Hopf lemma

Let Ω be a bounded domain in Rⁿ with smooth boundary. Let f be a real-valued function continuous on the closure of Ω and harmonic on Ω. If x is a boundary point such that f(x) > f(y) for all y in Ω sufficiently close to x, then the (one-sided) directional derivative of f in the direction of the outward pointing normal to the boundary at x is strictly positive.

Subtracting a constant, it can be assumed that f(x) = 0 and f is strictly negative at interior points near x. Since the boundary of Ω is smooth there is a small ball contained in Ω the closure of which is tangent to the boundary at x and intersects the boundary only at x. It is then sufficient to check the result with Ω replaced by this ball. Scaling and translating, it is enough to check the result for the unit ball in Rⁿ, assuming f(x) is zero for some unit vector x and f(y) < 0 if |y| < 1.

By Harnack's inequality applied to −f

${\displaystyle \displaystyle {-f(rx)\geq -{1-r \over (1+r)^{n-1}}f(0),}}$

for r < 1. Hence

${\displaystyle \displaystyle {{f(x)-f(rx) \over 1-r}={-f(rx) \over 1-r}\geq -{1 \over (1+r)^{n-1}}f(0)>-{f(0) \over 2^{n-1}}>0.}}$

Hence the directional derivative at x is bounded below by the strictly positive constant on the right hand side.

Consider a second order, uniformly elliptic operator of the form

${\displaystyle Lu=a_{ij}(x){\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}+b_{i}(x){\frac {\partial u}{\partial x_{i}}}+c(x)u,\qquad x\in \Omega .}$

Here ${\displaystyle \Omega }$ is an open, bounded subset of ${\displaystyle \mathbb {R} ^{n}}$.

The Weak Maximum Principle states that a solution of the equation ${\displaystyle Lu=0}$ in ${\displaystyle \Omega }$ attains its maximum value on the closure ${\displaystyle {\overline {\Omega }}}$ at some point on the boundary ${\displaystyle \partial \Omega }$. Let ${\displaystyle x_{0}\in \partial \Omega }$ be such a point, then necessarily

${\displaystyle {\frac {\partial u}{\partial \nu }}(x_{0})\geq 0,}$

where ${\displaystyle \partial /\partial \nu }$ denotes the outer normal derivative. This is simply a consequence of the fact that ${\displaystyle u(x)}$ must be nondecreasing as ${\displaystyle x}$ approach ${\displaystyle x_{0}}$. The Hopf Lemma strengthens this observation by proving that, under mild assumptions on ${\displaystyle \Omega }$ and ${\displaystyle L}$, we have

${\displaystyle {\frac {\partial u}{\partial \nu }}(x_{0})>0.}$

A precise statement of the Lemma is as follows. Suppose that ${\displaystyle \Omega }$ is a bounded region in ${\displaystyle \mathbb {R} ^{2}}$ and let ${\displaystyle L}$ be the operator described above. Let ${\displaystyle u}$ be of class ${\displaystyle C^{2}(\Omega )\cap C^{1}({\overline {\Omega }})}$ and satisfy the differential inequality

${\displaystyle Lu\geq 0,\qquad {\textrm {in}}~\Omega .}$

Let ${\displaystyle x_{0}\in \partial \Omega }$ be given so that ${\displaystyle 0\leq u(x_{0})=\max _{x\in {\overline {\Omega }}}u(x)}$. If (i) ${\displaystyle \Omega }$ is ${\displaystyle C^{2}}$ at ${\displaystyle x_{0}}$, and (ii) ${\displaystyle c\leq 0}$, then either ${\displaystyle u}$ is a constant, or ${\displaystyle {\frac {\partial u}{\partial \nu }}(x_{0})>0}$, where ${\displaystyle \nu }$ is the outward pointing unit normal, as above.

The above result can be generalized in several respects. The regularity assumption on ${\displaystyle \Omega }$ can be replaced with an interior ball condition: the lemma holds provided that there exists an open ball ${\displaystyle B\subset \Omega }$ with ${\displaystyle x_{0}\in \partial B}$. It is also possible to consider functions ${\displaystyle c}$ that take positive values, provided that ${\displaystyle u(x_{0})=0}$. For the proof and other discussion, see the references below.

• Hopf maximum principle

• Evans, Lawrence (2000), Partial Differential Equations, American Mathematical Society, ISBN 0-8218-0772-2
• Fraenkel, L. E. (2000), An Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge University Press, ISBN 978-0-521-461955
• Krantz, Steven G. (2005), Geometric Function Theory: Explorations in Complex Analysis, Springer, pp. 127–128, ISBN 0817643397
• Taylor, Michael E. (2011), Partial differential equations I. Basic theory, Applied Mathematical Sciences, 115 (2nd ed.), Springer, ISBN 9781441970541 (The Hopf lemma is referred to as "Zaremba's principle" by Taylor.)

• Hayk Mikayelyan, Henrik Shahgholian Hopf's lemma for a class of singular/degenerate PDE-s
• Hopf's lemma for a class of fractional singular/degenerate PDE-s
• D. E. Apushkinskaya, A. I. Nazarov A counterexample to the Hopf-Oleinik lemma (elliptic case)