Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 107 (Elliptic Partial Differential Equations)

The mean value property for harmonic functions says that whenever $\overline{B_r(x_0)}\subset \Omega$,If $u$ attains its maximum $M$ at an interior point, the average of the nonnegative function $M-u$ over every sufficiently small centred ball is zero. Continuity makes $u=M$ on each such ball, and connectedness propagates this equality throughout the domain. Applying the same argument to $-u$ proves the Strong maximum principle for harmonic functions.

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Differentiate the ball mean-value formula and use the divergence theorem:ConsequentlyThus one may take $C(n)=n$.

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Iteration gives the interior derivative estimate for a harmonic functionThe growth hypothesis bounds the right side by $C^{\prime }{R}^{\alpha -1}+o(1)$, which tends to zero as $R\to \infty$. Thus every derivative of order $k+1$ vanishes everywhere. The Taylor theorem makes $u$ a polynomial of degree at most $k$, proving the Polynomial-growth Liouville theorem for harmonic functions.

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Join the Laplace operator to the target operator byThe family has one ellipticity constant. The global Schauder estimate and the maximum principle give, uniformly in $t$,for zero boundary data. Let $I$ contain those $t$ for which $L_t:C_0^{2,\alpha }\to C^{\alpha }$ is onto. The assumed Laplace solvability gives $0\in I$; the bounded inverse theorem and small perturbations make $I$ open; and the uniform estimate plus compactness of lower Hölder embeddings makes $I$ closed. The method of continuity yields $I=[0,1]$. At $t=1$ this gives the required solution, and the maximum principle gives uniqueness.

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The strict maximum principle applied to $a_{ij}{\partial }_{ij}u=f>0$ with zero boundary data gives $u<0$ in $\Omega$. On ${\Omega }^{\prime }$ set $w=v-u$. Then $a_{ij}{\partial }_{ij}w=0$, while $w=-u\ge 0$ on $\partial {\Omega }^{\prime }$ and this boundary value is positive somewhere because ${\Omega }^{\prime }$ is proper. Hence $w>0$ inside and $w(y_0)=0$. The Hopf boundary point lemma at this boundary minimum givesTherefore $D_{\nu }v(y_0)-D_{\nu }u(y_0)<0$, proving the normal derivatives are unequal.

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The Campanato space ${\mathcal{L}}^{p,\mu }(\Omega )$ consists of $f\in L^p(\Omega )$ for whichwhere $f_E=|E{|}^{-1}{\int }_{E}f$. On a smooth bounded domain,with equivalent norms for $0<\alpha \le 1$.

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Fix $B_R(x_0)\subset B_1$, put $A_0=A(x_0)$, and let $v$ have the same boundary values as $u$ while solving $\mathrm{div}(A_0\nabla v)=0$. Then $w=u-v\in W_0^{1,2}(B_R)$ satisfiesSince $u\in C^1$ gives $|Du|\le L$ locally, the supplied energy estimate and Hölder continuity of $A$ yieldThe constant-coefficient decay estimate givesThus, for $\Phi (r)={\int }_{B_r}|\nabla u-(\nabla u{)}_{B_r}{|}^2$,Because $n+2>n+2\alpha$, the Campanato iteration lemma gives $\Phi (\rho )\le C{\rho }^{n+2\alpha }$ on balls in $B_{1/2}$. Hence $\nabla u\in {\mathcal{L}}^{2,n+2\alpha }=C^{0,\alpha }$ and $u\in C^{1,\alpha }(B_{1/2})$.

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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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