Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 107 (Elliptic Partial Differential Equations)

For a harmonic function $u$, letThe divergence theorem givesSince $M(r)\to u(x)$ as $r\downarrow 0$, $M(r)=u(x)$. Integrating the spherical averages in the radial variable gives the corresponding ball average, proving the mean value property for harmonic functions. If $u$ attains its maximum at an interior point, the average of the nonnegative function $\max u-u$ on every sufficiently small centred sphere is zero. Continuity makes $u$ constant on those spheres, and connectedness propagates that value through the domain. Thus the weak maximum principle for elliptic operators gives

For the derivative estimate, choose $r>0$ smaller than half the distance from ${\Omega }^{\prime }$ to $\partial \Omega$, and let ${\rho }_r$ be a smooth radial mollifier supported in $B_r(0)$. Writing its convolution in polar coordinates and using the spherical mean value property shows that $u\ast {\rho }_r=u$ on ${\Omega }^{\prime }$. Hence, for every multi-index $\alpha$,so Holder inequality gives

Solved by gpt-5.6-sol high.

Suppose $u_1$ and $u_2$ are weak solutions with the same trace, and put $w=u_1-u_2\in H_0^1(\Omega )$. The weak formulation permits $w$ itself as a test function, givingThus $w$ is almost everywhere constant, and its zero trace makes that constant zero. This proves uniqueness.

The weak identity also says that $\Delta u=0$ in the sense of distributions. The Weyl lemma therefore gives $u\in C^{\infty }(\Omega )$ and $\Delta u=0$ pointwise. The assumed continuity on $\overline{\Omega }$ retains the prescribed boundary values, so the weak solution is the unique classical solution in $C^{\infty }(\Omega )\cap C^0(\overline{\Omega })$.

Solved by gpt-5.6-sol high.

Fix a closed ball $\overline{B_R(a)}\subset \Omega$, and let $v$ solve the Dirichlet problem for the Laplace equation in this ball with boundary data $u$. Put $w=u-v$. The function $w$ is continuous, vanishes on the boundary, and inherits the restricted spherical mean identity because $v$ has the full mean value property for harmonic functions.

Suppose $M={\max }_{\overline{B_R(a)}}w>0$. Its maximum set $E$ is a nonempty compact subset of the open ball. Choose $x\in E$ maximizing $|x-a|$. For every sufficiently small radius in the sequence attached to $x$,Equality of the average with the maximum and continuity imply that the whole sphere belongs to $E$. Its point in the direction from $a$ through $x$ lies farther from $a$ than $x$ does; if $x=a$, any point on the sphere does. Both cases contradict the choice of $x$. Hence $w\le 0$, and applying the same argument to $-w$ gives $w=0$.

Thus $u=v$ on every relatively compact ball. It is consequently harmonic and smooth locally, proving the local converse to the mean value property.

Solved by gpt-5.6-sol high.

Write the flux asAfter expanding the divergence, the principal symbol of a partial differential equation along a candidate solution $u$ is determined byOnly the symmetric part $A_s=(A+A^T)/2$ contributes to $A^{ik}{D}_{ik}u$. The problem is elliptic along $u$ when ${\xi }^T{A}_s\xi \ge 0$ for every $\xi$, and it is strictly elliptic where this quantity is positive for every nonzero $\xi$. It is uniformly elliptic on a set when constants $0<\lambda \le \Lambda <\infty$, independent of the point, satisfyThese definitions separate pointwise positive definiteness from a quantitative lower and upper bound; a degenerate elliptic operator may lose strict ellipticity at some jets.

Solved by gpt-5.6-sol high.

For the p-energythe first variation in the direction $\psi \in C_c^{\infty }(\Omega )$ isAn integration by parts therefore gives the Euler-Lagrange equationwhich is the p-Laplacian equation. In the notation of the question one takes $a^{ij}(x,z,\eta )=|\eta {|}^{p-2}{\delta }^{ij}$.

For $\eta \ne 0$, the principal coefficient matrix isIts eigenvalue in directions orthogonal to $\eta$ is $|\eta {|}^{p-2}$, while its eigenvalue parallel to $\eta$ is $(p-1)|\eta {|}^{p-2}$. The coefficients are $C^{1,\alpha }$ away from $\eta =0$, and the condition number there is at most $p-1$. On every region where $0<m\le |Du|\le M$, this gives uniform ellipticity with constants depending on $m$, $M$, and $p$. At $Du=0$ all principal eigenvalues vanish, so the operator is degenerate there and is not strictly elliptic on a domain containing a critical point.

Solved by gpt-5.6-sol high.

The boundary data are encoded by the Affine Sobolev spaceA function $u\in {\mathcal{A}}_{\phi }$ is a weak solution of the homogeneous p-Laplacian equation when

For existence, take a minimizing sequence for the p-energy on ${\mathcal{A}}_{\phi }$. The Poincare inequality bounds $u-\phi$ in $W^{1,p}$ by its gradient, so the sequence is bounded in the reflexive Banach space $W^{1,p}$. A weakly convergent subsequence remains in the weakly closed affine space, and convexity of $|\eta {|}^p$ gives weak lower semicontinuity. The direct method in the calculus of variations therefore produces a minimizer, whose first variation is precisely the displayed weak equation.

Solved by gpt-5.6-sol high.

Seek a radial function $u=u(r)$ on the unit ball. The equation and regularity at the origin giveHence $u^{\prime }(r)=(r/n{)}^{1/(p-1)}$, and the zero Dirichlet boundary condition givesThe power $|x{|}^q$ is twice differentiable at the origin exactly when $q\ge 2$. Here $q=p/(p-1)$, so $u\in C^2$ at the origin exactly when $1<p\le 2$. Under the assumption $2<p<n$, this weak solution is never $C^2$ at the origin, exhibiting the regularity loss caused by degenerate ellipticity.

Solved by gpt-5.6-sol high.

Forthe weak maximum principle for elliptic operators states that $Lu\ge 0$ impliesIn particular, a solution of $Lu=0$ cannot have a positive interior maximum exceeding its boundary maximum.

Because the coefficient matrix is positive definite and the closure of the smooth bounded domain is compact, strict ellipticity supplies a uniform lower bound after restricting to $\overline{\Omega }$. Rotate and translate coordinates so that $\Omega$ is bounded in the $x_1$ direction, and set $h=e^{\gamma x_1}$. For sufficiently large $\gamma$,If $u+\epsilon h$ had a positive interior maximum, its gradient would vanish and its Hessian matrix would be negative semidefinite there, giving $L(u+\epsilon h)\le 0$. This contradicts $L(u+\epsilon h)=\epsilon Lh>0$. Comparing on the boundary and sending $\epsilon \downarrow 0$ proves the assertion.

Solved by gpt-5.6-sol high.

The Global Schauder estimate isThe constant depends on the dimension, $\alpha$, the domain and its boundary regularity, the ellipticity constants, and the $C^{0,\alpha }$ norms of the coefficients, but not on $u$, $f$, or $\phi$.

Solved by gpt-5.6-sol high.

Fix $v\in C^{1,\beta }(\overline{\Omega })$. Its jet map $x\mapsto (x,v(x),Dv(x))$ is $C^{0,\beta }$, while the barred coefficients are $C^{0,\alpha }$ in all their variables. Composition therefore makes$C^{0,\alpha \beta }$. The image of the jet map is compact, so strict ellipticity on the coefficient domain has a positive uniform lower bound on this image. The assumed linear Dirichlet problem theory now gives a uniquesolving ${\bar{a}}^{ij}(x,v,Dv)D_{ij}u+\bar{b}(x,v,Dv)=0$ with boundary value $\phi$. Thus $T:C^{1,\beta }\to C^{1,\beta }$ is well defined.

Solved by gpt-5.6-sol high.

Let $v$ range over a bounded subset of $C^{1,\beta }$. All jets $(x,v,Dv)$ then lie in one compact set, so the composed coefficients have uniform $C^{0,\alpha \beta }$ bounds and one ellipticity constant. The weak maximum principle for elliptic operators bounds $T(v)$ in $C^0$ by the boundary data and the bounded forcing term. The Global Schauder estimate consequently bounds $T(v)$ in $C^{2,\alpha \beta }$.

The compact embedding of Hölder spacesthen makes the image relatively compact. Hence $T$ is a compact operator on $C^{1,\beta }(\overline{\Omega })$.

Solved by gpt-5.6-sol high.

Suppose $v_k\to v$ in $C^{1,\beta }$ and write $u_k=T(v_k)$. The preceding uniform Schauder estimate bounds $(u_k)$ in $C^{2,\alpha \beta }$. Every subsequence therefore has a further subsequence converging in $C^{1,\beta }$ by the compact embedding of Hölder spaces. The composed coefficients converge uniformly, and lower-exponent Hölder compactness provides enough convergence of the second derivatives to pass to the linear equation. Every subsequential limit solves the problem defining $T(v)$.

Uniqueness of that linear Dirichlet problem forces every such limit to equal $T(v)$. Since every subsequence has a further subsequence with this same limit, the whole sequence converges to $T(v)$ in $C^{1,\beta }$. Thus $T$ is continuous.

Solved by gpt-5.6-sol high.

The Leray-Schauder fixed point theorem says that a continuous compact map $T$ on a Banach space has a fixed point if the homotopy setis bounded. If $u=tT(u)$ with $t>0$, multiplying the equation for $T(u)$ by $t$ shows that $u$ solvesThe case $t=0$ gives $u=0$. Therefore it is enough to prove one uniform $C^{1,\beta }$ estimate for all solutions of this family and all $t\in [0,1]$. The theorem then yields a fixed point $u=T(u)$, which solves the original quasilinear problem.

Initially the construction gives $u\in C^{2,\alpha \beta }$. This makes $u$ and $Du$ Lipschitz, so composing the original $C^{0,\alpha }$ coefficients with the jet of $u$ produces $C^{0,\alpha }$ coefficients. A second application of the Global Schauder estimate gives $u\in C^{2,\alpha }(\overline{\Omega })$. The fixed-point theorem is an existence result and supplies no uniqueness.

Solved by gpt-5.6-sol high.

The Hölder interpolation inequality says that for $u\in C^{l,\alpha }(B_R(x_0))$ and every $\epsilon >0$,More generally, each lower derivative norm can be bounded by an arbitrarily small multiple of the top $C^{l,\alpha }$ seminorm plus a constant multiple of the $C^0$ norm. The powers of $R$ make the inequality invariant under scaling of the ball.

Solved by gpt-5.6-sol high.

For every ${\Omega }^{\prime }⋐\Omega$, the Interior Schauder estimate isThe constant depends on $n$, $\alpha$, the ellipticity constants, the coefficient $C^{0,\alpha }$ norms, $\Omega$, ${\Omega }^{\prime }$, and in particular the distance from ${\Omega }^{\prime }$ to the boundary. It is independent of $u$ and $f$.

Solved by gpt-5.6-sol high.

For $y$ in the closed upper half-space, write $B_r^{+}(y)=B_r(y)\cap \overline{{\mathbb{R}}_{+}^n}$. The boundary Simon absorption lemma has the following scaled form. Let $S$ be a nonnegative set functional that is monotone and subadditive on finite covers by such half-balls. Given $\lambda \ge 0$ and $0<\theta <1$, there is ${\delta }_0={\delta }_0(n,\lambda ,\theta )>0$ such that, if $0<\delta \le {\delta }_0$ and every admissible nested pair satisfiesthenThe same conclusion holds for half-balls centred on the flat boundary and truncated balls meeting it. Iteration over a finite covering absorbs the small first term into the left-hand side.

Solved by gpt-5.6-sol high.

The Boundary Schauder estimate iswhere $C$ depends only on the dimension, $\alpha$, the ellipticity constants, and the coefficient Hölder norms.

Subtract a $C^{2,\alpha }$ extension of $\phi$ to reduce to a function $v$ with zero data on the flat boundary; this changes the forcing by a controlled $C^{0,\alpha }$ term. The key local estimate is that for every $\delta >0$,To prove it, argue by contradiction. A failing normalized sequence has points $x_k,y_k$ at which the second-derivative Hölder quotient stays nonzero. Set $r_k=|x_k-y_k|$, subtract the appropriate second-order Taylor polynomial, and rescale space by $r_k$ and the functions by $r_k^{2+\alpha }[D^2{v}_k{]}_{\alpha }$. Necessarily $r_k\to 0$.

If the rescaled distance to the flat boundary tends to infinity, the domains converge to all of ${\mathbb{R}}^n$; otherwise they converge to a half-space. Coefficient compactness freezes the principal matrix, while the normalized right-hand sides converge locally uniformly to zero. A linear change of variables turns every limiting equation into the Laplace equation. In the whole-space case the Polynomial-growth Liouville theorem for harmonic functions makes the limit a polynomial of degree at most two. In the half-space case the zero boundary data permit odd reflection across the flat boundary, after which the same theorem applies. The subtracted Taylor normalization forces that polynomial to vanish to second order, contradicting the nonzero limiting Hölder oscillation.

Apply the local estimate on all interior balls and boundary half-balls. The interior and boundary forms of the Simon absorption lemma absorb the term multiplied by $\delta$, while the Hölder interpolation inequality absorbs the remaining lower $C^2$ norm into the top seminorm and the $C^0$ norm. Restoring $\phi$ yields the displayed estimate.

Solved by gpt-5.6-sol high.