The Hopf boundary point lemma, also called the Hopf lemma, gives a strict normal derivative inequality at a boundary maximum of a solution of an elliptic differential inequality.
Let 
be a bounded open set, and let
 |
with bounded continuous coefficients. Assume 
and uniform ellipticity, so that 
for some constant 
and every 
, 
.
Suppose 
, 
, and 
satisfies 
for every 
.
If the boundary is continuously differentiable near
and an open ball contained in 
is tangent to the boundary
at 
, then
 |
where 
is the outward unit normal vector at 
(Gilbarg and Trudinger 2001, Carlotto 2021). The normal derivative
is the directional derivative along 
. Thus a nonconstant solution cannot have zero outward
normal derivative at such a boundary maximum.
If the operator includes an additional term 
with 
,
the same conclusion holds when 
.
Replacing 
by 
gives a negative outward normal derivative at a strict boundary minimum
when 
. With 
,
this version requires 
.
