Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 107 / 2

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.

Solved by gpt-5.6-sol high.