Introduction
Background. The classical maximum modulus principles are fundamental
properties of solutions to partial differential equations. They have numerous im-
portant applications to various problems in the theory of these equations, both
linear and nonlinear.
Let Ω be a bounded domain in the Euclidean space Rn. For the elliptic equation
(1)
n
j,k=1
ajk(x)
∂2u
∂xj∂xk

n
j=1
aj(x)
∂u
∂xj
a0(x)u = 0 , x Ω,
with bounded coefficients, positive-definite matrix ((ajk(x))), and with a0(x) 0,
two classical facts are basic: the weak and strong maximum modulus principles.
By the weak one, a solution u C2(Ω) C(Ω) to equation (1) satisfies
max
Ω
|u| = max
∂Ω
|u| .
According to the strong principle the maximum modulus of a non-constant solution
u is never attained inside Ω. Note that the last property is not dealt with in the
present book.
The weak and strong maximum modulus principles hold also for the parabolic
equation
(2)
∂u
∂t

n
j,k=1
ajk(x, t)
∂2u
∂xj∂xk
+
n
j=1
aj(x, t)
∂u
∂xj
+ a0(x, t)u = 0
with bounded coefficients, positive-definite matrix ((ajk(x, t))), and with a0(x, t)
0 in a cylinder QT = Ω × (0,T ]. By the weak maximum modulus principle, if
u C(2,1)(QT ) C(QT ) is a solution of (2), then
max
QT
|u| = max
ΓT
|u| ,
where ΓT = {(x, t) ∂QT : 0 t T }.
The maximum principles just mentioned were obtained for different classes of
solutions and under various assumptions about the coefficients, but are valid almost
exclusively for scalar equations of the second order (with weakly coupled systems
as the only exception).
The not so sharp but incomparably more general property of the same nature,
which holds for equations and systems of arbitrary order in smooth domains, is the
so-called Miranda-Agmon maximum principle. In particular, for a homogeneous
elliptic equation of order 2 , this principle is the estimate
(3) max
Ω
|∇
−1
u| c(Ω) max
∂Ω
|∇
−1
u|.
1
http://dx.doi.org/10.1090/surv/183/01
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