Past exam of the partial differential equations for pure and applied mathematicians and calculus of variations course of the University of Oxford / 2007 / B / 8

Superlinearity and continuity make $\Phi$ bounded below by a finite constant $M$. Consequently $I(u)\ge M(b-a)$ for every admissible function. The affine endpoint interpolant has finite energy by continuity on its compact range, so the infimum is finite. Some other admissible energies can be infinite; this does not affect the bounded-below or minimizing-sequence argument.

The De la Vallée-Poussin criterion characterizes uniform integrability by existence of a nonnegative increasing convex function $\Psi$ with $\Psi (s)/s\to \infty$ and a uniform bound on $\int \Psi (|v_j|)$. On a finite interval this gives a weakly convergent subsequence in $L^1$ by the Dunford-Pettis theorem.

After discarding a finite prefix of the minimizing sequence if needed, put $H(t)=\Phi (t)-M\ge 0$. Its derivative sequence has $\int H(u_j^{\prime })\le C$. The superlinear bound directly gives uniform integrability: for $H(t)\ge A|t|$ outside $[-R_A,R_A]$,If the convex version of the criterion is used, choose increasing $R_m\to \infty$ so $H(t)\ge m^2|t|$ for $|t|\ge R_m$, and set $\Psi (s)={\sum }_{m\ge 1}(s-R_m{)}_{+}$. This is locally a finite sum, convex and increasing. It is superlinear because for fixed $N$ and $s\ge 2R_N$, $\Psi (s)\ge Ns/2$. If exactly $m$ terms are active, $\Psi (s)\le ms\le H(t)$ for $s=|t|$, so its integrals have the same bound. The criterion therefore supplies $u_{j_k}^{\prime }\rightharpoonup v$ in $L^1$.

Define $u(x)=\alpha +{\int }_a^{x}v$. Interval-indicator tests give $u_{j_k}(x)\to u(x)$ for every $x$, including $b$, so its endpoint values are correct and $u\in W^{1,1}$. The common derivative $L^1$ bound makes these functions uniformly bounded. Uniform integrability also gives common equicontinuity, since an increment is bounded by the derivative integral on the intervening interval. The Arzelà-Ascoli theorem yields uniform convergence to the same identified limit, and in particularHere the printed admissible space is $W^{1,1}$, so this compactness argument gives precisely the needed membership.

Write $l={\inf }_{\mathcal{A}}I$. Since the limit is admissible, the assumed lower-semicontinuity inequality gives $l\le I(u)\le \mathrm{liminf}I(u_{j_k})=l$. Thus $I(u)=l$ and $u$ is a global minimizer.

The extra integrand hypothesis in the Tonelli existence theorem for one dimension is convexity in the derivative variable for each fixed $(x,u)$. Together with the stated continuity and coercive lower bound it gives the lower-semicontinuity property along the uniform-function/weak-derivative convergence just proved.

The assumed weak-star limit is the constant $m=\theta a+(1-\theta )b$. Each integer rescaling has exactly $j$ full periods on $(0,1)$, so a change of variables on those periods givesApply the asserted lower semicontinuity to this sequence. The integral of $f$ of its constant limit is $f(m)$, givingThis holds for every real $a,b$ and $0<\theta <1$; the endpoint values of $\theta$ give equality. Therefore $f$ is convex.