Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 107 / 2

The Lagrangian isThe Euler-Lagrange equation for $X$ isExpanding the divergence and simplifying givesThe equation for $Y$ is

For compactly supported smooth variations $(\varphi ,\psi )$, differentiating at $t=0$ givesThis is the weak formulation. Taking $\psi =0$ or $\varphi =0$ separates its two equations. When $X,Y$ are smooth, integration by parts transfers each derivative from the test function; the fundamental lemma of the calculus of variations then recovers exactly the two pointwise equations in part (a).

Take a minimizing sequence $(X_k,Y_k)$ in the affine Sobolev class $W$. The standing bounds $C^{-1}\le X_k\le C$ make the energy uniformly equivalent toThe fixed boundary values and the Poincare inequality therefore bound the sequence in $H^1(\Omega ;{\mathbb{R}}^2)$. By weak compactness in a reflexive Banach space, a subsequence converges weakly to $(X,Y)$. The assumed weak closedness keeps the limit in $W$, and the assumed weak lower semicontinuity givesThus the direct method in the calculus of variations produces a minimizer. Its first variation vanishes in every compactly supported direction, so part (b) makes it a weak solution of the system.

Let $u\in H_{\mathrm{loc}}^1(\Omega )$ weakly solvewhere $a^{ij},F^i\in C^{0,\alpha }$, $g\in L^p$ for a suitable $p>n/(1-\alpha )$, and $a^{ij}$ is uniformly elliptic. The interior divergence-form Schauder estimate states that for ${\Omega }^{\prime }⋐{\Omega }_1⋐\Omega$,For the homogeneous equation the final two terms vanish.

Put $h=\log X$. The first equation becomeswhile the second remainsThe assumed $C^{0,\alpha }$ regularity and the bounds away from zero make $X^{-2}$ a $C^{0,\alpha }$ uniformly elliptic coefficient. The divergence-form $C^{1,\alpha }$ estimate from part (d) first gives $Y\in C_{\mathrm{loc}}^{1,\alpha }$. Hence the right side of the equation for $h$ is $C^{0,\alpha }$, and the Interior Schauder estimate gives $h$, and therefore $X$, in $C_{\mathrm{loc}}^{2,\alpha }$.

Expanding the $Y$ equation givesThe higher-order Schauder estimates now alternate between the equations for $h$ and $Y$, gaining derivatives at each step. Induction yields $(X,Y)\in C^{\infty }({\Omega }^{\prime };H)$ for every ${\Omega }^{\prime }⋐\Omega$.