Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 107 / 4

A minimizing sequence in ${\mathrm{Lip}}_g(\Omega ,k)$ is uniformly bounded and equi-Lipschitz. The Arzela-Ascoli theorem gives a uniformly convergent subsequence with limit $u$ having the same boundary data and Lipschitz constant at most $k$. Its gradients have a weak-star convergent subsequence in $L^{\infty }$, with limit $Du$. Convexity of $F$ makes the integral functional weak-star lower semicontinuous, soThus $u$ attains the infimum by the direct method in the calculus of variations.

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Let $L=\mathrm{Lip}(u)<k$ and take $v\in {\mathrm{Lip}}_g(\Omega )$. For small $t>0$, $u_t=(1-t)u+tv$ has Lipschitz constant at most $(1-t)L+t\mathrm{Lip}(v)<k$. Constrained minimality and convexity giveCancellation gives $\mathcal{F}[u]\le \mathcal{F}[v]$, so $u$ is an unconstrained minimizer.

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Let $H={\sup }_{{\overline{B}}_1}\Vert D^{2}g\Vert$. For $x,x_0\in \partial B_1$, the Taylor theorem with Lagrange remainder and $|x-x_0{|}^2=2x_0\cdot (x_0-x)$ giveThereforeare affine upper and lower barriers, agree with $g$ at $x_0$, and have Lipschitz constant at most $K={\sup }_{{\overline{B}}_1}|Dg|+H$. Thus $g$ has the bounded slope condition.

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For $y\in \partial \Omega$, let $b_y^{-}\le g\le b_y^{+}$ be the affine barriers from the bounded slope condition. Their constant gradients satisfy the Euler-Lagrange equation, so they minimize the autonomous convex functional for their own boundary values. The comparison principle givesSince both barriers equal $u(y)$ at $y$ and are $K$-Lipschitz,for $x\in \Omega$ and $y\in \partial \Omega$. The supplied boundary-to-interior criterion now gives $\mathrm{Lip}(u)\le K$.

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Part c supplies a bounded slope condition constant $K$. Choose $k>K$. Parts a and d give a constrained minimizer with $\mathrm{Lip}(u)\le K<k$, and part b makes it a minimizer over all of ${\mathrm{Lip}}_g(B_1)$.

For $F(\xi )=\sqrt{1+|\xi {|}^2}$, the Euler-Lagrange equation is the minimal surface equation for a graphThe Lipschitz bound confines $Du$ to a compact set on which $D^{2}F$ is uniformly positive definite, so the equation is uniformly elliptic. Interior regularity gives $C^{1,\alpha }$ first, and repeated Schauder estimates then give smoothness in the interior.

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