Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 107 / 3

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.

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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 $\phi$.

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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.

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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 $C^{1,\beta }(\overline{\Omega })$.

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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.

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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.

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