Tonelli's Existence Theorem for an Integrand Depending on the Function

Contents

Statement
The Sobolev space in one dimension
The Arzelà–Ascoli theorem and the compact embedding
Weak lower semicontinuity with $y$ present
Proof of the theorem

This appendix proves Theorem 20.78 of Calculus of Variations: an integrand $f\left(x,y,q\right)$ that is convex in the slope $q$ and coercive in it makes the functional Equation (20.11) attain its minimum in the Sobolev class with prescribed endpoint values. The abstract half of the argument is already in the chapter — the direct method Theorem 20.75 and the convexity lemma Lemma 20.77 — and what that lemma leaves owed is the passage from an integrand $f\left(x,q\right)$ to one that also depends on $y$. That passage is what is carried out here, and it turns on one fact about the one-dimensional Sobolev space: a bounded set of $W^{1,r}(a,b)$ is precompact in the continuous functions with the uniform norm. That compact embedding is proved below in full (Theorem A41.8) from the fundamental theorem of calculus, Hölder's inequality and the Arzelà–Ascoli theorem, the last of which is not available anywhere else in this treatise and is therefore also proved here (Lemma A41.7).

Uniform convergence of the competitors is exactly the leverage the $y$-dependence needs: it makes the values $f\left(x,u_{n}(x),q\right)$ converge pointwise to $f\left(x,u(x),q\right)$ and confines them to a compact range of $y$, on which the growth hypothesis on $\pp f/\pp q$ can be applied uniformly. With that in hand the classical Scorza–Dragoni argument — which is the route taken when no such growth hypothesis is assumed — is not needed; Remark A41.11 states precisely what it would supply and why it is avoided here, and Remark A41.12 lists the analytic inputs that are quoted rather than proved, all of them from the Lebesgue theory and the duality of $L^{r}$ which this treatise declares as imports in Remark 16.1.

Statement

Throughout, $1<r<\infty$ and $r'=r/\left(r-1\right)$ is the conjugate exponent, so that $1/r+1/r'=1$; $a<b$ are real and $y_{a},y_{b}$ are the prescribed endpoint values of Equation (20.10).

Theorem A41.1 (Tonelli).

Let $f:[a,b]\times\R\times\R\longrightarrow\R$ satisfy

[(H1)] $f$ is continuous, and continuously differentiable with respect to its third argument, with derivative written $f_{q}=\pp f/\pp q$;
[(H2)] $q\longmapsto f\left(x,y,q\right)$ is convex for every $\left(x,y\right)$;
[(H3)] $f\left(x,y,q\right)\ge\alpha\abs{q}^{r}-\beta$ for constants $\alpha>0$ and $\beta\in\R$ and all $\left(x,y,q\right)$;
[(H4)] for every $R>0$ there is a constant $C_{R}$ with $\abs{f_{q}\left(x,y,q\right)} \le C_{R}\left(1+\abs{q}^{r-1}\right)$ whenever $x\in[a,b]$, $\abs{y}\le R$ and $q\in\R$.

Then the functional Equation (20.11) attains its minimum on

\begin{equation}\tag{A41.1} \mathcal{A}_{r} =\set{u\in W^{1,r}(a,b)\ \mid\ u(a)=y_{a},\ u(b)=y_{b}}\ec \end{equation}

that is, there is $u_{\ast}\in\mathcal{A}_{r}$ with $J\left[u_{\ast}\right]=\inf_{\mathcal{A}_{r}}J$, and the infimum is finite. Rests on Theorem 20.75, Lemma 20.77 and Equation (20.11).

Remark A41.2 (Where each hypothesis comes from).

(H1)–(H4) are the hypotheses of Theorem 20.78 as the chapter states it. (H2) and (H3) — convexity and coercivity in the slope — are Tonelli's own. (H1) and (H4) are the hypotheses under which the chapter's convexity lemma Lemma 20.77 was already stated in the $y$-free case: there $f$ is of class $C^{1}$ in $q$ and $\abs{\pp f/\pp q\left(x,q\right)}\le C\left(1+\abs{q}^{r-1}\right)$. (H4) is that same bound made locally uniform in $y$, which is the least that can be asked once $y$ enters, and it is automatic for every integrand met in this book — for a mechanical Lagrangian $\Lag=\tfrac12 m_{ij}(q)\dot q^{i}\dot q^{j}-V(q)$ the derivative in the velocities is linear in them, so (H4) holds with $r=2$ and $C_{R}$ the maximum of the mass matrix over $\abs{y}\le R$. The theorem as Tonelli stated it [Tonelli:1915] assumes only continuity and convexity; Remark A41.11 says what closing that last gap costs.

The Sobolev space in one dimension

Definition A41.3 (The space $W^{1,r}(a,b)$).

For $1<r<\infty$, $W^{1,r}(a,b)$ is the set of functions $u:[a,b]\longrightarrow\R$ for which there exists $v\in L^{r}(a,b)$ with

\begin{equation}\tag{A41.2} u(x)=u(a)+\int_{a}^{x}v(t)\,\dd t \qquad\text{for every }x\in[a,b]\ec \end{equation}

normed by $\norm{u}_{W^{1,r}}=\norm{u}_{L^{r}}+\norm{v}_{L^{r}}$. The function $v$ is unique as an element of $L^{r}$ and is written $u'$; every $u\in W^{1,r}(a,b)$ is continuous on $[a,b]$, by Equation (A41.2) and the absolute continuity of the Lebesgue integral. Rests on Definition 14.85 and Theorem 11.42.

Remark A41.4 (Why the space is defined by the integral formula).

Definition 14.85 introduces $H^{1}=W^{1,2}$ through distributional derivatives, and in one dimension the two definitions agree: if $u\in L^{r}(a,b)$ has a distributional derivative $v\in L^{r}$, then $w(x)=\int_{a}^{x}v$ has the same distributional derivative, so $u-w$ has distributional derivative zero and is therefore almost everywhere equal to a constant — which is the du Bois-Reymond lemma Lemma 20.19 in its integrable form, the one step of this identification that needs Lebesgue theory rather than the continuous argument given in the chapter. Nothing below uses the distributional definition, so Definition A41.3 is taken as the definition and the identification is recorded only to place the space where the reader expects it. In particular each element of $W^{1,r}(a,b)$ is here a genuine continuous function, not an equivalence class, so the boundary conditions in Equation (A41.1) have their literal meaning.

Lemma A41.5 (Young and Hölder).

Let $1<r<\infty$ and $r'=r/\left(r-1\right)$. For all $s,t\ge0$,

\begin{equation}\tag{A41.3} s\,t\le\frac{s^{r}}{r}+\frac{t^{r'}}{r'}\ec \end{equation}

and consequently, for measurable $g,h$ on $(a,b)$,

\begin{equation}\tag{A41.4} \int_{a}^{b}\abs{g\,h}\,\dd x \le\norm{g}_{L^{r'}}\,\norm{h}_{L^{r}}\ep \end{equation}

Rests on Theorem 11.38 and Definition A41.3.

Proof.

Derives Lemma A41.5. Young. For $s=0$ or $t=0$ the inequality is trivial, so let $s,t>0$ and put $\lambda=1/r\in(0,1)$, so $1-\lambda=1/r'$. The function $\Phi=-\log$ is of class $C^{2}$ on $(0,\infty)$ with $\Phi''(\tau)=\tau^{-2}>0$, hence convex there: writing $c=\lambda\,\sigma+\left(1-\lambda\right)\tau$ for $\sigma,\tau>0$, Taylor's theorem with Lagrange remainder (Theorem 11.38) about $c$ gives $\Phi(\sigma)=\Phi(c)+\Phi'(c)\left(\sigma-c\right) +\tfrac12\Phi''(\xi)\left(\sigma-c\right)^{2} \ge\Phi(c)+\Phi'(c)\left(\sigma-c\right)$ and the same with $\tau$ in place of $\sigma$; multiplying the first by $\lambda$, the second by $1-\lambda$ and adding annihilates the linear terms, because $\lambda\left(\sigma-c\right) +\left(1-\lambda\right)\left(\tau-c\right)=0$, and leaves $\lambda\,\Phi(\sigma)+\left(1-\lambda\right)\Phi(\tau)\ge\Phi(c)$. Taking $\sigma=s^{r}$ and $\tau=t^{r'}$ and undoing the minus sign,

\begin{equation*} \log\left(\frac{s^{r}}{r}+\frac{t^{r'}}{r'}\right) \ge\frac{1}{r}\log s^{r}+\frac{1}{r'}\log t^{r'} =\log\left(s\,t\right)\ec \end{equation*}

and $\log$ is increasing, which is Equation (A41.3).

Hölder. If $\norm{g}_{L^{r'}}=0$ or $\norm{h}_{L^{r}}=0$ then $gh=0$ almost everywhere and there is nothing to prove; if either norm is infinite the right side is $+\infty$. Otherwise replace $g$ by $g/\norm{g}_{L^{r'}}$ and $h$ by $h/\norm{h}_{L^{r}}$, which reduces the claim to the case $\norm{g}_{L^{r'}}=\norm{h}_{L^{r}}=1$. Apply Equation (A41.3) pointwise with $s=\abs{h(x)}$, $t=\abs{g(x)}$ and integrate:

\begin{equation*} \int_{a}^{b}\abs{g\,h}\,\dd x \le\frac{1}{r}\int_{a}^{b}\abs{h}^{r}\,\dd x +\frac{1}{r'}\int_{a}^{b}\abs{g}^{r'}\,\dd x =\frac{1}{r}+\frac{1}{r'}=1\ep \end{equation*}

∎

Lemma A41.6 (Uniform bound and uniform Hölder continuity).

Let $u\in W^{1,r}(a,b)$. Then for all $x,x'\in[a,b]$,

\begin{equation}\tag{A41.5} \abs{u(x)-u\left(x'\right)} \le\norm{u'}_{L^{r}}\,\abs{x-x'}^{1/r'}\ec \end{equation}

and consequently

\begin{equation}\tag{A41.6} \max_{[a,b]}\abs{u} \le\abs{u(a)}+\norm{u'}_{L^{r}}\left(b-a\right)^{1/r'}\ep \end{equation}

Rests on Definition A41.3 and Lemma A41.5.

Proof.

Derives Lemma A41.6. By Equation (A41.2), for $x'<x$, $u(x)-u\left(x'\right)=\int_{x'}^{x}u'(t)\,\dd t$. Apply Equation (A41.4) on the interval $\left(x',x\right)$ with $h=u'$ and $g$ the constant function $1$, whose $L^{r'}$ norm over that interval is $\left(x-x'\right)^{1/r'}$:

\begin{equation*} \abs{u(x)-u\left(x'\right)} \le\int_{x'}^{x}\abs{u'(t)}\,\dd t \le\left(x-x'\right)^{1/r'} \left(\int_{x'}^{x}\abs{u'}^{r}\,\dd t\right)^{1/r} \le\norm{u'}_{L^{r}}\left(x-x'\right)^{1/r'}\ep \end{equation*}

Taking $x'=a$ and adding $\abs{u(a)}$ gives Equation (A41.6).

∎

The Arzelà–Ascoli theorem and the compact embedding

Lemma A41.7 (Arzelà–Ascoli on an interval).

Let $\left(u_{n}\right)$ be a sequence of continuous functions on $[a,b]$ that is

uniformly bounded: $\abs{u_{n}(x)}\le K$ for all $n$ and all $x$; and
equicontinuous: for every $\varepsilon>0$ there is $\delta>0$, independent of $n$, such that $\abs{x-x'}<\delta$ implies $\abs{u_{n}(x)-u_{n}\left(x'\right)}<\varepsilon$ for every $n$.

Then some subsequence converges uniformly on $[a,b]$, and its limit is continuous. Rests on Theorems 11.7 and 11.8.

Proof.

Derives Lemma A41.7. A subsequence converging on a countable dense set. Enumerate the rational points of $[a,b]$ together with $a$ and $b$ as $q_{1},q_{2},\ldots$; the set is countable and dense in $[a,b]$. The numerical sequence $\left(u_{n}\left(q_{1}\right)\right)_{n}$ is bounded by $K$, so by Bolzano–Weierstrass (Theorem 11.7) there is a subsequence $\left(u^{(1)}_{n}\right)$ of $\left(u_{n}\right)$ for which $\left(u^{(1)}_{n}\left(q_{1}\right)\right)$ converges. Applying the same argument to $\left(u^{(1)}_{n}\left(q_{2}\right)\right)$ produces a subsequence $\left(u^{(2)}_{n}\right)$ of $\left(u^{(1)}_{n}\right)$ converging at $q_{1}$ and $q_{2}$, and so on, giving nested subsequences $\left(u^{(j)}_{n}\right)_{n}$, the $j$-th converging at $q_{1},\ldots,q_{j}$. The diagonal sequence $w_{k}=u^{(k)}_{k}$ is, from its $j$-th term onwards, a subsequence of $\left(u^{(j)}_{n}\right)_{n}$, so $\left(w_{k}\left(q_{j}\right) \right)_{k}$ converges for every $j$.

Uniform convergence. Let $\varepsilon>0$ and take $\delta>0$ from equicontinuity for $\varepsilon/3$. Partition $[a,b]$ into finitely many subintervals of length less than $\delta$ and pick in each one a point of the dense set, obtaining $q_{j_{1}},\ldots,q_{j_{m}}$ such that every $x\in[a,b]$ lies within $\delta$ of some $q_{j_{i}}$. Because each of the $m$ sequences $\left(w_{k}\left(q_{j_{i}}\right)\right)_{k}$ converges, it is Cauchy (Theorem 11.8), and there is an $N$ — the largest of the $m$ thresholds — with $\abs{w_{k}\left(q_{j_{i}}\right)-w_{l}\left(q_{j_{i}}\right)} <\varepsilon/3$ for all $k,l\ge N$ and all $i$. For an arbitrary $x\in[a,b]$ choose $i$ with $\abs{x-q_{j_{i}}}<\delta$; then, for $k,l\ge N$,

\begin{equation*} \abs{w_{k}(x)-w_{l}(x)} \le\abs{w_{k}(x)-w_{k}\left(q_{j_{i}}\right)} +\abs{w_{k}\left(q_{j_{i}}\right)-w_{l}\left(q_{j_{i}}\right)} +\abs{w_{l}\left(q_{j_{i}}\right)-w_{l}(x)} <\varepsilon\ec \end{equation*}

the outer terms by equicontinuity. The bound does not involve $x$, so $\left(w_{k}\right)$ is uniformly Cauchy; at each fixed $x$ it therefore converges (Theorem 11.8), to a limit $u(x)$, and letting $l\to\infty$ in the display gives $\abs{w_{k}(x)-u(x)}\le\varepsilon$ for every $k\ge N$ and every $x$ — uniform convergence.

Continuity of the limit. Given $\varepsilon>0$, pick $k$ with $\max_{[a,b]}\abs{w_{k}-u}<\varepsilon/3$ and $\delta$ from equicontinuity for $\varepsilon/3$; then $\abs{x-x'}<\delta$ gives

\begin{equation*} \abs{u(x)-u\left(x'\right)} \le\abs{u(x)-w_{k}(x)}+\abs{w_{k}(x)-w_{k}\left(x'\right)} +\abs{w_{k}\left(x'\right)-u\left(x'\right)}<\varepsilon\ep \end{equation*}

∎

Theorem A41.8 (Compact embedding of $W^{1,r}(a,b)$ into the continuous functions).

Let $\left(u_{n}\right)\subset W^{1,r}(a,b)$ with

\begin{equation}\tag{A41.7} \sup_{n}\norm{u_{n}'}_{L^{r}}\le M<\infty \qquad\text{and}\qquad \sup_{n}\abs{u_{n}(a)}\le M_{0}<\infty\ep \end{equation}

Then a subsequence of $\left(u_{n}\right)$ converges uniformly on $[a,b]$ to a continuous limit $u$. If in addition $u_{n}'\rightharpoonup v$ weakly in $L^{r}(a,b)$ along that subsequence — that is, $\int_{a}^{b}g\,u_{n}'\,\dd x\longrightarrow\int_{a}^{b}g\,v\,\dd x$ for every $g\in L^{r'}(a,b)$ — then $u\in W^{1,r}(a,b)$ with $u'=v$. Rests on Lemmas A41.6 and A41.7.

Proof.

Derives Theorem A41.8. Precompactness. By Equation (A41.6) the sequence is uniformly bounded by $K=M_{0}+M\left(b-a\right)^{1/r'}$, and by Equation (A41.5)

\begin{equation*} \abs{u_{n}(x)-u_{n}\left(x'\right)} \le M\,\abs{x-x'}^{1/r'} \end{equation*}

for every $n$, so the sequence is equicontinuous: given $\varepsilon>0$, the choice $\delta=\left(\varepsilon/M\right)^{r'}$ serves every $n$ at once, and it is here that $r>1$ is used, since $1/r'>0$. Lemma A41.7 now supplies a uniformly convergent subsequence, which we do not relabel, with continuous limit $u$.

Identification of the derivative. Fix $x\in[a,b]$ and let $\chi_{x}$ be the function equal to $1$ on $\left(a,x\right)$ and to $0$ on $\left(x,b\right)$. It belongs to $L^{r'}(a,b)$, being bounded on a bounded interval, so weak convergence gives

\begin{equation*} \int_{a}^{x}u_{n}'(t)\,\dd t =\int_{a}^{b}\chi_{x}\,u_{n}'\,\dd t \longrightarrow\int_{a}^{b}\chi_{x}\,v\,\dd t =\int_{a}^{x}v(t)\,\dd t\ep \end{equation*}

On the other hand Equation (A41.2) for $u_{n}$ says the left side equals $u_{n}(x)-u_{n}(a)$, which converges to $u(x)-u(a)$ by uniform — indeed pointwise — convergence. Hence $u(x)=u(a)+\int_{a}^{x}v$ for every $x$, which is Equation (A41.2) for $u$ with $v$ in the role of the derivative; since $v\in L^{r}$ by hypothesis, $u\in W^{1,r}(a,b)$ and $u'=v$.

∎

Weak lower semicontinuity with $y$ present

Theorem A41.9 (Convexity in $q$ implies weak lower semicontinuity).

Let $f$ satisfy (H1)–(H4) of Theorem A41.1. Let $\left(u_{n}\right)\subset W^{1,r}(a,b)$ and $u\in W^{1,r}(a,b)$ be such that

$\sup_{n}\norm{u_{n}'}_{L^{r}}<\infty$;
$u_{n}\longrightarrow u$ uniformly on $[a,b]$;
$u_{n}'\rightharpoonup u'$ weakly in $L^{r}(a,b)$.

Then

\begin{equation}\tag{A41.8} \int_{a}^{b}f\left(x,u,u'\right)\dd x \le\liminf_{n\to\infty} \int_{a}^{b}f\left(x,u_{n},u_{n}'\right)\dd x\ep \end{equation}

Rests on Lemma 20.77, Equation (20.66) and Theorem A41.8.

Proof.

Derives Theorem A41.9. Write $M=\sup_{n}\norm{u_{n}'}_{L^{r}}$ and $R=\sup_{n}\max_{[a,b]}\abs{u_{n}}$, the latter finite because a uniformly convergent sequence of continuous functions is uniformly bounded; enlarge $R$ if necessary so that also $\max_{[a,b]}\abs{u}\le R$.

Step 1: the tangent inequality, taken at the limit slope. Convexity of a $C^{1}$ function of $q$ puts its graph above every tangent, which is Equation (20.66); applied at the fixed value $y=u_{n}(x)$ of the second argument, with $q_{1}=u'(x)$ and $q_{2}=u_{n}'(x)$, it gives, for almost every $x$,

\begin{equation}\tag{A41.9} f\left(x,u_{n},u_{n}'\right) \ge f\left(x,u_{n},u'\right) +g_{n}(x)\left[u_{n}'(x)-u'(x)\right]\ec \qquad g_{n}=f_{q}\left(\cdot\,,u_{n},u'\right)\ep \end{equation}

Note that the tangent is taken at the limit slope $u'$ but at the approximating value $u_{n}$ of the function; this is the one departure from the $y$-free argument of Lemma 20.77, and it is what makes both terms on the right tractable. Integrating Equation (A41.9),

\begin{equation}\tag{A41.10} \int_{a}^{b}f\left(x,u_{n},u_{n}'\right)\dd x \ge\int_{a}^{b}f\left(x,u_{n},u'\right)\dd x +\int_{a}^{b}g_{n}\left(u_{n}'-u'\right)\dd x\ep \end{equation}

Both integrals on the right are well defined: the second because $g_{n}\in L^{r'}$ by Step 3 below and $u_{n}'-u'\in L^{r}$, so Equation (A41.4) applies; the first because its integrand is bounded below by $-\beta$ through (H3), so the integral exists in $\left(-\infty,+\infty\right]$.

Step 2: the first term. By (H1) the map $y\longmapsto f\left(x,y,u'(x)\right)$ is continuous, and $u_{n}(x)\longrightarrow u(x)$ for every $x$, so

\begin{equation*} f\left(x,u_{n}(x),u'(x)\right) \longrightarrow f\left(x,u(x),u'(x)\right) \qquad\text{for almost every }x\ep \end{equation*}

By (H3) the functions $f\left(x,u_{n},u'\right)+\beta$ are nonnegative and measurable, so Fatou's lemma applies to them and gives

\begin{equation}\tag{A41.11} \int_{a}^{b}f\left(x,u,u'\right)\dd x \le\liminf_{n\to\infty} \int_{a}^{b}f\left(x,u_{n},u'\right)\dd x\ec \end{equation}

the constants $\beta\left(b-a\right)$ cancelling from both sides. The inequality holds in $\left(-\infty,+\infty\right]$, so nothing is assumed about the finiteness of either side.

Step 3: the second term vanishes. Put $g=f_{q}\left(\cdot\,,u,u'\right)$. By (H4) with the constant $C_{R}$,

\begin{equation}\tag{A41.12} \abs{g_{n}(x)}\le C_{R}\left(1+\abs{u'(x)}^{r-1}\right) \qquad\text{and the same bound for }\abs{g}\ec \end{equation}

and the right-hand side lies in $L^{r'}(a,b)$, because $\left(\abs{u'}^{r-1}\right)^{r'}=\abs{u'}^{r}$ is integrable and constants are integrable on a bounded interval. In particular $g_{n},g\in L^{r'}$, as promised in Step 1. Moreover $g_{n}\longrightarrow g$ pointwise almost everywhere, by the continuity of $f_{q}$ in its second argument (H1) and the pointwise convergence $u_{n}\to u$. Since $\abs{g_{n}-g}^{r'}\le\left(2C_{R}\right)^{r'} \left(1+\abs{u'}^{r-1}\right)^{r'}\in L^{1}$ and tends to $0$ almost everywhere, the dominated convergence theorem gives

\begin{equation}\tag{A41.13} \norm{g_{n}-g}_{L^{r'}}\longrightarrow0\ep \end{equation}

Now split

\begin{equation*} \int_{a}^{b}g_{n}\left(u_{n}'-u'\right)\dd x =\int_{a}^{b}g\left(u_{n}'-u'\right)\dd x +\int_{a}^{b}\left(g_{n}-g\right)\left(u_{n}'-u'\right)\dd x\ep \end{equation*}

The first integral tends to $0$: $g\in L^{r'}$ and $u_{n}'\rightharpoonup u'$ in $L^{r}$, which is exactly the statement that $\int g\,u_{n}'\to\int g\,u'$. The second is bounded, by Hölder's inequality Equation (A41.4), by

\begin{equation*} \norm{g_{n}-g}_{L^{r'}}\,\norm{u_{n}'-u'}_{L^{r}} \le\left(M+\norm{u'}_{L^{r}}\right)\norm{g_{n}-g}_{L^{r'}}\ec \end{equation*}

which tends to $0$ by Equation (A41.13) and hypothesis (1). Hence

\begin{equation}\tag{A41.14} \int_{a}^{b}g_{n}\left(u_{n}'-u'\right)\dd x \longrightarrow0\ep \end{equation}

Conclusion. Take the lower limit in Equation (A41.10). The lower limit of a sum is at least the lower limit of the first summand plus the limit of the second when the latter exists, so by Equations (A41.11) and (A41.14)

\begin{equation*} \liminf_{n}\int_{a}^{b}f\left(x,u_{n},u_{n}'\right)\dd x \ge\liminf_{n}\int_{a}^{b}f\left(x,u_{n},u'\right)\dd x+0 \ge\int_{a}^{b}f\left(x,u,u'\right)\dd x\ec \end{equation*}

which is Equation (A41.8).

∎

Proof of the theorem

Proof of Theorem A41.1. Derives Theorem A41.1. The infimum is finite. The affine function $\ell(x)=y_{a}+\left(y_{b}-y_{a}\right)\left(x-a\right)/ \left(b-a\right)$ lies in $\mathcal{A}_{r}$, with constant derivative, and $f$ is continuous, so $J[\ell]$ is finite and $m=\inf_{\mathcal{A}_{r}}J\le J[\ell]<\infty$. By (H3), $J[u]\ge-\beta\left(b-a\right)$ for every admissible $u$, so $m>-\infty$.

Coercivity bounds a minimising sequence. Let $\left(u_{n}\right)\subset\mathcal{A}_{r}$ satisfy $J\left[u_{n}\right]\longrightarrow m$; discarding finitely many terms we may assume $J\left[u_{n}\right]\le m+1$ for every $n$. By (H3),

\begin{equation}\tag{A41.15} m+1\ge J\left[u_{n}\right] \ge\alpha\int_{a}^{b}\abs{u_{n}'}^{r}\,\dd x -\beta\left(b-a\right)\ec \end{equation}

so $\norm{u_{n}'}_{L^{r}}^{r} \le\left[m+1+\beta\left(b-a\right)\right]/\alpha=:M^{r}$, a bound independent of $n$. Since $u_{n}(a)=y_{a}$ for every $n$, the hypotheses Equation (A41.7) of Theorem A41.8 hold with $M_{0}=\abs{y_{a}}$, and by Equation (A41.6) the sequence is also bounded in $L^{r}$, hence bounded in $W^{1,r}(a,b)$.

Extraction of a limit. The sequence $\left(u_{n}'\right)$ is bounded in $L^{r}(a,b)$, a reflexive Banach space for $1<r<\infty$, so some subsequence converges weakly, $u_{n_{k}}'\rightharpoonup v$ in $L^{r}(a,b)$ — this is the compactness input of Theorem 20.75, quoted there in the same form (see Remark A41.12). Passing to a further subsequence, not relabelled, Theorem A41.8 makes $u_{n_{k}}$ converge uniformly on $[a,b]$ to a continuous $u_{\ast}$, and identifies $u_{\ast}\in W^{1,r}(a,b)$ with $u_{\ast}'=v$.

The limit is admissible. Uniform convergence is in particular pointwise at the endpoints, so $u_{\ast}(a)=\lim_{k}u_{n_{k}}(a)=y_{a}$ and $u_{\ast}(b)=\lim_{k}u_{n_{k}}(b)=y_{b}$: the class $\mathcal{A}_{r}$ is closed under this mode of convergence, which is the second place where the compact embedding earns its keep — weak convergence in $W^{1,r}$ alone says nothing about the value of a function at a point.

The limit minimises. The three hypotheses of Theorem A41.9 hold for $\left(u_{n_{k}}\right)$ and $u_{\ast}$, so

\begin{equation*} J\left[u_{\ast}\right] \le\liminf_{k}J\left[u_{n_{k}}\right]=m\ec \end{equation*}

while $J\left[u_{\ast}\right]\ge m$ because $u_{\ast}\in\mathcal{A}_{r}$. Hence $J\left[u_{\ast}\right]=m$ and the infimum is attained.

∎

Remark A41.10 (The shape of the argument, against the abstract direct-method theorem).

The proof just given is the abstract direct method with each of its three ingredients made concrete, and it is worth seeing which concrete statement plays which abstract role. Coercivity of $J$ in the sense of Theorem 20.75 is Equation (A41.15), and it controls only the derivative; that this suffices to control the function as well is Equation (A41.6), i.e. the boundary condition plus Hölder's inequality. Sequential weak lower semicontinuity is Theorem A41.9. The compactness that the abstract theorem imports from reflexivity is used here in two different topologies at once — weakly in $L^{r}$ for the derivatives, uniformly in $C^{0}$ for the functions — and the second is not a convenience but a necessity: without it neither the endpoint values nor the $y$-dependence of the integrand would survive the limit. That is the precise sense in which Lemma 20.77 left something owed.

Remark A41.11 (The route not taken: Scorza-Dragoni).

If (H4) is dropped and $f$ is asked only to be continuous and convex in $q$ — Tonelli's own hypotheses — Step 3 above breaks down, because $\abs{f_{q}\left(x,u_{n},u'\right)}$ can then grow with $n$ faster than any fixed $L^{r'}$ function, and the classical repair is a different one. It rests on the theorem of Scorza-Dragoni: a Carathéodory integrand — measurable in $x$, continuous in $\left(y,q\right)$ — is, after the deletion from $[a,b]$ of a set of arbitrarily small measure, continuous jointly in all its arguments on what remains; on that set the modulus of continuity in $y$ is then uniform in $q$ over compacta, which is what Step 3 needs, and the deleted set is handled by the coercivity bound (H3). This treatise's bibliography carries no entry for Scorza-Dragoni's paper, and the statement above is therefore quoted here without a citation — the honest record of a result that is named but neither used nor verified in this book. Nothing in Theorem A41.1 depends on it: the theorem proved above is the theorem under (H1)–(H4), and every integrand this treatise minimises satisfies (H4) (Remark A41.2).

Remark A41.12 (What is quoted here).

Three analytic inputs are used and not proved in this treatise. Each belongs to the measure theory and functional analysis that Remark 16.1 of Hilbert Spaces already declares as imports, and all three are in Reed and Simon [Reed:1972].

The Lebesgue integral and its convergence theorems. Fatou's lemma is used once, in Equation (A41.11), and the dominated convergence theorem once, in Equation (A41.13). The absolute continuity of the integral, used in Definition A41.3 to see that an element of $W^{1,r}$ is continuous, is of the same family.
The duality $\left(L^{r}\right)^{\ast}=L^{r'}$ and the reflexivity of $L^{r}$ for $1<r<\infty$, from which a bounded sequence in $L^{r}$ has a weakly convergent subsequence. This is the only compactness assumption in the proof, and it is the same one Theorem 20.75 makes in the chapter, where it is named as the Eberlein–Smulian theorem; the special case $r=2$ is the Hilbert-space statement of Hilbert Spaces.
The du Bois-Reymond lemma for integrable functions, used only in Remark A41.4 to identify Definition A41.3 with the distributional definition Definition 14.85. No step of the proof uses it; the continuous version proved in the chapter, Lemma 20.19, is not strong enough for the identification, which is why the space is defined here by Equation (A41.2).

Everything else — the Arzelà–Ascoli theorem, the compact embedding, Young's and Hölder's inequalities, and the semicontinuity theorem itself — is proved above from Real Analysis and the chapter.

Remark A41.13.

Theorem A41.1 discharges Theorem 20.78 of Section 20.5.2, completing the direct method of Theorem 20.75 for the concrete functional Equation (20.11). Its hypotheses are what Remark 20.79 tests against the Lagrangians of mechanics, and the answer given there is unchanged by anything proved here: convexity in the velocities holds for a kinetic energy with a positive definite mass matrix, while coercivity (H3) demands a potential bounded above, so the theorem certifies existence for a free or repelled particle and not for a confining one. The failure of convexity is equally untouched — the functional Equation (20.68) satisfies (H1), (H3) and (H4) but not (H2), and has no minimiser.