# Harnack Inequalities and ABP Estimates for Nonlinear Second-Order Elliptic Equations in Unbounded Domains

Authors: M. E. Amendola; L. Rossi; A. Vitolo
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

We are concerned with fully nonlinear uniformly elliptic operators with a superlinear gradient term. We look for local estimates, such as weak Harnack inequality and local maximum principle, and their extension up to the boundary. As applications, we deduce ABP-type estimates and weak maximum principles in general unbounded domains, a strong maximum principle, and a Liouville-type theorem.


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## Body

## 1. Introduction

The qualitative
theory of second-order  elliptic equations received a strong effort from Harnack
inequalities. Here, we will make use of this powerful technique to study
continuous viscosity solutionsu of fully nonlinear elliptic equations (
F
=
f
):(1.1)
F
(
x
,
u
(
x
)
,
D
u
(
x
)
,
D
2
u
(
x
)
)
=
f
(
x
)
,

x
∈
Ω
,in unbounded domains Ω of R
n,
where F is a real function of x
∈
Ω, t
∈
R, p
∈
R
n and  X
⁢ in the set      𝒮
n of n
×
n real symmetric matrices.We recall thatF is (degenerate) elliptic if F is nondecreasing in X and uniformly elliptic if there exist (ellipticity) constants λ and Λ such that 0
<
λ
≤
Λ and(1.2)
λ
tr
(
Y
)
≤
F
(
x
,
t
,
p
,
X
+
Y
)
−
F
(
x
,
t
,
p
,
X
)
≤
Λ
tr
(
Y
),for (1.2)
Y
≥
0,
that is (1.2)
Y is semidefinite positive, where (1.2)
tr
(
Y
) denotes the trace of the matrix (1.2)
Y.In the class of uniformly elliptic operators, there
are two extremal ones, well known as Pucci maximal and minimal operators,
respectively:(1.2)
P
λ
,
Λ
+
(
X
)
=
Λ
tr
(
X
+
)
−
λ
tr
(
X
−
)
,
P
λ
,
Λ
−
(
X
)
=
λ
tr
(
X
+
)
−
Λ
tr
(
X
−
)
,where X
± are the positive and negative parts of X,
which can be decomposed in a unique way as X
=
X
+
−
X
− with X
±
≥
0 and X
+
X
−
=
0.
Other examples of fully nonlinear uniformly elliptic operators can be found in
[1–3].Throughout this paper, we will consider elliptic
operators with the structure conditions(1.4)
F
(
x
,
t
,
p
,
X
)
≥
P
λ
,
Λ
−
(
X
)
−
b
(
x
)
|
p
|
q
,
(1.5)
F
(
x
,
t
,
p
,
X
)
≤
P
λ
,
Λ
+
(
X
)
+
b
(
x
)
|
p
|
q
,where 𝒫
± are the extremal Pucci operators, b
(
x
) is a continuous function and the exponent q
∈
[
1
,
2
],
so that the gradient term can have a superlinear, at most quadratic growth.Remark 1.1.
The above structure conditions
are exactly equivalent to the uniform ellipticity whenF is linear in the variable X
∈
𝒮
n.
In the nonlinear case they allow a slight generalization. Let us consider, for 0
<
λ
<
Λ and t
≥
0,
the function(1.6)
h
(
t
)
=
{
Λ
t
,
if
0
≤
t
≤
1
Λ
,
1
,
if
1
Λ
<
t
≤
1
λ
,
λ
t
,
if
t
>
1
λ
,then the operator F
=
h
(
tr
(
X
+
)
)
−
h
(
tr
(
X
−
)
) is elliptic and satisfies both the conditions
(1.4) and (1.5), even that it is not uniformly elliptic.
However, if (1.5) (resp., (1.4)) holds, then
subsolutions (resp., supersolutions) of the equation F
=
f are subsolutions (resp., supersolutions) of
uniformly elliptic equations, and this is needed to prove our results.We will be
concerned principally with the following topics in unbounded domains; see
[4–6] for classical results.[MP] maximum
principle for u.s.c. subsolutionsw of F
=
0 in the viscosity sense (v.s.), in the form(1.7)
F
≥
0
in
Ω
,
w
≤
0
on
∂
Ω
,
sup
⁡
Ω
w
<
+
∞
⟹
w
≤
0
in
Ω
;[LT] Liouville theorem for
continuous solutions ofF
=
0 v.s., in the form(1.8)
F
=
0
in
R
n
,
sup
⁡
R
n
w
<
+
∞
⟹
w
=
constant
in
R
n
.Concerning MP, it is worth to
note that the condition from above on the size of w can be weakened in the framework of the
Phragmén-Lindelöf theory (see, e.g., [7–9]) but not omitted at all, even for classical
subsolutions (see, e.g., [4, 10]). It is also well known that MP fails to hold in
general in exterior domains. In fact, due to the boundedness of the fundamental
solution u
(
x
)
=
|
x
|
2
−
n of the Laplace equation Δ
u
=
0,
the function w
=
1
−
u provides a counterexample to MP in Ω
=
R
n
∖
B
¯
1
(
0
).
Thus we introduce a local measure-geometric condition G
σ in Ω at y
∈
R
n,
which depends on the real parameter σ
∈
(
0
,
1
) : there exists a
ballB
=
B
(
y
) such that
(1.9)
y
∈
B
,

|
B
∖
Ω
y
|
≥
σ
|
B
|
,
whereΩ
y is the connected component of B
∖
∂
Ω containing y.IfG
σ is satisfied in Ω at all y
∈
Ω,
we simply say that Ω is a w
G domain (with parameter σ). This is a generalization of condition G of Cabré [10], which ultimately goes back to Berestycki et al. [11].LetR
(
y
) denote the radius of the ball B
=
B
(
y
) provided by condition w
G.
We will call domains of cylindrical
and conical type the w
G domains such that R
(
y
)
=
O
(
1
) and R
(
y
)
=
O
(
|
y
|
) as |
y
|
→
+
∞,
respectively. Examples of the first kind are domains with finite measure,
cylinders, slabs, complements of a periodic lattice of balls, whereas cones,
and complements, in the plane, of logarithmic spirals, are examples of the
second kind.In [12], it is shown that MP holds
true in a w
G domain for strong solutions of a linear
second-order uniformly elliptic operator F
=
tr
A
(
x
)
X;
see also [13, 14] for earlier
results and [15, 16] for viscosity
solutions of a fully nonlinear operator with linear and quadratic growth in the
gradient (i.e., in the case of the structure condition (1.5) with q
=
1 and q
=
2) provided that b
(
x
)
=
O
(
1
/
|
x
|
) and b
(
x
)
=
O
(
1
) as |
x
|
→
∞,
respectively.With the aim to find conditions on the coefficientb
(
x
) such that MP holds in w
G domains when 1
≤
q
≤
2,
our result is the following.Theorem 1.2 (MP).
Let0
<
σ
<
1 and 1
≤
q
≤
2.
Let Ω be a domain of R
n satisfying condition w
G or alternatively such that, for a closed
subset H of R
n,
(i)
M
P holds in each connected component of Ω
∖
H;
(ii)
conditionG
σ is satisfied in Ω at each y
∈
Ω
∩
H.
Suppose that w
∈
USC
(
Ω
¯
) is a viscosity solution of F
(
x
,
w
,
D
w
,
D
2
w
)
≥
0 and structure condition (1.5) holds with b
∈
C
(
Ω
¯
),
such that b
(
x
)
=
O
(
1
/
|
x
|
2
−
q
).
Ifw
≤
0 on ∂
Ω and sup
⁡
w
<
+
∞ in Ω,
then w
≤
0 in Ω.This yields indeed MP in a wider class of domains
thanw
G,
for example, the cut plane and more generally the complement of continuous
semi-infinite curves in R
2 and their generalizations to hypersurfaces in R
n.We also outline that the limit casesq
=
1 and q
=
2 of the above mentioned papers are obtained by
continuity from the intermediate cases 1
<
q
<
2,
as it follows from Theorem 1.2. Nonetheless, there are technical improvements
with respect to the previous works even in the limit cases.Consider in particular aparabolic shaped domain Ω,
satisfying condition w
G with R
(
y
)
=
O
(
|
y
|
α
), 0
<
α
<
1;
the limit cases α
=
0 and α
=
1 correspond to domains of cylindrical and
conical types, respectively.Based on an argument of [12], eventually passing to a
smaller r
y
≤
R
(
y
),
we can suppose that condition G
σ is satisfied with |
B
∖
Ω
y
|
=
σ
|
B
| exactly. We get the new following variant of
ABP estimate.Theorem 1.3 (ABP).
Let0
<
σ
,
τ
<
1, τ
′
>
1, R
0
,
β
≥
0, 1
≤
q
≤
2, and N
>
0.
Let Ω be a w
G domain, such that condition G
σ in Ω is fulfilled at each y
∈
Ω with R
(
y
)
≤
R
0
+
β
|
y
|
α, 0
≤
α
≤
1.
Assume that F satisfies the structure condition (1.5), with b
,
f
∈
C
(
Ω
¯
) and b
0
:
=
sup
⁡
Ω
|
b
(
x
)
|
(
1
+
|
x
|
α
(
2
−
q
)
)
<
+
∞.
Ifw
∈
USC
(
Ω
¯
) is a viscosity solution of F
(
x
,
w
,
D
w
,
D
2
w
)
≥
f such that w
≤
N in Ω and w
≤
0 on ∂
Ω,
then(1.10)
sup
⁡
Ω
w
≤
C
lim
⁡
ε
→
0
+
sup
⁡
y
∈
Ω
;
|
y
|
≥
ε
r
y
r
y
∥
f
−
∥
L
n
(
Ω
∩
B
τ
ε
r
y
,
τ
′
r
y
)
,where C is a positive constant depending on n, q, λ, Λ, b
0
N
q
−
1, σ, τ, τ
′, R
0, β.Note that in the case of a domain of cylindrical
type (α
=
0), it is sufficient to have b
(
x
)
=
O
(
1
),
for all q
∈
[
1
,
2
],
as well as in the case of a quadratic growth in the gradient variable (q
=
2), for all α
∈
[
0
,
1
].This result extends the previous ones contained in
[10, 14] for the linear case, and
[8, 16], dealing with fully
nonlinear equations, in the limit situations of cylindrical/conical domains
and linear/quadratic gradient terms.Remark 1.4.
In general, unlessq
=
1,
the above ABP type estimate is different from the so-called ABP maximum
principles since C depends on the upper bound N of w if b
0
>
0 and q
>
1.
For ABP-type estimates of this kind in bounded domains we refer to [17]. Counterexamples to the ABP
maximum principle can been found in [17–19].Consider nowΩ
=
R
n.
The classical Liouville theorem says that harmonic functions in the entire R
n,
which are bounded either above or below, are constant. This result continues to
hold for strong solutions of quasilinear uniformly elliptic equations; see
[20]. For viscosity
solutions of fully nonlinear uniformly elliptic equations with an additive
gradient term having linear growth, we refer to [21, 22]. Our result is the
following.Theorem 1.5 (Liouville theorem).
Letw
∈
C
(
R
n
) be such that F
(
x
,
w
,
D
w
,
D
2
w
)
=
0 in the viscosity sense, and assume structure
conditions (1.4) and (1.5), with b
∈
C
(
R
n
) such that b
(
x
)
=
O
(
1
/
|
x
|
2
−
q
) as |
x
|
→
+
∞.
If w is bounded either above or below, then w is constant.Remark 1.6.
Under some additional
assumptions, Liouville-type results also hold in un-bounded domains ofR
n containing balls of arbitrary large radius;
see [23].Our main tools
are Krylov-Safonov Harnack inequalities and local MP; see [20] for strong solutions of
quasilinear uniformly elliptic equations. For viscosity solutions and F satisfying the structure condition (1.4), they
can be found in [3]
if b
=
0 and in [24] if q
=
2;
see also [25]. In the
case of linear or superlinear, almost quadratic, growth in the gradient (1
≤
q
<
2), weak Harnack (wH) inequality and local MP
can be deduced using arguments of [17], in which a (full) Harnack inequality has been
established for L
p viscosity solutions; see also [26].Nevertheless, for convenience of the reader we
believed that it is worth to report systematically on this kind of inequalities
in Section3.As the previous ones, our approach follows the lines
of [3], based on the
methods of [27, 28] and on the ABP
maximum principle for viscosity solutions in bounded domains, due to Caffarelli
[29].Remark 1.7.
In deriving wH inequality and local MP, we only need the
Alexandroff-Bakelman-Pucci (ABP) estimate withq
=
1 and f continuous, so [30, Proposition 2.12] and also
[17, Theorem 4.1] in
the case of linear growth in the gradient term, are sufficient to our purpose.
But we notice that new ABP-type estimates have been established for L
p-viscosity solutions of equations with
discontinuous coefficients by Koike and Święch [19, 27] for q
∈
[
1
,
2
] and f
∈
L
p.Remark 1.8.
In the case of a superlinear
first-order term, wH inequality and local MP are obtained by interpolation
between the linear and quadratic cases, eliminating the square gradient term by
means of an exponential transformation used before by Trudinger [24], see Lemmas 3.1 and 3.2 below.
This kind of ideas have been also considered by Sirakov in [31].What we
definitely need are, for MP, the scaled boundary wH inequality (3.16), derived
in Section 3 by means of typical viscosity methods, and, for technical reasons,
its version in annular regions (3.24), and, for LT, the scaled Harnack
inequality (3.11). Moreover, using the interior wH inequality (3.7) and assuming
the structure condition (1.5), we also state a strong MP theorem, according to
which a subsolution u of equation F
=
0 cannot achieve a positive maximum inside any
domain (open connected set) of R
n unless it is
constant; see Theorem 5.1 below. For a different approach, based
on Hopf lemma, and more general versions see [32].The paper is
organized as follows. In Section2, we recall some basic results of elliptic
theory for viscosity solutions of second-order fully nonlinear equations with a
linear gradient term; in Section 3, we extend local maximum principle and weak
Harnack inequality, even up to the boundary, to the case of a superlinear
gradient term; these results are applied in Section 4 to get
Alexandroff-Bakelman-Pucci-type estimates and maximum principles, with the
proof of Theorems 1.2 and 1.3; finally, a strong maximum principle is derived and
the proof of Liouville theorem (Theorem 1.5) is given in Section 5. In the
appendix, for the sake of completeness, we show the basic weak Harnack
inequality and local MP for a uniformly elliptic operator with an additive
first-order term having linear growth in the gradient.


## 2. Basic Estimates (Linear Gradient Term)

LetΩ be a domain of R
n,
and denote by USC
(
Ω
) and LSC
(
Ω
),
respectively, the sets of the upper and lower semicontinuous functions in Ω.
The function u
∈
USC
(
Ω
) is said to be a viscosity subsolution of F
=
f if(2.1)
F
(
x
,
u
(
x
)
,
D
φ
(
x
)
,
D
2
φ
(
x
)
)
≥
f
(
x
)at any point x
∈
Ω and for all φ
∈
C
2
(
Ω
) such that φ
−
u has a local minimum in x.
Similarly, a viscosity supersolution u
∈
LSC
(
Ω
) of F
=
f satisfies(2.2)
F
(
x
,
u
(
x
)
,
D
φ
(
x
)
,
D
2
φ
(
x
)
)
≤
f
(
x
)at any point x
∈
Ω and for all φ
∈
C
2
(
Ω
) such that u
−
φ has a local minimum in x.We may also assume thatφ
(
x
)
=
u
(
x
) in the above definition, that is the graph of
the test function φ touches that one of u from above for subsolutions and from below for
supersolutions [3]. Moreover,
if F is continuous in the matrix-variable, as for
uniformly elliptic operators, then we may assume that φ
(
x
) is a paraboloid, that is a quadratic
polynomial.We will make
use of the following version of the ABP estimate, in whichΓ
u
+ denotes the upper contact set(2.3)
Γ
u
+
=
{
x
∈
Ω
/
∃
p
∈
R
n
s.t.
u
(
y
)
≤
u
(
x
)
+
p
⋅
(
y
−
x
)
for
x
∈
Ω
}of the graph of the function u.
Using [30, Proposition 2.12] or [17, Theorem 4.1], we have the following.Lemma 2.1 (ABP
estimate).
Letu
∈
LSC
(
B
¯
) be a viscosity supersolution of the
equation(2.4)
P
λ
,
Λ
−
(
D
2
u
)
−
b
0
|
D
u
|
=
fin a ball B of unit radius, such that u
≥
0 on ∂
B,
where f
∈
L
n
(
B
)
∩
C
(
B
),
for some constant b
0
≥
0.
Then(2.5)
sup
⁡
B
u
−
≤
C
∥
f
+
∥
L
n
(
Γ
u
−
+
)
,for a positive constant C
=
C
(
n
,
λ
,
Λ
,
b
0
).
Similarly, if u
∈
USC
(
B
¯
) is a viscosity subsolution of the
equation(2.6)
P
λ
,
Λ
+
(
D
2
u
)
+
b
0
|
D
u
|
=
fsuch that u
≤
0 on ∂
B,
then(2.7)
sup
⁡
B
u
+
≤
C
∥
f
−
∥
L
n
(
Γ
u
+
+
)
.From Lemma2.1, we obtain the following results, see the appendix, which extend  [3, Theorem 4.8, (1) and (2)]; see also [15].Here we denote byB
r a ball centered at x
0
∈
R
n of radius r
>
0.Lemma 2.2 (wH inequality).
Letb
0
≥
0 and 0
<
τ
<
1.
Suppose that u
∈
LSC
(
B
1
/
τ
) is a viscosity supersolution of (2.4), with f
∈
C
(
B
¯
1
/
τ
),
and u
≥
0 in B
1
/
τ.
Then(2.8)
(
1
|
B
1
|
∫
B
1
u
p
0
)
1
/
p
0
≤
C
(
inf
⁡
B
1
u
+
∥
f
+
∥
L
n
(
B
1
/
τ
)
)
,where C and p
0 are positive numbers, depending on n
,
λ
,
Λ
,
b
0
, and τ.Lemma 2.3 (local MP).
Letb
0
≥
0 and 0
<
τ
<
1.
Suppose that u
∈
USC
(
B
1
) is a viscosity subsolution of (2.6) with f
∈
C
(
B
¯
1
).
Then for all p
>
0
,
(2.9)
sup
⁡
B
τ
u
≤
C
(
(
1
|
B
1
|
∫
B
1
(
u
+
)
p
)
1
/
p
+
∥
f
−
∥
L
n
(
B
1
)
)
,where C is a positive constant,  depending on n
,
λ
,
Λ
,
b
0
,
τ
, and p.


## 3. Interior and Boundary Harnack Estimates and Local MP (Superlinear Gradient Term)

Firstly, we extend interior estimates (2.8) and (2.9)
to fully nonlinear operators F with a superlinear first-order term, such
that, respectively, (1.4) and (1.5) hold.Lemma 3.1 (wH inequality).
Letb
0
≥
0, 0
<
τ
<
1
, and 1
≤
q
≤
2.
Suppose that u
∈
LSC
(
B
1
/
τ
) is a viscosity solution of F
(
x
,
u
,
D
u
,
D
2
u
)
≤
f,
under structure condition (1.4) with b
≤
b
0, f
∈
C
(
B
¯
1
/
τ
),
and 0
≤
u
≤
1 in B
1
/
τ.
Then (2.8) holds with positive constants C and p
0,
depending on n
,
λ
,
Λ
,
b
0
,
τ
, and q.Lemma 3.2 (local MP).
Letb
0
≥
0, 0
<
τ
<
1
, and 1
≤
q
≤
2.
Suppose that u
∈
USC
(
B
1
) is a viscosity solution of F
(
x
,
u
,
D
u
,
D
2
u
)
≥
f,
under structure condition (1.5), with b
≤
b
0, f
∈
C
(
B
¯
1
),
and u
≤
1.
Then (2.9) holds for all p
>
0 with a positive constant C,
depending on n
,
λ
,
Λ
,
b
0
,
q
,
τ
, and p.Proof
of Lemmas3.1 and 3.2.
.
We only show the proof of Lemma3.2, since that one of Lemma 3.1
is similar. By the
structure condition (1.5), we have(3.1)
P
λ
,
Λ
+
(
D
2
u
)
+
b
0
|
D
u
|
q
≥
f
(
x
)and also, in the viscosity
sense,(3.2)
P
λ
,
Λ
+
(
D
2
u
+
)
+
b
0
|
D
u
+
|
q
≥
−
f
−
(
x
)
.From this, by Young's inequality,
it follows that(3.3)
P
λ
,
Λ
+
(
D
2
u
+
)
+
b
1
|
D
u
+
|
+
b
2
|
D
u
+
|
2
≥
−
f
−
(
x
)with(3.4)
b
1
=
(
2
−
q
)
b
0
1
/
q
,
b
2
=
(
q
−
1
)
b
0
2
/
q
.Using the transformation u
+
=
(
λ
/
b
2
)
log
⁡
(
1
+
(
b
2
/
λ
)
v
),
then the USC function v
=
(
λ
/
b
2
)
(
exp
⁡
(
(
b
2
/
λ
)
u
+
)
−
1
) satisfies the differential inequality(3.5)
P
λ
,
Λ
+
(
D
2
v
)
+
b
1
|
D
v
|
≥
−
f
−
(
x
)
(
1
+
b
2
λ
v
(
x
)
)in B
1
/
τ.
Therefore, we can apply Lemma 2.3 to the subsolution v.
To conclude the proof of Lemma 3.2, it is sufficient to observe that(3.6)
u
+
≤
v
≤
λ
b
2
(
exp
⁡
(
b
2
λ
)
−
1
)
u
+
.Rescaling variables
and functions, we highlight the dependence on geometric parameters.Theorem 3.3 (scaled wH inequality).
Letb
0
≥
0, 0
<
τ
<
1, N
>
0
, and 1
≤
q
≤
2.
Suppose that u
∈
LSC
(
B
R
/
τ
) is a viscosity solution of F
(
x
,
u
,
D
u
,
D
2
u
)
≤
f,
under structure condition (1.4), with b
≤
b
0, f
∈
C
(
B
¯
R
/
τ
),
and 0
≤
u
≤
N in B
R
/
τ.
Then(3.7)
(
1
|
B
R
|
∫

B
R
u
p
0
)
1
/
p
0
≤
C
(
inf
⁡
B
R
u
+
R
∥
f
∥
L
n
(
B
R
/
τ
)
)
,with positive constants C and p
0,
depending on n
,
λ
,
Λ
,
q
,
τ
, and b
0
N
q
−
1
R
2
−
q.Proof.
Considering, fory
∈
B
1
/
τ,
the function v
(
y
),
defined by u
(
x
)
=
N
v
(
x
/
R
),
we have(3.8)
P
λ
,
Λ
−
(
D
2
v
)
−
b
0
N
q
−
1
R
2
−
q
|
D
v
|
q
≤
R
2
N
−
1
f
+
(
R
y
)
.Thus, applying Lemma 3.1, we get(3.9)
(
1
|
B
1
|
∫

B
1
v
p
0
)
1
/
p
0
≤
C
(
inf
⁡
B
1
v
+
R
2
N
−
1
∥
f
+
(
R
y
)
∥
L
y
n
(
B
1
/
τ
)
)
,with C
=
C
(
n
,
λ
,
Λ
,
q
,
τ
,
b
0
N
q
−
1
R
2
−
q
),
from which the assert follows.Note that
constantsp
0 and C of the above wH inequality depend in general
on the upper bound N for the supersolution and on the radius R of the ball, but in the case q
=
1 there is no dependence on N and in the case q
=
2 no dependence on R.In the same
manner as in Theorem3.3 for wH inequality, we make the dependence on the
geometric constants explicit in the following local MP.Theorem 3.4 (scaled local MP).
Letb
0
≥
0, 0
<
τ
<
1, N
>
0
, and 1
≤
q
≤
2.
Suppose that u
∈
USC
(
B
R
) is a viscosity solution of F
(
x
,
u
,
D
u
,
D
2
u
)
≥
f,
under structure condition (1.5), with b
≤
b
0, f
∈
C
(
B
¯
R
),
and u
≤
N.
Then for all p
>
0
,
(3.10)
sup
⁡
B
τ
R
u
≤
C
(
(
1
|
B
R
|
∫

B
R
(
u
+
)
p
)
1
/
p
+
R
∥
f
−
∥
L
n
(
B
R
)
)
,with a positive constant C,
depending on n
,
λ
,
Λ
,
q
,
τ
,
b
0
N
q
−
1
R
2
−
q,
and p.Combining Theorems3.3 and 3.4, we get the full
Harnack inequality for solutions.Theorem 3.5 (Harnack inequality).
Letb
0
≥
0, 0
<
τ
<
1, N
>
0
, and 1
≤
q
≤
2.
Suppose that u
∈
C
(
B
R
/
τ
) is a viscosity solution of F
(
x
,
u
,
D
u
,
D
2
u
)
=
f in B
R
/
τ,
under structure conditions (1.4) and (1.5), with b
≤
b
0, f
∈
C
(
B
¯
R
/
τ
),
and 0
≤
u
≤
N.
Then(3.11)
sup
⁡
B
R
u
≤
C
(
inf
⁡
B
R
u
+
R
∥
f
∥
L
n
(
B
R
/
τ
)
)
,with a positive constant C
=
C
(
n
,
λ
,
Λ
,
q
,
τ
,
b
0
N
q
−
1
R
2
−
q
).We wish to extend the above estimates up to the
boundary, that is, to balls intersecting the boundary of the domainA
⊂
R
n,
where the solutions are defined. For this purpose we will need suitable
extensions of such solutions outside A.
Precisely, take concentric balls B
τ
R
⊂
B
R
⊂
B
R
/
τ such that B
τ
R
∩
A
≠
∅ and B
R
/
τ
∖
A
≠
∅.
For a nonnegative viscosity supersolution u
∈
LSC
(
A
¯
) of equation F
(
x
,
u
,
D
u
,
D
2
u
)
=
f in A,
we put(3.12)
m
=
inf
⁡
B
R
/
τ
∩
∂
A
u
;

u
m
−
(
x
)
=
{
min
⁡
(
u
(
x
)
,
m
)
,
if
x
∈
A
,
m
,
i
f
x
∉
A
,where 0
<
τ
<
1.
Similarly, for a viscosity subsolution u
∈
USC
(
A
¯
),
we put(3.13)
M
=
sup
⁡
B
R
∩
∂
A
u
+
;

u
M
+
(
x
)
=
{
max
⁡
(
u
+
(
x
)
,
M
)
,
if
x
∈
A
,
M
,
if
x
∉
A
.Denote also by f
0
+ and f
0
− the continuations of f
+ and f
− vanishing outside A,
respectively. Following [3, Proposition 2.8] and using the structure conditions
(1.4) and (1.5), we have(3.14)
P
λ
,
Λ
−
(
D
2
u
m
−
)
−
b
0
|
D
u
m
−
|
q
≤
f
0
+in B
R
/
τ for a viscosity supersolution u
∈
LSC
(
A
¯
),
and(3.15)
P
λ
,
Λ
+
(
D
2
u
τ
+
)
+
b
0
|
D
u
τ
+
|
q
≥
−
f
0
−in B
R for a viscosity subsolution u
∈
USC
(
A
¯
).Observe that, iff
+
=
0 on ∂
A,
then f
0
+ is continuous, and then we can apply Theorem 3.3
to get a boundary wH inequality. Similarly, if f
−
=
0 on ∂
A,
we can use Theorem 3.4 to deduce a boundary local MP.Nevertheless, even in the general case, whenf
0
+ and f
0
− are not necessarily continuous, we can get
boundary estimates by means of an approximation process, as shown here below,
where we use the notations defined just above.Theorem 3.6 (boundary
wH inequality).
Letb
0
≥
0, 0
<
τ
<
1, N
>
0
, and 1
≤
q
≤
2.
Suppose that u
∈
LSC
(
A
¯
) is a viscosity solution of F
(
x
,
u
,
D
u
,
D
2
u
)
≤
f,
under structure condition (1.4), with b
(
x
)
≤
b
0, f
∈
C
(
A
¯
),
and 0
≤
u
≤
N in A.
Then(3.16)
(
1
|
B
R
|
∫

B
R
(
u
m
−
)
p
0
)
1
/
p
0
≤
C
(
inf
⁡
B
R
∩
A
u
+
R
∥
f
∥
L
n
(
B
R
/
τ
∩
A
)
)
,with positive constants C and p
0,
depending on n
,
λ
,
Λ
,
q
,
τ
, and b
0
N
q
−
1
R
2
−
q.Proof.
Forε
>
0
, set(3.17)
m
ε
=
inf
⁡
I
ε
(
∂
A
)
u
,

I
ε
(
∂
A
)
=
{
x
∈
B
R
/
τ
∩
A
¯
:
d
ist
(
x
,
∂
A
)
≤
ε
}
,and, for x
∈
B
¯
R
/
τ,(3.18)
u
m
ε
−
(
x
)
=
{
min
⁡
(
u
(
x
)
,
m
ε
)
,
if
x
∈
A
,
m
ε
,
if
x
∉
A
,
f
ε
(
x
)
=
f
+
(
x
)
ρ
(
d
ist
(
x
,
R
n
∖
A
)
ε
)
,where(3.19)
ρ
(
t
)
=
{
t
,
if
0
≤
t
<
1
,
1,
if
t
≥
1.It is easy to check that u
m
ε
−
∈
LSC
(
B
R
/
τ
), 0
≤
u
m
ε
−
≤
N, f
ε
∈
C
(
B
¯
R
/
τ
)
, and(3.20)
P
λ
,
Λ
−
(
D
2
u
m
ε
−
)
−
b
0
|
D
u
m
ε
−
|
q
≤
f
ε
(
x
)in B
R
/
τ.
Therefore, we can apply Theorem 3.3 with u
m
ε
− instead of u and f
ε instead of f to get(3.21)
(
1
|
B
R
|
∫
B
R
(
u
m
ε
−
)
p
0
)
1
/
p
0
≤
C
(
inf
⁡
B
R
u
m
ε
−
+
R
∥
f
ε
∥
L
n
(
B
R
/
τ
)
)
.Note that inf
⁡
B
R
u
m
ε
−
≤
inf
⁡
B
R
∩
A
u and 0
≤
f
ε
≤
f
+ in A, f
ε
=
0 outside A.
Also observing that, by lower semicontinuity,(3.22)
m
≤
liminf
⁡
ε
→
0
m
εand therefore, by Fatou's lemma,(3.23)
∫

B
R
(
u
m
−
)
p
0
≤
liminf
⁡
ε
→
0
∫

B
R
(
u
m
ε
−
)
p
0
,from inequality (3.21) we get the
assert.In the sequel,
we will make also use of a version of boundary wH inequality for annular
regionsB
ε
R
,
R
=
B
R
∖
B
ε
R
(
0
), 0
<
ε
<
1,
which can be deduced by Theorem 3.6 reasoning as in [10, Theorem 3.1].In this casem
=
inf
⁡
∂
A
∩
B
ε
τ
R
,
τ
′
R
u,
where 0
<
ε
≤
1
/
2, 0
<
τ
<
1, τ
′
>
1.Corollary 3.7 (boundary wH inequality).
Let0
<
τ
<
1, τ
′
>
1, N
>
0
, and 1
≤
q
≤
2.
Suppose that u
∈
LSC
(
A
¯
) is a viscosity solution of F
(
x
,
u
,
D
u
,
D
2
u
)
≤
f,
under structure condition (1.4), with f
∈
C
(
A
¯
),
and 0
≤
u
≤
N in A.
Then(3.24)
(
1
|
B
ε
R
,
R
|
∫
B
ε
R
,
R
(
u
m
−
)
p
0
)
1
/
p
0
≤
C
(
inf
⁡
A
∩
B
ε
R
,
R
u
+
R
∥
f
∥
L
n
(
A
∩
B
ε
τ
R
,
τ
′
R
)
)
,with positive constants C and p
0,
depending on n
,
λ
,
Λ
,
q
,
τ
,
τ
′,
and N
q
−
1
R
2
−
q
∥
b
∥
L
∞
(
A
∩
B
ε
τ
R
,
R
/
τ
).In a similar manner, we extend the local MP
up to the boundary.Theorem 3.8 (boundary local MP).
Letb
0
≥
0, 0
<
τ
<
1, N
>
0
, and 1
≤
q
≤
2.
Suppose that u
∈
USC
(
A
¯
) is a viscosity solution of F
(
x
,
u
,
D
u
,
D
2
u
)
≥
f,
under structure condition (1.5), with f
∈
C
(
A
¯
) and u
≤
N in A.
Then for all p
>
0
,
(3.25)
sup
⁡
B
τ
R
∩
A
u
≤
C
(
(
1
|
B
R
|
∫
B
R
(
u
M
+
)
p
)
1
/
p
+
R
∥
f
−
∥
L
n
(
B
R
∩
A
)
)
,with a positive constant C,
depending on n
,
λ
,
Λ
,
q
,
τ
,
b
0
N
q
−
1
R
2
−
q,
and p.


## 4. ABP-Type Estimates and Maximum Principles

Here we use
boundary estimates of previous section to obtain MP in unbounded domainsΩ of R
n for viscosity subsolutions u
∈
USC
(
Ω
¯
),
bounded above, of equation F
(
x
,
u
,
D
u
,
D
2
u
)
=
0 under structure condition (1.5).We will make use of the measure-geometric conditionG
σ, 0
<
σ
<
1,
given in the introduction. By a continuity argument, see [12], eventually passing to a
smaller R,
which we will call r
y,
we can assume that condition G
σ is satisfied with |
B
∖
Ω
y
|
=
σ
|
B
| exactly.We also recall thatΩ is a w
G domain (with parameter σ) if each point y
∈
Ω satisfies condition G
σ in Ω.
In particular, if R
(
y
) is the radius of the ball B
=
B
(
y
) provided by condition G
σ,
we define domains of cylindrical and conical type as w
G domains such that R
(
y
)
=
O
(
1
) and R
(
y
)
=
O
(
|
y
|
),
respectively as |
y
|
→
∞.


### 4.1. Domains of Cylindrical Type

We start with
the conditionG of Cabré [10]. Let σ
<
1, τ
<
1
, and R
0 be positive real numbers. We say that an open
connected set Ω of R
n is a G domain if to each y
∈
Ω we can associate  a ball B
=
B
R
(
x
y
) of radius R
≤
R
0 such that(4.1)
y
∈
B
τ
R
(
x
y
)
,

|
B
∖
Ω
y
|
≥
σ
|
B
|
,


(
G
σ
,
τ
)where Ω
y is the connected component of Ω
∩
B containing y.SinceG
σ
≡
G
σ
,
1,
then a G domain of R
n is of cylindrical type, like domains of finite
Lebesgue n-dimensional measure, subdomains of ω
×
R
n
−
k,
where ω has finite Lebesgue  k-dimensional measure, the complement of the
spiral of equation r
=
θ in polar
coordinates of R
2.Given a differential operator with structure
conditions, like (1.4) and (1.5), Ω will be called a narrow domain when, for given τ and R
0,
condition G
σ
,
τ is satisfied for σ suitably close to 1,
depending on the structure constants and the remaining geometric constants.A straightforward application of Theorem3.8 yields MP
in narrow domains. Indeed, assume that u
≤
N and F
(
x
,
u
,
D
u
,
D
2
u
)
≥
0 in Ω.
Then, by (1.5), we have(4.2)
P
+
(
D
2
u
+
)
+
b
0
|
D
u
+
|
q
≥
0.Suppose that u
≤
0 on ∂
Ω and set M
=
sup
⁡
Ω
u
+.
Applying Theorem 3.8 in A
=
Ω
y with p
=
1,
we obtain(4.3)
u
(
y
)
≤
sup
⁡
Ω
y
∩
B
τ
R
(
x
y
)
u
≤
C
|
B
|
∫

Ω
y
∩
B
u
+
≤
C
M
|
Ω
y
∩
B
|
|
B
|
≤
C
M
(
1
−
σ
)
.From this, taking the supremum
over y
∈
Ω,
we get M
≤
0,
that is u
≤
0 in Ω,
provided σ
>
1
−
1
/
C,
and hence MP holds in this case.In order to pass from narrow domains to arbitrary
cylindrical domains we will use Theorem3.6, from which the following ABP-type
estimate follows.Theorem 4.1 (ABP estimate).
Letσ
,
τ
<
1,
let R
0 and N be positive real numbers, and 1
≤
q
≤
2.
Let Ω be a cylindrical domain such that condition G
σ in Ω is satisfied at each y
∈
Ω with R
(
y
)
≤
R
0.
Suppose thatw
∈
USC
(
Ω
¯
) is a viscosity solution of F
(
x
,
w
,
D
w
,
D
2
w
)
≥
f,
under the structure condition (1.5), with b
≤
b
0 and f
∈
C
(
Ω
¯
).
Ifw
≤
N in Ω and w
≤
0 on ∂
Ω,
then(4.4)
sup
⁡
Ω
w
≤
C
R
0
sup
⁡
y
∈
Ω
∥
f
∥
L
n
(
B
R
(
y
)
/
τ
(
x
y
)
∩
Ω
)
,where C depends on n
,
λ
,
Λ
,
σ
,
τ
, and b
0
R
0
2
−
q
N
q
−
1.Proof.
It is enough to show the result forτ
→
1
−.
SetM
=
sup
⁡
Ω
w
+ and u
=
M
−
w.
Let y
∈
Ω and B
=
B
R of radius R,
provided by condition G
σ in y.
We choose R
=
r
y such that |
B
∖
Ω
y
|
=
σ
|
B
|
; see the beginning of this section. We also denote by B
τ
R the concentric ball of radius τ
R.
Now we apply Theorem3.6 to u in A
=
Ω
y with B
τ
R
(
x
y
) instead of B
R and τ close enough to 1 in such a way that |
B
τ
R
(
x
y
)
∖
Ω
y
|
≥
(
σ
/
2
)
|
B
| and |
B
τ
R
(
x
y
)
∩
Ω
y
|
≥
(
(
1
−
σ
)
/
2
)
|
B
|.
Since w
≤
0 on ∂
Ω,
then m
≥
M,
hence we get(4.5)
(
σ
2
)
1
/
p
0
M
≤
(
1
|
B
τ
R
(
x
y
)
|
∫
B
τ
R
(
x
y
)
(
u
m
−
)
p
0
)
1
/
p
0
≤
C
(
M
−
sup
⁡
B
τ
R
(
x
y
)
∩
Ω
y
w
+
R
∥
f
∥
L
n
(
B
∩
Ω
)
)
,from which, for x
∈
B
τ
R
(
x
y
)
∩
Ω
y,
we obtain the pointwise inequality(4.6)
w
(
x
)
≤
sup
⁡
B
τ
R
(
x
y
)
∩
Ω
y
w
≤
t
M
+
R
∥
f
∥
L
n
(
B
∩
Ω
)
,with 0
<
t
<
1.
On the other hand, setting K
=
t
M
+
R
∥
f
∥
L
n
(
B
∩
Ω
) and Ω
K
=
{
x
∈
Ω
/
w
(
x
)
>
K
},
by virtue of (4.6) we have B
∖
Ω
K
⊃
B
τ
R
(
x
y
)
∩
Ω
y and therefore, by our choice of R and τ,(4.7)
|
B
∖
Ω
K
|
|
B
|
≥
|
B
τ
R
(
x
y
)
∩
Ω
y
|
|
B
|
≥
1
−
σ
2
.A further application of Theorem 3.6 to u
=
M
−
w in A
=
Ω
K yields(4.8)
(
1
−
σ
2
)
1
/
p
0
(
M
−
K
)
≤
(
1
|
B
|
∫
B
(
u
m
−
)
p
0
)
1
/
p
0
≤
C
(
M
−
sup
⁡
B
∩
Ω
K
w
+
R
∥
f
∥
L
n
(
B
R
/
τ
(
x
y
)
∩
Ω
)
)
,since in this case m
≤
M
−
K.
From this we deduce that, for x
∈
B
∩
Ω
K
(4.9)
w
(
x
)
≤
sup
⁡
B
∩
Ω
K
w
≤
(
1
−
t
′
)
M
+
t
′
K
+
R
∥
f
∥
L
n
(
B
R
/
τ
(
x
y
)
∩
Ω
)
≤
(
1
−
t
′
+
t
t
′
)
M
+
2
R
∥
f
∥
L
n
(
B
R
/
τ
(
x
y
)
∩
Ω
)
,with 0
<
t
′
<
1.
From the definition of Ω
K,
see also (4.6), it follows that(4.10)
w
(
x
)
≤
t
′
′
M
+
2
R
∥
f
∥
L
n
(
B
R
/
τ
(
x
y
)
∩
Ω
)
,with 0
<
t
′
′
<
1,
for all x
∈
Ω
∩
B and hence also for x
=
y.
Finally, passing to the supremum overy
∈
Ω,
we get the result.


### 4.2. General Domains

Firstly, we
considerw
G domains Ω,
such that condition G
σ in Ω holds at each y
∈
Ω without bounds for the radii R
(
y
) of the balls provided by G
σ.Note that in general the ABP-type estimate of Theorem4.1 is useless unless b
0
=
0,
see [13], since the the
constant C of ABP estimate blows up when R
→
+
∞.
This is why we assume b
(
x
)
=
O
(
1
/
|
x
|
2
−
q
) as |
x
|
→
+
∞ in the structure condition (1.5). Moreover, to
take advantage from the decay of b
(
x
),
it is convenient to use the boundary wH inequality for annular regions of
Corollary 3.7 rather than Theorem 3.6.Reasoning as in the proof of Theorem4.1, but quite
more carefully with the aid of (3.24) instead of (3.16), see [16], we get the following
ABP-type estimate.Theorem 4.2 (ABP).
Letσ and N be positive real numbers and 1
≤
q
≤
2.
Let Ω be a w
G domain (with parameter σ
<
1).
Suppose thatw
∈
USC
(
Ω
¯
) is a viscosity solution of F
(
x
,
w
,
D
w
,
D
2
w
)
≥
f,
under the structure condition (1.5), with b
,
f
∈
C
(
Ω
¯
) such that(4.11)
b
q
:
=
sup
⁡
y
∈
Ω
;
|
y
|
≥
ε
r
y
r
y
2
−
q
∥
b
∥
L
∞
(
Ω
∩
B
τ
ε
r
y
,
τ
′
r
y
)
<
+
∞
,for all ε
>
0 small enough, all τ
<
1 sufficiently close to 1, and some τ
′
>
1.
Ifw
≤
N in Ω and w
≤
0 on ∂
Ω,
then(4.12)
sup
⁡
Ω
w
≤
C
sup
⁡
y
∈
Ω
;
|
y
|
≥
ε
r
y
r
y
∥
f
−
∥
L
n
(
Ω
∩
B
τ
ε
r
y
,
τ
′
r
y
)
+
sup
⁡
y
∈
Ω
;
|
y
|
≤
ε
r
y
C
y
r
y
∥
f
−
∥
L
n
(
Ω
∩
B
ε
r
y
)for possibly smaller ε
>
0 and larger τ
<
1,
depending on n and σ.
HereC and C
y are positive constants depending on n, q, λ, Λ, b
q
N
q
−
1, σ, ε, τ, τ
′,
while C
y also depends on N
q
−
1
r
y
2
−
q
∥
b
∥
L
∞
(
Ω
∩
B
ε
r
y
).Proof  of Theorem1.2.
In the case ofw
G domains, Theorem 1.2 follows at once letting f
=
0 in Theorem 4.2. Suppose now that Ω can be split by a closed set H
⊂
R
n in components where MP holds and each y
∈
H satisfies condition G
σ in Ω.
By MP in the components, since we assume that w
≤
0 on ∂
Ω,
then for x
∈
Ω we have(4.13)
w
(
x
)
≤
sup
⁡
(
Ω
∩
H
)
∪
∂
Ω
w
+
=
sup
⁡
Ω
∩
H
w
+
.Reasoning as above for (4.10),
but using Corollary 3.7 instead of Theorem 3.6 as before to obtain Theorem 4.2, from
condition G
σ we deduce for y
∈
Ω
∩
H that(4.14)
w
(
y
)
≤
t
sup
⁡
Ω
w
+
,where t
∈
]
0
,
1
[ is independent of y.
Inserting this inequality in the former one, and taking the supremum over Ω,
we get the result.Examples
Provided thatb
(
x
)
=
O
(
1
/
|
x
|
2
−
q
) as |
x
|
→
∞,
this last result yields MP in very general domains such as, for instance:
(i)
w
G domains, like a proper cone Ω such that Ω
¯
≠
R
n and in general a domain of conical type, like
the complement Ω in R
n of Γ
×
R
n
−
2,
where Γ is a logarithmic spiral of equation r
=
e
θ in polar
coordinates, or also complement of a larger spiral
of equation r
=
s
(
θ
),
with s a positive increasing function.
(ii)
Domains which can be split inw
G subdomains by a suitable closed set H of R
n,
like the cut plane in R
2 or in general the complement in R
n of a graph {
(
x
,
y
)
∈
R
n
−
1
×
R
|
x
i
≥
0
,
i
=
1
,
…
,
n
−
1,
y
=
f
(
x
)
} such that |
f
(
x
)
|
≤
h
+
k
|
x
| for positive constants h and k.
As a further example, we show a repeated application
of Theorem1.2. Look at the complement Ω in R
2 of a sequence of balls B
r
(
k
), k
=
(
k
x
,0
)
,
k
x
∈
N,
with 0
<
r
<
1
/
3.
Consider the nonnegative x -axis as H,
then Ω
H
=
Ω
∖
H is connected. If K is the half-line of equations y
=
(
1
/
2
)
x, x
≥
0,
then we have the following:
(i)
Ω
H
∖
K has two components, which are domains of
conical type, where MP holds;
(ii)
each point ofΩ
H
∩
K satisfies condition G
1
/
2 in Ω
H.
Thus MP holds in Ω
∖
H by Theorem 1.2. Also, each point of H satisfies condition G
σ in Ω for some σ
∈
]
0
,
1
[ depending on r.
Therefore, again by Theorem 1.2, we conclude that MP holds in Ω.


### 4.3. Parabolic Shaped Domains

For a parabolic
shapedw
G domain, condition G
σ at y
∈
Ω holds with R
(
y
)
=
O
(
|
y
|
α
) as |
y
|
→
∞,
for some 0
≤
α
≤
1,
the limit cases α
=
0 and α
=
1 representing, respectively, the cylindrical
and the conical cases. Hence r
y
≤
R
(
y
)
≤
R
0
+
β
|
y
|
α for all y
∈
Ω with positive constants R
0 and β.
Then, choosing ε sufficiently small in Theorem 4.2, if |
y
|
≤
ε
r
y
, then(4.15)
|
y
|
≤
ε
R
0
+
ε
β
|
y
|
α
≤
ε
(
R
0
+
β
)
+
ε
β
|
y
|
,so that the supremum in the
second term of the right-hand side of (4.12) is taken over a bounded subset of y
∈
Ω,
in which r
y
≤
R
1 for some positive constant R
1.
Thus(4.16)
sup
⁡
y
∈
Ω
;
|
y
|
≤
ε
r
y
r
y
∥
f
−
∥
L
n
(
Ω
∩
B
ε
r
y
)
≤
R
1
sup
⁡
y
∈
Ω
;
r
y
≤
R
1
∥
f
−
∥
L
n
(
Ω
∩
B
ε
r
y
)
.Proof  of Theorem1.3.
Since conditionw
G holds with r
y
=
O
(
|
y
|
α
), 0
≤
α
≤
1,
the assumption b
(
x
)
=
O
(
1
/
|
x
|
α
(
2
−
q
)
) as |
x
|
→
+
∞ implies the finiteness of b
q in (4.11). Taking account of (4.16), by
continuity  of f the estimate (1.10) follows letting ε
→
0.


## 4.1. Domains of Cylindrical Type

We start with
the conditionG of Cabré [10]. Let σ
<
1, τ
<
1
, and R
0 be positive real numbers. We say that an open
connected set Ω of R
n is a G domain if to each y
∈
Ω we can associate  a ball B
=
B
R
(
x
y
) of radius R
≤
R
0 such that(4.1)
y
∈
B
τ
R
(
x
y
)
,

|
B
∖
Ω
y
|
≥
σ
|
B
|
,


(
G
σ
,
τ
)where Ω
y is the connected component of Ω
∩
B containing y.SinceG
σ
≡
G
σ
,
1,
then a G domain of R
n is of cylindrical type, like domains of finite
Lebesgue n-dimensional measure, subdomains of ω
×
R
n
−
k,
where ω has finite Lebesgue  k-dimensional measure, the complement of the
spiral of equation r
=
θ in polar
coordinates of R
2.Given a differential operator with structure
conditions, like (1.4) and (1.5), Ω will be called a narrow domain when, for given τ and R
0,
condition G
σ
,
τ is satisfied for σ suitably close to 1,
depending on the structure constants and the remaining geometric constants.A straightforward application of Theorem3.8 yields MP
in narrow domains. Indeed, assume that u
≤
N and F
(
x
,
u
,
D
u
,
D
2
u
)
≥
0 in Ω.
Then, by (1.5), we have(4.2)
P
+
(
D
2
u
+
)
+
b
0
|
D
u
+
|
q
≥
0.Suppose that u
≤
0 on ∂
Ω and set M
=
sup
⁡
Ω
u
+.
Applying Theorem 3.8 in A
=
Ω
y with p
=
1,
we obtain(4.3)
u
(
y
)
≤
sup
⁡
Ω
y
∩
B
τ
R
(
x
y
)
u
≤
C
|
B
|
∫

Ω
y
∩
B
u
+
≤
C
M
|
Ω
y
∩
B
|
|
B
|
≤
C
M
(
1
−
σ
)
.From this, taking the supremum
over y
∈
Ω,
we get M
≤
0,
that is u
≤
0 in Ω,
provided σ
>
1
−
1
/
C,
and hence MP holds in this case.In order to pass from narrow domains to arbitrary
cylindrical domains we will use Theorem3.6, from which the following ABP-type
estimate follows.Theorem 4.1 (ABP estimate).
Letσ
,
τ
<
1,
let R
0 and N be positive real numbers, and 1
≤
q
≤
2.
Let Ω be a cylindrical domain such that condition G
σ in Ω is satisfied at each y
∈
Ω with R
(
y
)
≤
R
0.
Suppose thatw
∈
USC
(
Ω
¯
) is a viscosity solution of F
(
x
,
w
,
D
w
,
D
2
w
)
≥
f,
under the structure condition (1.5), with b
≤
b
0 and f
∈
C
(
Ω
¯
).
Ifw
≤
N in Ω and w
≤
0 on ∂
Ω,
then(4.4)
sup
⁡
Ω
w
≤
C
R
0
sup
⁡
y
∈
Ω
∥
f
∥
L
n
(
B
R
(
y
)
/
τ
(
x
y
)
∩
Ω
)
,where C depends on n
,
λ
,
Λ
,
σ
,
τ
, and b
0
R
0
2
−
q
N
q
−
1.Proof.
It is enough to show the result forτ
→
1
−.
SetM
=
sup
⁡
Ω
w
+ and u
=
M
−
w.
Let y
∈
Ω and B
=
B
R of radius R,
provided by condition G
σ in y.
We choose R
=
r
y such that |
B
∖
Ω
y
|
=
σ
|
B
|
; see the beginning of this section. We also denote by B
τ
R the concentric ball of radius τ
R.
Now we apply Theorem3.6 to u in A
=
Ω
y with B
τ
R
(
x
y
) instead of B
R and τ close enough to 1 in such a way that |
B
τ
R
(
x
y
)
∖
Ω
y
|
≥
(
σ
/
2
)
|
B
| and |
B
τ
R
(
x
y
)
∩
Ω
y
|
≥
(
(
1
−
σ
)
/
2
)
|
B
|.
Since w
≤
0 on ∂
Ω,
then m
≥
M,
hence we get(4.5)
(
σ
2
)
1
/
p
0
M
≤
(
1
|
B
τ
R
(
x
y
)
|
∫
B
τ
R
(
x
y
)
(
u
m
−
)
p
0
)
1
/
p
0
≤
C
(
M
−
sup
⁡
B
τ
R
(
x
y
)
∩
Ω
y
w
+
R
∥
f
∥
L
n
(
B
∩
Ω
)
)
,from which, for x
∈
B
τ
R
(
x
y
)
∩
Ω
y,
we obtain the pointwise inequality(4.6)
w
(
x
)
≤
sup
⁡
B
τ
R
(
x
y
)
∩
Ω
y
w
≤
t
M
+
R
∥
f
∥
L
n
(
B
∩
Ω
)
,with 0
<
t
<
1.
On the other hand, setting K
=
t
M
+
R
∥
f
∥
L
n
(
B
∩
Ω
) and Ω
K
=
{
x
∈
Ω
/
w
(
x
)
>
K
},
by virtue of (4.6) we have B
∖
Ω
K
⊃
B
τ
R
(
x
y
)
∩
Ω
y and therefore, by our choice of R and τ,(4.7)
|
B
∖
Ω
K
|
|
B
|
≥
|
B
τ
R
(
x
y
)
∩
Ω
y
|
|
B
|
≥
1
−
σ
2
.A further application of Theorem 3.6 to u
=
M
−
w in A
=
Ω
K yields(4.8)
(
1
−
σ
2
)
1
/
p
0
(
M
−
K
)
≤
(
1
|
B
|
∫
B
(
u
m
−
)
p
0
)
1
/
p
0
≤
C
(
M
−
sup
⁡
B
∩
Ω
K
w
+
R
∥
f
∥
L
n
(
B
R
/
τ
(
x
y
)
∩
Ω
)
)
,since in this case m
≤
M
−
K.
From this we deduce that, for x
∈
B
∩
Ω
K
(4.9)
w
(
x
)
≤
sup
⁡
B
∩
Ω
K
w
≤
(
1
−
t
′
)
M
+
t
′
K
+
R
∥
f
∥
L
n
(
B
R
/
τ
(
x
y
)
∩
Ω
)
≤
(
1
−
t
′
+
t
t
′
)
M
+
2
R
∥
f
∥
L
n
(
B
R
/
τ
(
x
y
)
∩
Ω
)
,with 0
<
t
′
<
1.
From the definition of Ω
K,
see also (4.6), it follows that(4.10)
w
(
x
)
≤
t
′
′
M
+
2
R
∥
f
∥
L
n
(
B
R
/
τ
(
x
y
)
∩
Ω
)
,with 0
<
t
′
′
<
1,
for all x
∈
Ω
∩
B and hence also for x
=
y.
Finally, passing to the supremum overy
∈
Ω,
we get the result.


## 4.2. General Domains

Firstly, we
considerw
G domains Ω,
such that condition G
σ in Ω holds at each y
∈
Ω without bounds for the radii R
(
y
) of the balls provided by G
σ.Note that in general the ABP-type estimate of Theorem4.1 is useless unless b
0
=
0,
see [13], since the the
constant C of ABP estimate blows up when R
→
+
∞.
This is why we assume b
(
x
)
=
O
(
1
/
|
x
|
2
−
q
) as |
x
|
→
+
∞ in the structure condition (1.5). Moreover, to
take advantage from the decay of b
(
x
),
it is convenient to use the boundary wH inequality for annular regions of
Corollary 3.7 rather than Theorem 3.6.Reasoning as in the proof of Theorem4.1, but quite
more carefully with the aid of (3.24) instead of (3.16), see [16], we get the following
ABP-type estimate.Theorem 4.2 (ABP).
Letσ and N be positive real numbers and 1
≤
q
≤
2.
Let Ω be a w
G domain (with parameter σ
<
1).
Suppose thatw
∈
USC
(
Ω
¯
) is a viscosity solution of F
(
x
,
w
,
D
w
,
D
2
w
)
≥
f,
under the structure condition (1.5), with b
,
f
∈
C
(
Ω
¯
) such that(4.11)
b
q
:
=
sup
⁡
y
∈
Ω
;
|
y
|
≥
ε
r
y
r
y
2
−
q
∥
b
∥
L
∞
(
Ω
∩
B
τ
ε
r
y
,
τ
′
r
y
)
<
+
∞
,for all ε
>
0 small enough, all τ
<
1 sufficiently close to 1, and some τ
′
>
1.
Ifw
≤
N in Ω and w
≤
0 on ∂
Ω,
then(4.12)
sup
⁡
Ω
w
≤
C
sup
⁡
y
∈
Ω
;
|
y
|
≥
ε
r
y
r
y
∥
f
−
∥
L
n
(
Ω
∩
B
τ
ε
r
y
,
τ
′
r
y
)
+
sup
⁡
y
∈
Ω
;
|
y
|
≤
ε
r
y
C
y
r
y
∥
f
−
∥
L
n
(
Ω
∩
B
ε
r
y
)for possibly smaller ε
>
0 and larger τ
<
1,
depending on n and σ.
HereC and C
y are positive constants depending on n, q, λ, Λ, b
q
N
q
−
1, σ, ε, τ, τ
′,
while C
y also depends on N
q
−
1
r
y
2
−
q
∥
b
∥
L
∞
(
Ω
∩
B
ε
r
y
).Proof  of Theorem1.2.
In the case ofw
G domains, Theorem 1.2 follows at once letting f
=
0 in Theorem 4.2. Suppose now that Ω can be split by a closed set H
⊂
R
n in components where MP holds and each y
∈
H satisfies condition G
σ in Ω.
By MP in the components, since we assume that w
≤
0 on ∂
Ω,
then for x
∈
Ω we have(4.13)
w
(
x
)
≤
sup
⁡
(
Ω
∩
H
)
∪
∂
Ω
w
+
=
sup
⁡
Ω
∩
H
w
+
.Reasoning as above for (4.10),
but using Corollary 3.7 instead of Theorem 3.6 as before to obtain Theorem 4.2, from
condition G
σ we deduce for y
∈
Ω
∩
H that(4.14)
w
(
y
)
≤
t
sup
⁡
Ω
w
+
,where t
∈
]
0
,
1
[ is independent of y.
Inserting this inequality in the former one, and taking the supremum over Ω,
we get the result.Examples
Provided thatb
(
x
)
=
O
(
1
/
|
x
|
2
−
q
) as |
x
|
→
∞,
this last result yields MP in very general domains such as, for instance:
(i)
w
G domains, like a proper cone Ω such that Ω
¯
≠
R
n and in general a domain of conical type, like
the complement Ω in R
n of Γ
×
R
n
−
2,
where Γ is a logarithmic spiral of equation r
=
e
θ in polar
coordinates, or also complement of a larger spiral
of equation r
=
s
(
θ
),
with s a positive increasing function.
(ii)
Domains which can be split inw
G subdomains by a suitable closed set H of R
n,
like the cut plane in R
2 or in general the complement in R
n of a graph {
(
x
,
y
)
∈
R
n
−
1
×
R
|
x
i
≥
0
,
i
=
1
,
…
,
n
−
1,
y
=
f
(
x
)
} such that |
f
(
x
)
|
≤
h
+
k
|
x
| for positive constants h and k.
As a further example, we show a repeated application
of Theorem1.2. Look at the complement Ω in R
2 of a sequence of balls B
r
(
k
), k
=
(
k
x
,0
)
,
k
x
∈
N,
with 0
<
r
<
1
/
3.
Consider the nonnegative x -axis as H,
then Ω
H
=
Ω
∖
H is connected. If K is the half-line of equations y
=
(
1
/
2
)
x, x
≥
0,
then we have the following:
(i)
Ω
H
∖
K has two components, which are domains of
conical type, where MP holds;
(ii)
each point ofΩ
H
∩
K satisfies condition G
1
/
2 in Ω
H.
Thus MP holds in Ω
∖
H by Theorem 1.2. Also, each point of H satisfies condition G
σ in Ω for some σ
∈
]
0
,
1
[ depending on r.
Therefore, again by Theorem 1.2, we conclude that MP holds in Ω.


## 4.3. Parabolic Shaped Domains

For a parabolic
shapedw
G domain, condition G
σ at y
∈
Ω holds with R
(
y
)
=
O
(
|
y
|
α
) as |
y
|
→
∞,
for some 0
≤
α
≤
1,
the limit cases α
=
0 and α
=
1 representing, respectively, the cylindrical
and the conical cases. Hence r
y
≤
R
(
y
)
≤
R
0
+
β
|
y
|
α for all y
∈
Ω with positive constants R
0 and β.
Then, choosing ε sufficiently small in Theorem 4.2, if |
y
|
≤
ε
r
y
, then(4.15)
|
y
|
≤
ε
R
0
+
ε
β
|
y
|
α
≤
ε
(
R
0
+
β
)
+
ε
β
|
y
|
,so that the supremum in the
second term of the right-hand side of (4.12) is taken over a bounded subset of y
∈
Ω,
in which r
y
≤
R
1 for some positive constant R
1.
Thus(4.16)
sup
⁡
y
∈
Ω
;
|
y
|
≤
ε
r
y
r
y
∥
f
−
∥
L
n
(
Ω
∩
B
ε
r
y
)
≤
R
1
sup
⁡
y
∈
Ω
;
r
y
≤
R
1
∥
f
−
∥
L
n
(
Ω
∩
B
ε
r
y
)
.Proof  of Theorem1.3.
Since conditionw
G holds with r
y
=
O
(
|
y
|
α
), 0
≤
α
≤
1,
the assumption b
(
x
)
=
O
(
1
/
|
x
|
α
(
2
−
q
)
) as |
x
|
→
+
∞ implies the finiteness of b
q in (4.11). Taking account of (4.16), by
continuity  of f the estimate (1.10) follows letting ε
→
0.


## 5. Strong Maximum Principle and Liouville Theorem

The weak Harnack inequality of Theorem3.3 can be used
to show the following strong MP.Theorem 5.1 (strong MP).
LetΩ be a domain of R
n.
Let w
∈
C
(
Ω
) be such that F
(
x
,
w
,
D
w
,
D
2
w
)
≥
0 in the viscosity sense, and assume structure
condition (1.5), with b
∈
C
(
Ω
¯
).
If x
0
∈
Ω and M
:
=
w
(
x
0
)
≥
w
(
x
)
, for all x
∈
Ω,
then w
≡
M in Ω.Proof.
Following [33], set Ω
1
=
w
−
1
(
{
M
}
) and Ω
2
=
Ω
∖
Ω
1.
By assumption Ω
1
≠
∅.
By continuity of w,
it turns out that Ω
2
=
w
−
1
(
]
−
∞
,
M
[
) is an open subset of R
n.
Moreover, plainly, Ω
=
Ω
1
∪
Ω
2 and Ω
2
∩
Ω
1
=
∅.
Recall thatΩ is an open connected set. Thus it is
sufficient to show that Ω
1 is in turn an open subset to have Ω
=
Ω
1,
as claimed in the statement of the theorem. Indeed, let x
1
∈
Ω
1,  that is, w
(
x
1
)
=
M,
and set u
=
M
−
w,
then u is a nonnegative viscosity solution of F
(
x
,
u
,
D
u
,
D
2
u
)
≤
0.
Applying (3.7) in a ball B
R
:
=
B
R
(
x
1
)
⊂
B
R
/
τ
(
x
1
)
⊂
⊂
Ω,
we get(5.1)
(
1
|
B
R
|
∫
B
R
(
M
−
w
)
p
0
)
1
/
p
0
≤
C
inf
⁡
B
R
u
=
0
,from which, by continuity, u
≡
M in B
R
(
x
1
).
This shows that Ω
1 is an open subset of R
n and concludes the proof.The Liouville
type result of Theorem1.5 is instead based on Harnack inequality (3.11) of
Theorem 3.5. It is convenient to consider its version in annular regions B
R
,2
R
=
B
2
R
(
0
)
∖
B
¯
R
(
0
) to take advantage of the decay of b
(
x
),
obtained in standard way, using inequality (3.11) in a chain of linked balls.
This yields, for continuous solutions u
∈
C
(
B
¯
R
/
2
,
4
R
) of equation F
=
f, 0
≤
u
≤
N,
under the structure conditions (1.4) and (1.5), with b
,
f
∈
C
(
B
¯
R
/
2
,
4
R
),
the following inequality:(5.2)
sup
⁡
B
R
,2
R
u
≤
C
(
inf
⁡
B
R
,2
R
u
+
R
∥
f
∥
L
n
(
B
R
/
2
,
4
R
)
)
,with a positive constant C
=
C
(
n
,
λ
,
Λ
,
q
,
τ
,
∥
b
∥
L
∞
(
B
R
/
2
,
4
R
)
N
q
−
1
R
2
−
q
).Proof  of Theorem1.5.
By the strong MP of Theorem5.1, we know that w can achieve neither a maximum nor a minimum at
a point of R
n unless it is constant,
in which case we should be done.
Suppose for
instance thatw
≤
M
:
=
sup
⁡
w
<
+
∞.
Let R
k be an increasing sequence of positive numbers
such that lim
⁡
k
→
∞
R
k
=
∞.
Set M
k
=
sup
⁡
∂
B
R
k
w and m
k
=
inf
⁡
∂
B
R
k
w.
By weak maximum principle, M
k is increasing and m
k is decreasing; thus(5.3)
lim
⁡
k
→
∞
M
k
=
M
,


lim
⁡
k
→
∞
m
k
=
m
∈
[
−
∞
,
+
∞
[
.Then, using Harnack inequality (5.2), with u
=
M
−
w,
we get(5.4)
M
−
m
k
=
sup
⁡
∂
B
R
k
(
M
−
w
)
≤
C
inf
⁡
∂
B
R
k
(
M
−
w
)
=
C
(
M
−
M
k
)
,from which(5.5)
M
≤
C
(
M
−
M
k
)
+
m
k
≤
C
(
M
−
M
k
)
+
m
kand, letting k
→
∞,
we get M
=
m,
as we  wanted to show.



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*Source: 178534-2008-05-08.xml*