The strong maximum principle is a result for elliptic partial differential equations that forces a function attaining an interior maximum to be constant, under the appropriate differential inequality and sign hypotheses.
Let 
be a bounded, connected open
set in 
, and consider
 |
Assume the coefficients are bounded and continuous, 
, and the principal part is uniformly elliptic,
meaning that a constant 
satisfies
 |
for every 
and 
. If 
, 
, and 
, then attaining its nonnegative maximum
over 
at an interior point forces 
to be constant throughout 
. If 
,
the maximum need not be nonnegative (Gilbarg and Trudinger
2001, Carlotto 2021). Here 
means that the second partial
derivatives are continuous.
For 
and 
continuous on the closure
of 
, the weak maximum principle bounds 
by its boundary values. The strong
principle adds the rigidity conclusion at an interior maximum.
Applied to 
, it gives the corresponding minimum
principle for 
. A harmonic
function is a basic example, with 
the Laplacian.
