli theorem.)

Here is how to fix this. Using, as you did, the Arzelà–Ascoli theorem and then passing to a subsequence, without loss of generality (wlog) we may assume that $$u_n\to \bar u$$ uniformly. Also, you showed that the sequence $$(u_n')$$ is bounded in (the reflexive Banach space) $$L^r$$.

So, by the Eberlein–Shmulyan theorem (Kôsaku Yosida, Functional Analysis, Springer 1980, Chapter V, Appendix, section 4; alternatively, see e.g. this version), passing again to a subsequence, wlog we may assume that $$u_n'\to v$$ for some $$v\in L^r$$ in the weak topology of $$L^r$$.

Further, by Mazur's lemma, for each natural $$n$$ there exist a natural $$N_n\ge n$$ and nonnegative real numbers $$a_{n,k}$$ for $$k\in\{n,\dots,N_n\}$$ such that $$\sum_{k=n}^{N_n}a_{n,k}=1$$ and $$\begin{equation*} v_n:=\sum_{k=n}^{N_n}a_{n,k} u_k'\to v \tag{0} \end{equation*}$$ in $$L^r$$. For $$x\in[a,b]$$, let now $$\begin{equation*} w_n(x):=u_n(a)+\int_0^x v_n(t)\,dt =u_n(a)-\sum_{k=n}^{N_n}a_{n,k}u_k(a)+\sum_{k=n}^{N_n}a_{n,k}u_k(x). \tag{1} \end{equation*}$$

Since $$u_n\to \bar u$$ uniformly and the $$u_n$$'s are uniformly bounded, we see that $$w_n\to \bar u$$ uniformly and the $$w_n$$'s are uniformly bounded. Therefore and because $$f$$ is continuous, we have $$\begin{equation*} J_1[w_n] := \int_a^b f(x,w_n(x))\, dx\to J_1[\bar u]=\lim_n J_1[u_n]. \end{equation*}$$ Also, by the convexity of $$g(x,\xi)$$ in $$\xi$$, $$\begin{equation*} J_2[w_n] := \int_a^b g(x,w_n'(x))\, dx \le\sum_{k=n}^{N_n}a_{n,k}J_2[u_k]. \end{equation*}$$ Also, $$J[w_n]=J_1[w_n]+J_2[w_n]$$. So, \begin{equation*} \begin{aligned} \limsup_n J[w_n]&\le \lim_n J_1[w_n]+\limsup_n J_2[w_n] \\ &\le \lim_n J_1[u_n]+\sum_{k=n}^{N_n}a_{n,k}\limsup_n J_2[u_n] \\ &= \lim_n J_1[u_n]+\limsup_n J_2[u_n] \\ &= \limsup_n (J_1[u_n]+J_2[u_n]) \\ &= \limsup_n J[u_n]= \lim_n J[u_n]=\inf_{u\in X} J[u]. \end{aligned} \end{equation*} So, passing to a subsequence, wlog we may assume that $$\begin{equation*} J[w_n]\to\inf_{u\in X} J[u]. \end{equation*}$$

Recall that $$w_n\to \bar u$$ uniformly. So, in view of (1) and (0), $$\begin{equation*} \bar u(x)=\bar u(a)+\int_0^x v(t)\,dt \end{equation*}$$ for $$x\in[a,b]$$, so that $$\bar u\in AC$$ and $$\bar u'=v$$ almost everywhere (a.e.). It also follows that $$w_n'=v_n\to v=\bar u'$$ in $$L^r$$ and hence in measure. So, by the continuity of $$f$$ and $$g$$ and the Fatou lemma, $$$$J[\bar u]=J[\lim_n w_n]\le\liminf_n J[w_n]=\lim_n J[w_n]=\inf_{u\in X} J[u].$$$$ It is also easy to see that $$\bar u\in X$$. Thus, $$\bar u$$ is a minimizer of $$J[u]$$ over $$u\in X$$.

• I think the OP refers to Tonelli's theorem in the Calculus of Variations, not to the measure theoretic theorem en.wikipedia.org/wiki/Tonelli%27s_theorem_(functional_analysis) Commented Mar 17, 2023 at 11:15
• @PietroMajer : Thank you for this information. Commented Mar 17, 2023 at 12:59