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