Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 107 / 2

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.

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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.

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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.

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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.

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