Past exam of the partial differential equations for pure and applied mathematicians and calculus of variations course of the University of Oxford / 2008 / B / 8 / a

The continuous superlinear function $\Phi$ tends to positive infinity at both ends, so it has a finite global minimum $M$. Hence $I(u)\ge M(b-a)$ for every admissible function. An affine function with the prescribed endpoints is Lipschitz and has finite energy by continuity of the integrand on its compact range. Thuswhere the strict left inequality simply records finiteness of $M$.

A minimizing sequence has an energy upper bound $C$. Set $H(t)=\Phi (t)-M\ge 0$. Then $\int H(u_j^{\prime })\le C-M(b-a)=C_1$. Since $H(t)/|t|\to \infty$, for every $A>0$ there is $R_A$ such that $H(t)\ge A|t|$ for $|t|>R_A$. Consequently, for every measurable $E$,Taking first $A$ large and then $|E|$ small proves uniform integrability; taking $E=(a,b)$ also gives a uniform $L^1$ bound. The Dunford-Pettis theorem on a finite-measure space identifies bounded uniformly integrable subsets of $L^1$ with relatively weakly compact ones. Thus a subsequence satisfies

The compactness conclusion justified by the hypotheses is $u\in W^{1,1}$ with the prescribed traces; the printed claim $u\in W^{1,\infty }$ needs additional regularity. DefineThen $u\in W^{1,1}$ and $u^{\prime }=v$. Weak $L^1$ convergence tested against $1_{(a,x)}$ gives $u_{j_k}(x)\to u(x)$ for every $x$, including the endpoint trace at $b$. Uniform integrability of the derivatives makes the functions equicontinuous: their increment over an interval is bounded by the integral of $|u_j^{\prime }|$ on that interval, uniformly small for short intervals. Their fixed initial trace and common derivative $L^1$ bound also make them uniformly bounded. The Arzelà-Ascoli theorem therefore upgrades the convergence to uniform convergence, and in particularUniform integrability does not bound the derivative in $L^{\infty }$. The natural compact class here is absolutely continuous functions, rather than Lipschitz functions. This distinction can affect attainment: smooth strictly convex superlinear integrands can have non-Lipschitz minimizing limits, so it is not repaired merely by adding convexity in the last variable. If one additionally has a uniform Lipschitz bound on the minimizing sequence, weak testing on measurable subsets passes that bound to $v$ and proves the printed $u\in \mathcal{A}$. Alternatively one can formulate the direct method from the outset in the fixed-trace $W^{1,1}$ class. The proof above gives all the asserted convergence and trace conclusions without assuming the missing bound.

Provided the limit belongs to the original admissible class, the definition of the infimum and the assumed inequality giveThe needed fundamental property is sequential lower semicontinuity for the convergence in the compactness argument: uniform, or strong $L^1$, convergence of functions together with weak $L^1$ convergence of derivatives. The extra integrand hypothesis in the classical Tonelli existence theorem is convexity in the derivative variable $v$ for every fixed $(x,u)$, together with the stated coercivity and continuity. In one dimension the resulting integral is lower semicontinuous under the uniform/weak-derivative convergence above.

This lower-semicontinuity assertion does not itself supply the missing Lipschitz membership in (iii). With the natural fixed-trace $W^{1,1}$ admissible class and its own infimum, it completes Tonelli's direct existence argument. Retaining the printed Lipschitz class requires the additional membership estimate; changing admissible classes can otherwise encounter the Lavrentiev phenomenon.