Weak maximum principle for elliptic operators

For a second-order elliptic operator $Lu=a^{ij}{D}_{ij}u+b^i{D}_{i}u+cu$ with $c\le 0$, the inequality $Lu\ge 0$ on a bounded domain impliesAn exponential barrier reduces the non-strict inequality to a contradiction at a positive interior maximum.

Let $h:\mathbb{R}\to \mathbb{R}$ be continuous, odd, strictly increasing and onto. Suppose $u\in C^2({\mathbb{R}}^d)$ tends to zero at infinity and satisfies $-\Delta u+h(u)=f$, with bounded $f$. At a positive maximum $m$ of $u$, $-\Delta u\ge 0$, so $h(m)\le f\le \Vert f{\Vert }_{\infty }$. Apply the same argument to $-u$ to obtain the displayed bound, using the maximum-point argument of the weak maximum principle for elliptic operators. For $h(s)=s+\sin s$, strict monotonicity follows by integrating $h^{\prime }=1+\cos s$, whose zeros are isolated. Thus $\Vert u{\Vert }_{\infty }\le h^{-1}(\Vert f{\Vert }_{\infty })\le \Vert f{\Vert }_{\infty }+1$. The stronger bound by $\Vert f{\Vert }_{\infty }$ holds when that norm is at most $\pi$, but need not hold for larger sources.

If $c\le 0$, $\Delta u+cu=f$, and $u=\psi$ on the boundary of a bounded domain inside $\{|x_1|<d\}$, let $M={\sup }_{\partial \Omega }|\psi |$ and $F={\sup }_{\Omega }|f|$. The nonnegative barrier $v=M+(e^{2d}-e^{x_1+d})F$ satisfies $(\Delta +c)v\le -F$. Apply the weak maximum principle for elliptic operators to $u-v$ and $-u-v$ to get $\Vert u{\Vert }_{\infty }\le M+e^{2d}F$.

For a locally uniformly elliptic operator $L=a^{ij}{D}_{ij}+b^i{D}_i+c$ with locally bounded coefficients and $c\le 0$, a $C^2$ function satisfying $Lu\ge 0$ on a connected domain cannot attain a nonnegative global maximum inside unless it is constant. If $c=0$, the maximum can have either sign. If the maximum set were proper, choose an interior ball in its complement tangent to that set. The Hopf boundary point lemma gives a nonzero normal derivative at the contact point, contradicting the vanishing gradient at an interior maximum.

The strong minimum principle for elliptic operators follows by applying this to $-u$. For nonnegative $u$ with $Lu\le 0$, replacing $c$ by $\min (c,0)$ retains the inequality, so the strict positivity-or-zero conclusion does not require an original sign restriction on $c$.

For $Lu=a^{ij}{D}_{ij}u+b^i{D}_{i}u$ with $a^{ij}{\xi }_i{\xi }_j\ge \lambda |\xi {|}^2$ and $|b^1|<\lambda \ell$, the function $q(x)=e^{-\ell x_1}$ satisfiesIf $Lu\ge 0$ had a strict interior maximum above the boundary maximum, then $u+\epsilon q$ would retain an interior maximum for small $\epsilon >0$, where ellipticity gives $L(u+\epsilon q)\le 0$. This contradicts its strictly positive image under $L$.

A positive zeroth-order coefficient can destroy the weak maximum principle. On $(0,1)$, the function $u(x)=\sin (\pi x)$ has zero boundary values and a positive interior maximum, but

The Alexandrov–Bakelman–Pucci estimate bounds the positive maximum of a function by its boundary maximum and an $L^n$ norm of the negative part of a uniformly elliptic operator applied to it. It yields a maximum principle and a sup-norm estimate for elliptic equations with measurable principal coefficients.

Let $L$ be uniformly elliptic with bounded coefficients and nonpositive zeroth-order coefficient. If $u\ge 0$, $Lu\le 0$, and $u$ vanishes at an interior point of a connected domain, then $u$ vanishes identically. This is the minimum form of the strong maximum principle.