On Cordes and Campanato Conditions

On Cordes and Campanato Conditions
Antonio Tarsia
∗
Keywords: nonvariational elliptic equations; Cordes condition; Campanato
condition.
AMS 1991 Subject classifications: 35j60; 35j25
Let Ω be an open bounded set in Rn with a sufficiently regular boundary,
and let A(x) = {aij }i,j=1,···,n be a real matrix, with coefficients aij ∈ L∞ (Ω).
We consider the following problem:

1,2
 u ∈ H 2,2 ∩ H0 (Ω)
(1)
 Pn
a.e. in Ω.
i,j=1 aij (x)Di Dj u(x) = f (x),
If f ∈ L2 (Ω), it is known1 that problem (1) is not well posed with only
hypothesis of uniform ellipticity on the matrix A(x):
there exists a positive constant ν such that
n
X
aij (x)ηi ηj ≥ νkηk2Rn ,
a.e. in Ω,
∀η = (η1 , . . . , ηn ) ∈ Rn .
(2)
i,j=1
When n > 2 , the proof of existence and uniqueness of the solution for
problem (1.1) needs hypotheses on A(x) stronger than uniform ellipticity. In this
paper we compare some of these hypotheses: namely Campanato and Cordes
ones. We display below these conditions.
Condition 1 (Condition of Cordes, see [10], [15]).
Let A(x) = {aij (x)}i,j=1,···,n be a matrix such that kA(x)k n2 6= 0 , a.e. in
R
Ω. We say that A(x) satisfies the Condition of Cordes if there exists ε ∈ (0, 1)
such that
Pn
2
( i=1 aii (x))
Pn
≥ n − 1 + ε, a.e. in Ω.
(3)
2
i,j=1 aij (x)
Condition 2 (Condition A, see [2]).
∗ Dipartimento di Matematica, Università di Pisa, via F. Buonarroti, 2. 56127 PISA,
ITALY. e-mail: [email protected]
1 See the counterexamples in [11] and in [14].
1
There exist three real constants α, γ, δ with α > 0, γ > 0, δ ≥ 0 and γ +δ < 1
such that:

1/2
n
n
n
n
X X
X
X
2
ξii − α
aij (x)ξij ≤ γ 
ξij
+δ
ξii ,
(4)
i=1
i,j=1
i,j=1
i=1
2
∀ξ = {ξij }i,j=1,···,n ∈ Rn , a.e. in Ω.
Condition 3 (Condition Ax , see [2] and [5]).
There exist three real constants σ, γ, δ and a function a(x) ∈ L∞ (Ω), with
σ > 0 , γ > 0, δ ≥ 0, γ + δ < 1, a(x) ≥ σ > 0 , such that

1/2
n
X
n
n
X X
X
n
2
ξii − a(x)
aij (x)ξij ≤ γ 
ξij  + δ ξii ,
i=1
i,j=1
i,j=1
i=1
(5)
2
∀ξ = {ξij }i,j=1,···,n ∈ Rn , a.e. in Ω.
In my paper I will show that Condition Ax is equivalent to the Cordes
Condition (see Section 3), while Condition A is stronger than Condition Ax ,
even if n = 2 (see Section 2). If n = 2, we will show that Condition Ax is
equivalent to the uniform ellipticity (see Section 2). So if n = 2 a matrix is
unformly elliptic if and only if satisfies Cordes Condition. The usefulness of
these results is in the fact that to verify when a matrix satisfies Condition Ax
can be very hard, while it is quite easy to verify whether a matrix satisfies Cordes
Condition. On the other side Condition Ax is useful for showing in a simple
manner, namely by means of Campanato Theory of Near Operators (see [2] and
[7] ), existence and uniqueness (and also regularity) of solution of Problem (1);
the reader can also see [12], for more results and details about the theory of
nonvariational elliptic equations .
References
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