Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 107 / 2

Uniform ellipticity means that there is $\lambda >0$ such that the symmetric part of the principal coefficient matrix satisfiesfor every $x\in \Omega$ and $\xi \in {\mathbb{R}}^n$. This is the defining coercive bound for a uniformly elliptic operator.

Assume $c\le 0$, the coefficients are bounded, and $u$ is bounded above. Choose $\kappa$ so large that the bounded positive function $q(y)=e^{\kappa y}$ satisfiesAlso set $\psi (x)=\log (1+|x{|}^2)$. Boundedness of the coefficients, $x\cdot D\psi \le 2$, and $c\le 0$ give a global upper bound $L\psi \le C$.

For $\epsilon >0$ and $0<\delta <\epsilon m/C$, the functiontends to $-\infty$ as $|x|\to \infty$ and satisfies $Lv>0$. If $v$ exceeded both zero and its values on $y=\pm 1$, it would attain a positive interior maximum. At that point $Dv=0$ and $D^{2}v\le 0$, whence $Lv\le cv\le 0$, a contradiction. Letting $\delta \downarrow 0$ and then $\epsilon \downarrow 0$ provesThis is the weak maximum principle for elliptic operators on the slab. The boundedness or a comparable growth condition is necessary because the domain is unbounded.

The weak maximum principle for elliptic operators fails without a condition at infinity. The functionis harmonic on the upper half-space, continuous on its closure, and vanishes on the boundary, but it is positive and unbounded in the interior.

For $n\ge 3$, the radial functionis harmonic on ${\mathbb{R}}^n\setminus \overline{B_1(0)}$, vanishes on the unit sphere, and is positive in the domain. It therefore violates the weak maximum principle for elliptic operators. In two dimensions the corresponding counterexample is $u(x)=\log |x|$, since the fundamental solution of the Laplace equation changes from a power to a logarithm.

TakeIts principal symbol is ${\xi }^4$, so it is elliptic, and $a_4{a}_0=-1<0$. Yet $u(x)=\sin x$ satisfies $Lu=0$ on $(0,\pi )$ and vanishes at both boundary points while remaining positive inside. Thus second-order ellipticity is essential to the usual weak maximum principle for elliptic operators.