Università degli studi di Roma
La Sapienza
FACOLTÀ DI SCIENZE MATEMATICHE, FISICHE E NATURALI
Corso di Laurea magistrale in Matematica per le applicazioni
Existence methods for semilinear
elliptic equations with an absorption
term and measure data
Relatori:
Luigi Orsina
Francesco Petitta
Laureando:
Francescantonio Oliva
Matricola: 1130269
Sessione:
Marzo 2013
Anno Accademico 2012-2013
Introduction
The aim of this thesis is to provide a collection of different tools and strategies which can be applied in order to obtain the existence of a solution to the
following semilinear Dirichlet problem
(
−∆u + g(u) = µ in Ω,
(1)
u=0
on ∂Ω,
where Ω ⊂ RN (N ≥ 3) is a bounded domain, g : R → R is a continuous
function that satisfies an absorption condition (i.e. g(s)s ≥ 0), and µ is a
general measure.
The first phisical motivation for studying problem (1) arises from the study of
the dynamic of a fluid or a gas through porous medium that is modeled by the
following equation
∂v
− ∆(|v|m−1 v) = 0,
(2)
∂t
where m > 0 represents the porosity of the medium.
Indeed, when the Crandall-Liggett theory (see [1], [8]) is applied to (2), then
the question is whether the following equation has a solution
v − λ∆(|v|m−1 v) = f,
(3)
with f ∈ L1 and λ > 0. It can be observed that, up to a change of variable
and scaling out λ, the equation (3) is essentially a model of the problem (1).
The second motivation relates to the Thomas-Fermi equation
X
3
mi δai ,
−∆v + 4π[(v − ω)+ ] 2 =
(4)
where δai is the Dirac’s delta concentrated on ai , that gives a model for the
electrostatic of some particles around the nucleus (for istance, the fermions).
These motivations give some insights on the importance of studying problem
(1) where g is a nonlinear function and µ is a general measure.
i
The thesis will be organized as follows.
In Chapter 1, some definitions and basic tools are introduced along with some
basic knowledges concerning the linear Dirichlet problem
(
−∆u = f in Ω,
(5)
u=0
on ∂Ω,
that is fundamental in order to understand the semilinear problem (1).
In Chapter 2, we approach problem (1) in some regular cases. More precisely,
one can define the functional
Z
Z
Z
1
2
|∇u| + G(u) − f u,
J(u) =
2 Ω
Ω
Ω
where f ∈ (W01,2 (Ω))0 , and the function G is the primitive of g. It is proved
that J admits a minimum in W01,2 (Ω) that, in case g(u) ∈ L1 (Ω), is proved to
be a solution to (1) with datum f .
In the same chapter, it is also presented a readaptation of Stampacchia linear
duality method in order to find a solution to the semilinear problem (1), where
the term g is linear and µ is a general measure.
The beginning of Chapter 3 concerns the different approaches to the existence
of the solution to (1) whether µ is a function in L1 (Ω) or it is a general measure.
The L1 -case was studied in [10] by Gallouët and Morel and it relies on an
approximation scheme for the datum µ.
The measure case was discussed in [4] by Bénilan and Brezis. They presented
an example that establish the nonexistence of the solution when µ is a general
measure, namely a Dirac delta and the function g grows too fast.
Therefore, in this case, it would make sense to understand what happens when
g has different growths. This is the aim of the central part of the third chapter.
The first situation requires g to satisfy some growth condition whose model is
g(t) = |t|p−1 t,
where p > 0.
In [2], Baras and Pierre showed that the critical value for the growth of g
was p = NN−2 . Again, in this case, their strategy is based on an approximation
scheme on the term µ.
They approached also the critical and supercritical growth. According to the
example provided by Bénilan and Brezis, the only possible way is to restrict
the set of measures that can be chosen in the problem in order to get a solution.
By mean of the method of sub and supersolutions (see [11] by Montenegro and
ii
Ponce), one gets the existence of a solution when the measure is absolutely
0
continuous with respect to the W 2,p capacity.
Later, in the special case N = 2, it is briefly described a faster growth case
that can be modeled as
g(t) = (et − 1).
The main result was proved in [16] by Vázquez. He shows how much a concentrated measure can be tall in order to get a solution, that is µ({x}) ≤ 4π for
every x ∈ Ω. This is proved by mean of the method of sub and supersolutions.
At the end of the same chapter we prove that, on one hand, problem (1),
where the measure is diffuse (i.e. absolutely continuous with respect to the
W 1,2 capacity), always admits solution. This result was proved in [13] by
Orsina and Ponce via approximation of the term g.
On the other hand, we show that if problem (1) admits solution for every
nondecreasing function g, then the measure must be diffuse. The result was
obtained in [6] by Brezis, Marcus and Ponce.
Therefore, we can provide a picture that summarizes what we present:
iii
Thus, up to Chapter 3, we present the following strategies:
ˆ the variational method,
ˆ the solution for duality,
ˆ approximation scheme for the term µ,
ˆ the method of sub and supersolutions,
ˆ approximation scheme for the term g.
In Chapter 4, in order to deal with general measures µ and functions g’s, the
concepts of good measures and reduced measures are introduced.
Briefly, it can be said that a measure µ is good if the problem (1) admits
solution.
The reduced measures are introduced in [6] by Brezis, Marcus and Ponce in
order to understand what happens when forcing the problem (1) to get a solution when it does not admit one. The reduced measure µ∗ is defined as a
measure that satisfies
Z
Z
Z
∗
∗
(6)
ϕdµ∗ , ∀ϕ ∈ C0∞ (Ω),
− u ∆ϕ + g(u )ϕ =
Ω
Ω
Ω
where u∗ is the largest subsolution to (1). We prove that µ∗ is the largest good
measure below µ and that it is also a good approximation, in the capacitary
sense, of the same measure. This translates into the fact that u∗ is a good
approximation of a solution to (1) in this “new sense”.
Two different approximation schemes are presented in order to get to the
largest subsolution to (1). The first one relies on approximating the term g
by bounded functions and the other one on approximating the measure via
convolution. Both methods are proved to be consistent with the concept of
reduced measures.
iv
Contents
Introduction
1 Basic tools
1.1 Measures . . . . . . .
1.2 Marcinkiewicz spaces
1.3 Sobolev spaces . . .
1.4 Sobolev capacity . .
1.5 Real analysis tools .
1.6 The linear theory . .
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1
. 1
. 4
. 6
. 9
. 10
. 13
2 The problem in some regular cases
19
2.1 Existence through variational method . . . . . . . . . . . . . . . 20
2.2 A linear case proved via duality . . . . . . . . . . . . . . . . . . 25
3 Measure data
3.1 L1 (Ω) and measure data . . . . . . . . . . . . . . . . . . . . .
3.2 Polynomial growth . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Subcritical growth when p < NN−2 approximating the
measure . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Method of sub and supersolutions . . . . . . . . . . . .
3.2.3 Critical and supercritical growth when p ≥ NN−2 through
sub and supersolutions method . . . . . . . . . . . . .
3.3 Exponential growth when N = 2 . . . . . . . . . . . . . . . . .
3.4 Diffuse measure data . . . . . . . . . . . . . . . . . . . . . . .
4 Good measures and reduced measures
4.1 Good measures . . . . . . . . . . . . .
4.2 Reduced Measures . . . . . . . . . . .
4.2.1 Properties of reduced measures
4.2.2 General nonlinearities g . . . .
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29
. 29
. 35
. 36
. 40
. 43
. 47
. 50
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59
60
64
67
71
A Uniqueness
73
Bibliography
75
v
Chapter 1
Basic tools
In this first chapter we recall some definitions and tools in order to state
and to prove the existence theorems in the next chapters.
From now on, unless explicitly stated, let Ω be a bounded subset of RN , and
N ≥ 3.
1.1
Measures
Definition 1.1. The σ-algebra Σ is a family of subsets of a set Ω such that:
ˆ ∅ ∈ Σ,
ˆ if E ∈ Σ, then {E ∈ Σ,
ˆ if (En )n∈N ⊆ Σ, then
+∞
[
En ∈ Σ,
n=1
then (Ω, Σ) is called measurable space.
Remark 1. For istance, (Ω, P(Ω)) is a measurable space, where we have denoted by P(Ω) the power set of Ω.
Definition 1.2. The collection B(Ω) of Borel sets is the smallest σ-algebra
that contains all of the open sets.
Definition 1.3. A nonnegative measure is a set function µ : Σ → [0, +∞]
such that:
ˆ µ(∅) = 0,
ˆ µ(
+∞
[
n=1
En ) =
+∞
X
µ(En ), for every sequence (En )n∈N ⊆ Σ of disjoint sets,
n=1
then (Ω, Σ, µ) is called measure space.
1
1.1 Measures
1. Basic tools
Remark 2. We refer to the second property in the previous definition as σaddivity.
Remark 3. We note that a σ-addivity measure µ is also σ-subaddivity, that is
µ(
+∞
[
En ) ≤
n=1
+∞
X
µ(En ), for every sequence (En )n∈N ⊂ Σ.
n=1
Remark 4. We also note that a nonnegative measure µ is also monotone, that
is
µ(A) ≤ µ(B), for every A ⊆ B.
Definition 1.4. A nonnegative measure µ defined on B(Ω) is called Borel
measure. The support of a Borel measure is the complement of the largest set
A ⊆ Ω such that µ(A) = 0, that is


[
supp(µ) = { 
A .
A:µ(A)=0
Definition 1.5. A Borel measure µ is said to be regular if for every E ∈ B(Ω)
and for every > 0 there exist an open set A , and a closed set C such that
C ⊆ E ⊆ A , µ(A \ C ) ≤ .
Definition 1.6. A measure µ is said to be bounded if µ(Ω) < +∞.
In the rest of this work we could ask for the measure µ to take positive and
negative values.
For istance, we can obtain it defining µ3 = µ1 − µ2 . But some difficulties arises
from the fact that µ3 is not defined when µ1 = µ2 = +∞.
This takes us to the next definition.
Definition 1.7. A signed measure on a measurable space (Ω, Σ) is a set function µ : Σ → [−∞, +∞] such that:
ˆ µ(∅) = 0,
ˆ µ(
+∞
[
n=1
En ) =
+∞
X
µ(En ), for every sequence (En )n∈N ⊆ Σ of disjoint sets,
n=1
ˆ µ assumes at most one of the values +∞, −∞.
2
1. Basic tools
1.1 Measures
We denote by M+ (Ω) the set of nonnegative, regular, bounded measures
on Ω.
Definition 1.8. We define the set of bounded Radon measures on Ω as
M(Ω) = {µ1 − µ2 , µi ∈ M+ (Ω)}.
For the proofs of this section see [15].
Theorem 1.1.1. Let µ ∈ M(Ω), then there exists a unique pair (µ+ , µ− ) ∈
[M+ (Ω)]2 such that
µ = µ+ − µ− ,
and such that there exist disjoint sets E + , E − ∈ B(Ω) such that
µ+ (E) = µ(E ∪ E + ), µ− (E) = µ(E ∪ E − ).
Remark 5. The measures µ+ , µ− are called positive and negative of the measure
µ.
We define, for every µ ∈ M(Ω),
||µ||M(Ω) = |µ|(Ω),
where |µ|(Ω) = µ+ + µ− is called total variation of the measure µ.
If M(Ω) is equipped with the norm just defined, then it becomes a Banach
space which is the dual of C 0 (Ω).
Let µ1 , µ2 ∈ M(Ω), then we can provide that
sup{µ1 , µ2 } =
µ1 + µ2 + |µ1 + µ2 |
.
2
It can be proved that
µ+ = sup{µ, 0},
and
µ− = inf{µ, 0}.
We provide some properties that will be usefuls in some measure decompositions.
Definition 1.9. The measure µ ∈ M(Ω) is said to be concentrated on a Borel
set E if, for every Borel set B,
µ(B) = µ(B ∪ E).
We denote this property by µbE.
3
1.2 Marcinkiewicz spaces
1. Basic tools
Definition 1.10. Let µ, λ ∈ M(Ω). µ is said to be absolutely continuous with
respect to λ if
λ(E) = 0 implies µ(E) = 0.
We denote this property by µ λ.
Definition 1.11. The measure µ, λ ∈ M(Ω) are said to be orthogonal if there
exists a set E such that
µ(E) = 0, and λ = λbE.
We denote this property by µ⊥λ.
We provide the following decomposition theorem.
Theorem 1.1.2. Let µ, λ ∈ M(Ω), then there exists a unique pair (µ0 , µ1 ) ∈
[M(Ω)]2 such that
µ = µ0 + µ1 , where µ0 λ, µ1 ⊥λ.
Example 1.1. We provide the following examples of bounded Radon measures:
ˆ the lebesgue measure LN concentrated on a bounded set of RN ,
ˆ the Dirac’s delta concentrated at x0
(
1
δx0 (E) =
0
if x0 ∈ E,
otherwise,
ˆ the measure defined by
Z
µ(E) =
f (x)dx,
E
where f ∈ L1 (Ω).
By last point we can deduce that, formally, L1 (Ω) ⊂ M(Ω).
1.2
Marcinkiewicz spaces
In this section we will introduce the Marcinkiewicz spaces M p (Ω, µ), where
µ is a bounded measure.
4
1. Basic tools
1.2 Marcinkiewicz spaces
Definition 1.12. Let 0 < p < +∞, then
M p (Ω, µ) = {u : u is µ measurable, ∃ c > 0 such that µ({|u| > t}) ≤ ct−p , ∀ t > 0}.
Moreover, M p (Ω, µ) can be equipped with the seminorm
kukM p (Ω,µ) = inf{c : µ({|u| > t}) ≤ ct−p , ∀t > 0}.
Now we can prove where “these spaces are”.
For every 1 < p < +∞, 1 ≤ q < p we have Lp (Ω, µ) ⊂ M p (Ω, µ) ⊂ Lq (Ω, µ).
i) Lp (Ω, µ) ⊂ M p (Ω, µ)
For every u ∈ Lp (Ω, µ), we have
p
t µ({|u| > t}) = t
Z
Z
p
p
p
Z
|u| dµ ≤
t dµ ≤
dµ =
{|u|>t}
Z
{|u|>t}
{|u|>t}
|u|p dµ,
Ω
that is
kukM p (Ω,µ) ≤ kukLp (Ω,µ)
ii) M p (Ω, µ) ⊂ Lq (Ω, µ)
For every u ∈ M p (Ω, µ), and for every q < p, we have
Z
+∞
Z
q
q−1
|u| dµ = q
Ω
t
Z
0
Z
+∞
q−1
+q
t
+q
tq−1 µ({|u| > t})dt
0
Z
µ({|u| > t})dt ≤ q
1
Z
1
µ({|u| > t})dt = q
1
tq−1 µ(Ω)dt
0
+∞
Z
q−1−p p
t
t µ({|u| > t})dt ≤ qµ(Ω) + qc
1
+∞
tq−1−p < +∞,
0
since q − 1 − p < −1.
If µ is the Lebesgue measure, then we could refer to M p (Ω, µ) as M p (Ω).
Remark 6. The space M p (Ω) is a Lorentz space. It is an interpolation space
that, in many cases, is called weak Lp (Ω).
5
1.3 Sobolev spaces
1.3
1. Basic tools
Sobolev spaces
Definition 1.13. Let u ∈ Lp (Ω). The function u has a weak derivative (or
distributional derivative) if there exists a function v = (v1 , ..., vN ) ∈ (Lp (Ω))N
such that
Z
Z
∂ϕ
= − vi ϕ, ∀i = 1, 2, ..., N, ∀ϕ ∈ C0∞ (Ω).
u
Ω
Ω ∂xi
The function vi is the weak derivative in the direction xi that is denoted by
∂u
. If u has a weak derivative in every direction, then we denote the weak
∂xi
gradient as the vector
∂u
∂u
∇u =
, ...,
.
∂x1
∂xN
Now we are ready to introduce the Sobolev spaces.
Definition 1.14. If p ≥ 1, then the Sobolev spaces W 1,p (Ω) is defined by
W 1,p (Ω) = {u ∈ Lp (Ω) : ∇u ∈ (Lp (Ω))N }.
If W 1,p (Ω) is equipped with the norm
||u||W 1,p (Ω) = ||u||Lp (Ω) + ||∇u||(Lp (Ω))N ,
then it becomes a Banach space.
If W 1,2 (Ω) is equipped with the product scalar
Z
Z
(u|v)W 1,2 (Ω) =
uv + ∇u · ∇v,
Ω
Ω
then it becomes a Hilbert space.
Remark 7. In many cases, W 1,2 (Ω) is denoted by H 1 (Ω).
For the proofs of this section see [5].
Proposition 1.3.1. W 1,p (Ω) is reflexive for every 1 < p < +∞, W 1,p (Ω) is
separable for every 1 ≤ p < +∞. W 1,2 (Ω) is a separable Hilbert space.
At this point we want to define Sobolev functions that are zero on the boundary
of Ω, but functions in W 1,p (Ω) are only defined through almost equivalence
equivalence.
This takes us to the following definition.
Definition 1.15. We define W01,p (Ω) as the closure of C0∞ (Ω) with respect to
the norm of W 1,p (Ω).
6
1. Basic tools
1.3 Sobolev spaces
Remark 8. Again, W01,2 (Ω) is denoted by H01 (Ω).
Remark 9. We recall that p0 is said to be the conjugates of p if
1
1
+ 0 = 1.
p p
If s, t ≥ 0, then we have the Young inequality
0
sp t p
+ 0.
st ≤
p
p
(1.1)
0
Lastly, if f ∈ Lp (Ω), g ∈ Lp (Ω), then we have the Hölder inequality
Z
|f g| ≤ ||f ||Lp (Ω) ||g||Lp0 (Ω) .
(1.2)
Ω
The first important result arises by considering that a function in W01,2 (Ω)
is, in some sense, zero at the boundary. Thus, we can control the norm of the
function with the norm of its gradient.
Theorem 1.3.2 (Poincaré). Let p ≥ 1, then
||u||Lp (Ω) ≤ C||∇u||(Lp (Ω))N ,
∀u ∈ W01,p (Ω),
where C[Ω, N, p] > 0.
Remark 10. It follows from the previous theorem that we can consider on
W01,p (Ω) the following equivalent norm
||u||W 1,p (Ω) = ||∇u||(Lp (Ω))N .
0
We state the important Sobolev embedding theorem.
Theorem 1.3.3 (Sobolev). Let 1 ≤ p < N , and let p∗ =
||u||Lp∗ (Ω) ≤ S||u||W 1,p (Ω) ,
0
Np
,
N −p
then
∀u ∈ W01,p (Ω),
where S[N, p] > 0.
Remark 11. p∗ =
have:
Np
N −p
ˆ p∗ =
Np
N −p
> 1,
ˆ 1∗ =
N
N −1
≤
Np
N −p
is called the Sobolev embedding exponent. Thus, we
= p∗ < +∞,
ˆ p∗ > p.
Moreover, we have the following theorem.
7
1.3 Sobolev spaces
1. Basic tools
Theorem 1.3.4. Let 1 ≤ p ≤ ∞, and let p∗ =
continuous embeddings:
Np
,
N −p
then we have the following
∗
ˆ if 1 ≤ p < N , then W 1,p (Ω) ⊂ Lp (Ω),
ˆ if p = N , then W 1,p (Ω) ⊂ Lq (Ω), for every p ≤ q < +∞,
ˆ if p > N , then W 1,p (Ω) ⊂ L∞ (Ω).
Moreover, if p > N , we have, for every u ∈ W 1,p (Ω),
|u(x) − u(y)| ≤ CkukW 1,p (Ω) |x − y|λ , a.e. x, y ∈ Ω
where λ = 1 −
N
,
p
C[Ω, N, p] ≥ 0. Thus, we have W 1,p (Ω) ⊂ C(Ω).
Remark 12. The embeddings we have stated in the previous theorem are only
continuous. We recall that a map G : A → B, with A, B Banach spaces, is
said to be compact if the closure of G(Z) is compact in B for every bounded
set Z ⊂ B.
The next theorem relates to compact embedding of Sobolev spaces.
Theorem 1.3.5 (Rellich-Kondrachov). Let 1 ≤ p ≤ ∞, then we have the
following compact embeddings:
ˆ if p < N , then W 1,p (Ω) ,→,→ Lq (Ω), for every 1 ≤ q < p∗ ,
ˆ if p = N , then W 1,p (Ω) ,→,→ Lq (Ω), for every 1 ≤ q < +∞,
ˆ if p > N , then W 1,p (Ω) ,→,→ L∞ (Ω).
Remark 13. We denote by ,→,→ the compact embedding.
Before provide some informations on the dual of a Sobolev space, we state
a final result concerning the composition of Sobolev functions with regular
functions.
Theorem 1.3.6 (Stampacchia). Let F : R → R be a lipschitz continuous
function such that F (0) = 0. If u ∈ W01,p (Ω) then:
ˆ F (u) ∈ W01,p (Ω),
ˆ ∇F (u) = F 0 (u)∇u, a.e. in Ω.
Now we provide some tools to represent an element of the dual of a Hilbert
space.
8
1. Basic tools
1.4 Sobolev capacity
Theorem 1.3.7 (Riesz). Let H be a separable Hilbert space, and let T ∈ H 0
such that
|hT, xi| ≤ ||x||, ∀x ∈ H.
Then there exists a unique y ∈ H such that
hT, xi = (y|x),
∀x ∈ H.
We can present a representation theorem for the dual of a Sobolev space.
Theorem 1.3.8. Let p > 1, and let T ∈ (W01,p (Ω))0 , then there exists F ∈
0
(Lp (Ω))N such that
Z
hT, ui =
F · ∇u, ∀u ∈ W01,p (Ω).
Ω
0
Remark 14. We denote by W −1,p (Ω) the dual of W01,p (Ω).
1.4
Sobolev capacity
Definition 1.16. Let k ∈ N∗ , let 1 ≤ p < +∞, and let K ⊂ RN be a compact
set. The W k,p capacity is defined as
capW k,p (K) = inf{||ϕ||W k,p (RN ) , ϕ ∈ C0∞ (RN ) and ϕ ≥ 1 in K}.
It can be considered the capacity of any subset in the following way.
If A ⊂ RN is open, then let
capW k,p (A) = sup{capW k,p (K) : K ⊂ RN is compact and K ⊂ A},
and, more generally, if A ⊂ RN is any Borel set, then let
capW k,p (A) = sup{capW k,p (U ) : U ⊂ RN is open and A ⊂ U },
where is assumed inf ∅ = +∞ by convention.
The proofs of this section are in [12].
We give the following theorem that concerns some basic properties of capacity.
Theorem 1.4.1. The map capW k,p : E ⊂ RN → capW k,p (E) satisfies:
ˆ capW k,p (∅) = 0,
ˆ if E1 ⊆ E2 , then capW k,p (E1 ) ≤ capW k,p (E2 ),
9
1.5 Real analysis tools
ˆ if E =
+∞
[
1. Basic tools
En , then capW k,p (E) ≤
n=1
+∞
X
capW k,p (En ).
n=1
Remark 15. The map capW k,p is defined on Borel sets and gets [0, +∞] as
values. Thus, by the previous theorem, we have that capW k,p is an outer
measure.
Now we clarify what we mean by saying that a measure is absolutely continuous
with respect to capacity.
Definition 1.17. Let µ ∈ M(Ω), then µ is an absolutely continuous measure
with respect to the W k,p capacity if for every Borel set A ⊂ RN such that
capW k,p (A) = 0, then |µ|(A) = 0. This property is denoted by µ capW k,p .
Remark 16. In this case, by absolutely continuous, we intend that the measure
µ does not charge set of zero W k,p capacity.
We give a final property concerning capacity.
Proposition 1.4.2. Let k ∈ N∗ and let 1 < p < +∞ then, for every α > 0,
capW k,p ({|ϕ| ≥ α}) ≤
C
||ϕ||pW k,p (RN ) ,
αp
∀ϕ ∈ C0∞ (RN ),
for some constant C[N, k, p] > 0.
Lastly, we give a convergence result.
Proposition 1.4.3. Let k ∈ N∗ and 1 < p < +∞, and let (ϕn )n∈N ⊂ C0∞ (RN ).
If (ϕn )n∈N converges in W k,p (RN ), then there exists a subsequence (ϕnk )k∈N
that converges pointwise in RN \ E for some Borel set E ⊂ RN such that
capW k,p (E) = 0.
1.5
Real analysis tools
Definition 1.18. Let (Ω, Σ, µ) be a measure space, and let (fn )n∈N be a sequence of µ-measurable functions. The sequence (fn )n∈N is said to converge
almost everywhere to f if (fn )n∈N converges fn in Ω \ E, where E is such that
µ(E) = 0.
Definition 1.19. Let (Ω, Σ, µ) be a measure space, and let (fn )n∈N be a sequence of µ-measurable functions. The sequence (fn )n∈N is said to converge to
f in measure if, for every > 0,
lim µ({x ∈ Ω : |fn (x) − f (x)| > }) = 0.
n→+∞
Definition 1.20. Let (Ω, Σ, µ) be a measure space, and let (fn )n∈N be a sequence of µ-measurable functions. The sequence (fn )n∈N is said to converge to
f in Lp (Ω, µ), for 1 ≤ p ≤ +∞, if
lim kfn − f kLp (Ω,µ) = 0.
n→+∞
10
1. Basic tools
1.5 Real analysis tools
For the proofs of this section see [5], [15].
Theorem 1.5.1. Let (Ω, Σ, µ) be a measure space such that µ(Ω) < +∞. Let
(fn )n∈N be a sequence of µ-measurable functions converging almost everywhere
to f . If (fn )n∈N , f are bounded almost everywhere, then The sequence (fn )n∈N
converges to f in measure.
Theorem 1.5.2. Let (Ω, Σ, µ) be a measure space, and let (fn )n∈N be a sequence of µ-measurable functions that converges to f in measure. Then there
exists a subsequence (fnk )k∈N (Ω, Σ, µ) thet converges almost everywhere to f .
Theorem 1.5.3 (Egorov). Let (Ω, Σ, µ) be a measure space, let A ⊂ Ω such
that µ(Ω) ≤ +∞, (fn )n∈N be a sequence of µ-measurable functions that converges almost everywhere to f on A. Then, for every δ > 0, there exists a set
Aδ ⊆ A such that µ(Aδ ) ≤ δ and such that (fn )n∈N converges uniformly to f
on A \ Aδ .
Theorem 1.5.4 (Fatou). Let (Ω, Σ, µ) be a measure space, where µ is a nonnegative measure. Let (fn )n∈N be a sequence of nonnegative µ-measurable functions that converges almost everywhere to f . Then
Z
Z
f dµ ≤ lim inf
fn dµ.
n→+∞
Ω
Ω
Theorem 1.5.5 (Monotone convergence). Let (Ω, Σ, µ) be a measure space,
where µ is a nonnegative measure. Let (fn )n∈N be a sequence of nonnegative
µ-measurable functions such that fn ≤ fn+1 for every n ∈ N. Thus, if
f (x) = lim fn (x),
n→+∞
then
Z
Z
f dµ = lim
Ω
n→+∞
fn dµ.
Ω
Theorem 1.5.6 (Dominated convergence). Let (Ω, Σ, µ) be a measure space,
let g ∈ L1 (Ω, µ), and let (fn )n∈N be a sequence of µ-measurable functions such
that |fn | ≤ g almost everywhere and such that (fn )n∈N converges almost everywhere to f in Ω. Then f ∈ L1 (Ω, µ) and
Z
Z
f dµ = lim
Ω
n→+∞
11
fn dµ.
Ω
1.5 Real analysis tools
1. Basic tools
Theorem 1.5.7 (Vitali). Let (Ω, Σ, µ) be a measure space. Let (fn )n∈N be
a sequence of µ-measurable functions that converges almost everywhere to f .
Then
Z
|fn − f | dµ = 0,
lim
n→+∞
Ω
if and only if, for every > 0, there exist δ > 0, such that, if µ(E) < δ, then
Z
sup |fn | dµ < .
(1.3)
n∈N
E
This last condition is called equi-integrability.
Definition 1.21. Let (fn )n∈N be a sequence of functions in Lp (Ω) with 1 <
p < +∞. Then (fn )n∈N converges weakly to f in Lp (Ω) if
Z
Z
0
lim
fn gdx =
f gdx, ∀ g ∈ Lp (Ω).
n→+∞
Ω
Ω
Definition 1.22. Let (fn )n∈N be a sequence of functions in L1 (Ω). Then
(fn )n∈N converges weakly to f in L1 (Ω) if
Z
Z
lim
fn g dx =
f g dx, ∀ g ∈ L∞ (Ω).
n→+∞
Ω
Ω
Definition 1.23. Let (fn )n∈N be a sequence of functions in W01,p (Ω) with
1 ≤ p ≤ +∞. Then (fn )n∈N converges to f in W01,p (Ω) if
lim kfn − f kW 1,p (Ω) = 0.
n→+∞
0
0
Definition 1.24. Let (Fn )n∈N be a sequence of functions in W −1,p (Ω). Then
0
(Fn )n∈N converges to F in W −1,p (Ω) if
lim kFn − F kLp0 (Ω)N = 0.
n→+∞
Definition 1.25. Let (fn )n∈N be a sequence of functions in W01,p (Ω) with
1 < p < +∞. Then (fn )n∈N converges weakly to f in W01,p (Ω) if, for every
0
T ∈ W −1,p (Ω),
lim hT, fn i = hT, f i.
n→+∞
Now, we present a consequence of the Banach-Alaoglu-Bourbaki theorem.
Theorem 1.5.8. Let (fn )n∈N be a sequence of functions bounded in W01,p (Ω)
with 1 < p < +∞. Then, there exists subsequence (fnk )k∈N that converges
weakly to f in W01,p (Ω).
Theorem 1.5.9. Let (fn )n∈N be a sequence of functions bounded in W01,p (Ω),
then:
12
1. Basic tools
1.6 The linear theory
ˆ if (fn )n∈N converges to f in W01,p (Ω), then there exists a subsequence
(fnk )k∈N that converges weakly to f ,
ˆ if (fn )n∈N converges weakly to f in W01,p (Ω), then ||fn ||W 1,p (Ω) is bounded
0
and
kf kW 1,p (Ω) ≤ lim inf kfn kW 1,p (Ω) ,
n→+∞
0
0
ˆ if (fn )n∈N converges weakly to f in W01,p (Ω), and if (gn )n∈N converges to
0
g in W −1,p (Ω)0 , then
lim hgn , fn i = hg, f i.
n→+∞
Definition 1.26. Let (µn )n∈N be a sequence of measures in M(Ω).The sequence (µn )n∈N converges weak-∗ to the measure µ if
Z
Z
lim
ϕ dµn =
ϕ dµ, ∀ϕ ∈ C0 (Ω).
n→+∞
Ω
Ω
Definition 1.27. Let (µn )n∈N be a sequence of measures in M(Ω).The sequence (µn )n∈N converges narrow to the measure µ if
Z
Z
lim
ϕ dµn =
ϕ dµ, ∀ϕ ∈ Cb (Ω),
n→+∞
Ω
Ω
where we have denoted by Cb (Ω) the bounded functions on Ω
We provide the following fixed point theorem.
Theorem 1.5.10 (Schauder). Let X be a Banach space, and let G ⊆ X be
convex, bounded and closed. Thus, if the map S : G → G is continuous and
such that S(G) is compact, then S has, at least, a fixed point.
Lastly, we give the following definition.
Definition 1.28. The map a : Ω × RN → RN is called Carathéodory function
if a(·, ξ) is measurable for every ξ ∈ RN and if a(x, ·) is continuous for almost
every x ∈ Ω.
1.6
The linear theory
This section concerns the linear Dirichlet problem that is fundamental to
understand the semilinear problem.
Let Ω ⊂ RN be a bounded domain, where N ≥ 3.
Let the problem be
(
−∆u = f
u=0
in Ω,
on ∂Ω.
We define the weak solution to (1.4).
13
(1.4)
1.6 The linear theory
1. Basic tools
2N
Definition 1.29. Let f ∈ L N +2 . A function u : Ω → R is a solution to (1.4)
if:
ˆ u ∈ W01,2 (Ω),
Z
Z
ˆ
∇u · ∇v =
f v,
Ω
∀v ∈ W01,2 (Ω).
Ω
Remark 17. We recall that
2N
N +2
Z
∗ 0
= (2 ) so that the term
f v is well defined
Ω
by the Hölder inequality.
Now we state some regularity results, due to Stampacchia, on the solution to
(1.4).
For the proofs of this section see [5], [14].
Theorem 1.6.1. Let f ∈ Lp (Ω). Then the solution u to (1.4) is such that
u ∈ W01,2 (Ω). Moreover:
ˆ if p >
that
N
,
2
then u ∈ L∞ (Ω) and there exists a constant C[Ω, N, p] such
||u||L∞ (Ω) ≤ C||f ||Lp (Ω) ,
≤p≤
ˆ if N2N
+2
such that
N
,
2
∗∗
then u ∈ Lp (Ω) and there exists a constant C[Ω, N, p]
||u||Lp∗∗ (Ω) ≤ C||f ||Lp (Ω) .
Now we show the solution via duality that was presented in order to obtain
uniqueness of the solution to some Dirichlet problem for divergence operators.
In reality, we could apply classic theory to the laplacian operator but we still
want to present the following approach that will be utilized later in this work.
Thus, we introduce the duality solution.
Let us consider the following problems
(
−∆u = f in Ω,
u=0
on ∂Ω.
(
−∆v = g
v=0
in Ω,
on ∂Ω,
where f, g ∈ L∞ (Ω). It follows from Theorem 1.6.1 that u, v ∈ W01,2 (Ω).
Therefore, we can take v, u as test functions, respectively, for the first and the
second problem above.
We have
Z
Z
Z
Z
ug =
∇v · ∇u =
∇u · ∇v =
vf.
Ω
Ω
Ω
Ω
Therefore, we have
Z
Z
ug =
Ω
vf.
Ω
14
1. Basic tools
1.6 The linear theory
Since by Theorem 1.6.1 g ∈ L∞ (Ω) means v ∈ L∞ (Ω), we can state that the
two integrals are well defined if f, u ∈ L1 (Ω).
This took Guido Stampacchia to give the following definition.
Definition 1.30. Let f ∈ L1 (Ω). A function u ∈ L1 (Ω) is a duality solution
to (1.4) if, for every g ∈ L∞ (Ω), then
Z
Z
ug =
f v,
Ω
Ω
where v is the solution to
(
−∆v = g
v=0
in Ω,
on ∂Ω.
We state some regularity on the duality solution to (1.4).
Theorem 1.6.2. Let f ∈ Lp (Ω). Then the duality solution u to (1.4) is such
that:
ˆ if 1 < p <
2N
,
N +2
∗∗
then u ∈ Lp (Ω),
ˆ if p = 1, then u ∈ Lq (Ω) for every q <
N
.
N −2
We still does not know anything about the gradient of the solution or what
happens when treating a measure.
We provide the following definition.
Definition 1.31. Let f ∈ M(Ω). A function u ∈ L1 (Ω) is a duality solution
to (1.4) if, for every g ∈ L∞ (Ω), then
Z
Z
ug =
vdf,
Ω
Ω
where v is the solution to
(
−∆v = g
v=0
in Ω,
on ∂Ω.
Theorem 1.6.3. Let f ∈ M(Ω). Then the duality solution u to (1.4) belongs
to Lq (Ω) for every q < NN−2 .
Lastly, we only have to state some regularity for the gradient of the solution.
Theorem 1.6.4. Let f ∈ Lp (Ω). Then
the duality solution u to (1.4) is such
1,p∗
2N
that, if 1 < p < N +2 , then u ∈ W0 (Ω).
15
1.6 The linear theory
1. Basic tools
Theorem 1.6.5. Let f ∈ M(Ω). Then the duality solution u to (1.4) is such
that u ∈ W01,q (Ω), for every q < NN−1 .
From what we have stated we can say that in general:
N
ˆ u∈
/ L N −2 (Ω),
N
ˆ ∇u ∈
/ L N −1 (Ω).
Thus, we could be interested to understand which are the “nearest” spaces the
solution belongs to.
Theorem 1.6.6. Let f ∈ M(Ω). Then the solution u to (1.4) is such that,
for every t > 0:
ˆ |{|u| > t}|
N −2
N
ˆ |{|∇u| > t}|
≤
N −1
N
N
C
||µ||M(Ω) that is u ∈ M N −2 (Ω),
t
≤
N
C0
||µ||M(Ω) that is ∇u ∈ M N −1 (Ω),
t
for some constants C[N ], C 0 [N ] > 0.
Remark 18. Lastly, we recall that the solution to (1.4) is unique. This result
is proved in [5].
Thus, we conclude the chapter with some tools that will be usefuls in order to
prove various existence theorems in the next chapter.
We start by a version of the classical weak maximum principle.
Theorem 1.6.7. Let u ∈ L1 (Ω). then:
Z
∞
ˆ if, for every ϕ ∈ C0 (Ω),
u∆ϕ ≥ 0, then u ≤ 0 in Ω,
Ω
ˆ if, for every ϕ ∈ C0∞ (Ω),
Z
u∆ϕ ≤ 0, then u ≥ 0 in Ω.
Ω
It is useful have a tool that permits to pass from an inequality in the sense of
distributions (i.e. Cc (Ω)0 ) to an inequality in the sense of C0∞ (Ω)0 .
Theorem 1.6.8. Let u ∈ L1 (Ω) and µ ∈ M(Ω), then the following are equivalent:
Z
Z
∞
ˆ for every ϕ ∈ C0 (Ω), − u∆ϕ ≤
ϕdµ,
Ω
Ω
16
1. Basic tools
1.6 The linear theory
ˆ for every ϕ ∈ Cc (Ω), −
Z
Z
u∆ϕ ≤
Ω
1
lim
→0 Z
ϕdµ, and
Ω
u+ = 0.
{x∈Ω:d(x,∂Ω)<}
Now we present Kato’s inequality that can be seen as the maximum principle
for functions not twice differentiable.
Theorem 1.6.9. If u ∈ L1 (Ω) is such that ∆u ∈ L1 (Ω), then
Z
Z
+
− u ∆ϕ ≤
χ{u>0} u∆ϕ, ∀ϕ ∈ Cc (Ω).
Ω
Ω
We provide the following Kato’s inequality up to the boundary.
Theorem 1.6.10. Let f ∈ L1 (Ω). If u ∈ L1 (Ω) is such that
Z
Z
∆uϕ ≥
f ϕ, ∀ϕ ∈ C0∞ (Ω),
Ω
then
Z
+
Ω
Z
∆u ϕ ≥
Ω
χ{u>0} f ϕ,
∀ϕ ∈ C0∞ (Ω).
Ω
Lastly, we present the inverse maximum principle.
Theorem 1.6.11. If u ∈ L1 (Ω) is such that ∆u ∈ M(Ω). If u ≥ 0 in Ω, then
the concentrated part of ∆u with respect to the W 1,2 capacity satisfies
(∆u)c ≤ 0.
17
1.6 The linear theory
1. Basic tools
18
Chapter 2
The problem in some regular
cases
Let Ω ⊂ RN be a bounded domain. We want to study the existence and, in
some cases, the uniqueness of the solution to the following semilinear Dirichlet
problem
(
−∆u + g(u) = f in Ω,
(2.1)
u=0
on ∂Ω,
where g : R → R is a continuous function that satisfies the sign condition
when some regularity is assumed on f .
Definition 2.1. We say that g : R → R satisfies the sign condition if
g(t)t ≥ 0,
∀t ∈ R.
(2.2)
Now we need to define what we mean by “solution”. If the function g and
the datum f are regular enough, we can formally talk of weak solution to (2.1)
in the sense that
Z
Z
Z
∇u · ∇v + g(u)v =
f v, ∀v ∈ W01,2 (Ω).
Ω
Ω
Ω
Later we will give a definition to specify what we mean by “regular enough”.
Our approach will rely in setting up the regularity on the right hand side
(the term we call datum from now on); thus, we will assume conditions on
the growth of the term g in order to obtain the existence of the solution to
(2.1). The function g, when satisfying the sign condition, is an absorption
term meaning that we have
Z
Z
|g(u)| ≤
|f |.
Ω
Ω
As we will see various times, this can be proved using a regularized version of
the sign function as a test function. In particular, when the last inequality is
19
2.1 Existence through variational method
2. The problem in some regular cases
assumed, then we have tools and estimates arising from the linear theory.
In this chapter we start presenting the existence of the solution in two regular
cases through two different methods. We will deal with cases where either the
datum is smooth or the absorption term is linear. As we will see, there is, at
least formally, a sort of balance between the datum and the absorption term
in order to obtain the existence of the solution.
We will deal with two cases:
ˆ the absorption term g is a continuous function satisfying the sign condition and the datum is a function in Lp (Ω); if p ≥ (2∗ )0 we obtain the
existence of the solution via variational method,
ˆ the absorption term g is linear (g(u) = u) and the datum is a general
measure; in this case the existence is proved via duality.
2.1
Existence through variational method
We briefly give a naive idea of the variational method in order to find
solutions of PDEs as in (2.1).
We begin to establish a convenient weak formulation for the problem. Then
we will write
L(u) = 0,
and we try to identify the real functional J that formally satisfies
J 0 (u) = L(u).
In this way the problem of finding a solution to PDEs turns out into looking
for a critical point of the functional J. It is not always possible to approach
the problem in this way. As we will see, we can apply this method only when
∗ 0
f ∈ L(2 ) (Ω) in (2.1), more generally when f ∈ (W01,2 (Ω))0 . This will be better
explained later but it is a consequence of the fact that only in this case we can
bound from below the functional associated to the problem.
We want to establish the existence and the uniqueness of the solution to
(2.1) through the variational method. We have already stated what we mean
by “solution” and this leads us to define the functional J in the following way
Z
Z
Z
1
2
|∇u| + G(u) − f u,
J(u) =
2 Ω
Ω
Ω
where
ˆ u ∈ W01,2 (Ω),
ˆ G : R → R is defined by G(t) =
Z
g(s)ds, ∀t ∈ R.
0
20
t
2. The problem in some regular cases
2.1 Existence through variational method
We assume that g satisfies the sign condition (2.2) so that we note
G(t) ≥ 0,
∀t ∈ R.
∗ 0
Remark 19. If f ∈ L(2 ) (Ω) then the functional J is well defined. Moreover,
J(u) 6≡ −∞.
This can be proved observing that:
ˆ G(t) ≥ 0,
∀t ∈ R,
∗
1,2
(Ω) ,→ L2 (Ω) by Sobolev embedding (see Theorem 1.3.3) then
ˆ W0Z
− f u ≥ −||f ||L(2∗ )0 (Ω) ||u||L2∗ (Ω) (Hölder inequality (1.2)).
Ω
Now we prove the existence of a minimizer for J through a standard minimization technique. At first, we show that minimazing sequences are bounded
in W01,2 (Ω).
Lemma 2.1.1. Let g : R → R be a continuous function that satisfies the sign
∗ 0
condition (2.2). If f ∈ L(2 ) (Ω) then
2
2
||v||W 1,2 (Ω) ≤ C[N ] J(v) + ||f ||L(2∗ )0 (Ω) , ∀v ∈ W01,2 (Ω).
0
Proof. Let v ∈ W01,2 (Ω). We have
Z
Z
Z
1
2
|∇v| + G(v) − f v = J(v),
2 Ω
Ω
Ω
so that,
1
2
Z
Z
2
|∇v| ≤ J(v) +
Ω
f v,
Ω
as G(v) ≥ 0.
Now we consider the second term on the right hand side of the last inequality
Z
Hölder
Sobolev
f v ≤ ||f ||L(2∗ )0 (Ω) ||v||L2∗ (Ω) ≤ ||f ||L(2∗ )0 (Ω) S||∇v||L2 (Ω)
Ω
Y oung
≤ C ||f ||2L(2∗ )0 (Ω) + S||∇v||2L2 (Ω)
so that, combining the last two inequalities we have
1
− S ||∇v||2L2 (Ω) ≤ J(v) + C ||f ||2L(2∗ )0 (Ω) .
2
The conclusion holds choosing such that 12 − S > 0.
21
2.1 Existence through variational method
2. The problem in some regular cases
Remark 20. We have just proved that J is bounded from below.
Proposition 2.1.2. Let g : R → R be a continuous function that satisfies the
∗ 0
sign condition (2.2). If f ∈ L(2 ) (Ω) then there exists a function u ∈ W01,2 (Ω)
such that
J(u) ≤ J(v),
∀v ∈ W01,2 (Ω)
Proof. Let (un )n∈N be a sequence in W01,2 (Ω).
If for every n ∈ N we have J(un ) < ∞, then the sequence (un )n∈N is bounded
in W01,2 (Ω) by Lemma 2.1.1.
It follows from the Banach-Alaoglu theorem (see Theorem 1.5.8) that there
exists a subsequence (unk )k∈N that weakly converges to some function u in
W01,2 (Ω). In particular, by Sobolev embedding, (unk )k∈N converges weakly to
∗ 0
∗
u in L2 (Ω). Since f ∈ L(2 ) (Ω) we have
Z
Z
f u = lim
f unk .
k→∞
Ω
Ω
By the Rellich-Kondrachov theorem (see Theorem 1.3.5), we can assume that
the not relabeled (unk )k∈N converges almost everywhere to u.
Observing that G(u) ≥ 0 and applying the Fatou’s lemma (see Lemma 1.5.4)
we obtain
Z
Z
G(u) ≤ lim inf
G(unk ),
k→∞
Ω
Ω
and by weakly lower semicontinuity of the norms we have
Z
Z
2
|∇u| ≤ lim inf
|∇unk |2 .
k→∞
Ω
Ω
Combining the previous inequalities
J(u) ≤ lim inf J(unk ).
k→∞
We choose (un )n∈N as a minimizing sequence of J and since u ∈ W01,2 (Ω) we
have
J(u) ≤ lim inf J(unk ) = inf
J(v),
1,2
k→∞
v∈W0 (Ω)
so that u is a minimizer for J.
Now we prove that the minimum for the functional J is a solution to (2.1).
We suppose that u is the minimum for J
∀t ∈ R, ∀v ∈ W01,2 (Ω),
J(u) ≤ J(u + tv),
that is
1
2
Z
Z
Z
1
|∇u| + G(u) − f u ≤
2
Ω
Ω
Ω
2
22
Z
2
Z
|∇u| + t
Ω
∇u · ∇v +
Ω
2. The problem in some regular cases
t2
+
2
Z
2
Z
Z
|∇v| +
Ω
2.1 Existence through variational method
G(u + tv) −
Ω
Z
fu + t
Ω
f v.
Ω
cancelling equal terms, we get
Z
Z Z
Z
t2
G(u + tv) − G(u)
v − t fv +
|∇v|2 .
0 ≤ t ∇u · ∇v + t
tv
2
Ω
Ω
Ω
Ω
We note that the second term on the right hand side is the incremental quotient
of G. So if t ≥ 0, dividing by t and letting t tend to zero, we obtain
Z
Z
Z
0≤
∇u · ∇v + g(u)v − f v,
Ω
Ω
Ω
while if t ≤ 0 we can reason in the same way in order to obtain the reverse
inequality. This implies
Z
Z
Z
∇u · ∇v + g(u)v =
f v,
Ω
Ω
Ω
and so we have proved that u is the solution to (2.1).
There is a problem in the proof we have completed; we have understimated
what regularity we need in order to let t tend to zero and to pass to the limit
into the integral of G.
In order to define well all the terms in the weak formulation we must ask for
further regularity on g(u) while restricting the test functions. This argument
takes us to the following definition.
Definition 2.2. Let g : R → R be a continuous function and let f ∈ Lp (Ω),
with p ≥ (2∗ )0 . A function u : Ω → R is a solution to (2.1) if:
ˆ u ∈ W01,2 (Ω),
ˆ g(u) ∈ L1 (Ω),
Z
Z
Z
ˆ
∇u · ∇v + g(u)v =
f v,
Ω
Ω
∀v ∈ W01,2 (Ω) ∩ L∞ (Ω).
Ω
At this point, we can discuss the uniqueness of the solution that, in the variational setting, is proved when is assumed further regularity on the function
g.
Proposition 2.1.3. Let g : R → R be a nondecreasing continuous function
∗ 0
that satisfies the sign condition (2.2). If f ∈ L(2 ) (Ω) then the problem (2.1)
has a unique solution.
23
2.1 Existence through variational method
2. The problem in some regular cases
Proof. Suppose that u, v are solutions to (2.1), taking the truncation at the
level k of the difference of the solutions as a test function we have
Z
Z
Z
∇u · ∇(u − v)k + g(u)(u − v)k =
f (u − v)k ,
Ω
and
Ω
Z
Ω
Z
Z
∇v · ∇(u − v)k +
Ω
g(v)(u − v)k =
Ω
f (u − v)k ,
Ω
combining the previous equalities we have
Z
Z
Z
Z
∇u · ∇(u − v)k − ∇v · ∇(u − v)k + g(u)(u − v)k − g(v)(u − v)k = 0,
Ω
Ω
Ω
Ω
so that, since ∇u · ∇(u − v)k = 0 a.e. on |u − v| ≥ k
Z
Z
2
|∇(u − v)k | + (g(u) − g(v))(u − v)k = 0.
Ω
Ω
Recalling that g is not decreasing we can drop the second term which is nonnegative. This implies (u − v)k = 0 and the same argument holds for every
k, so that we can let k tend to infinity and, since (u − v) ∈ W01,2 (Ω), we have
u = v.
So we have proved what is summarized in the following picture.
24
2. The problem in some regular cases
2.2 A linear case proved via duality
Remark 21. When g(u) ∈ L1 (Ω) the existence of the minimum for the functional implies the existence of the solution to the problem (2.1). For istance,
= 2∗ , then the minimum u for the
if |g(s)| ≤ C(|s|q + 1), where q ≤ N2N
−2
functional J belongs to Lq (Ω) by Sobolev embedding. This implies that
g(u) ∈ L1 (Ω) and u is the solution to the problem. If g is also a nondecreasing
function then sthe solution to the problem is unique.
2.2
A linear case proved via duality
In this section we are going to deal with the existence of the solution via
duality.
The case we are approaching is
(
−∆u + u = µ in Ω,
(2.3)
u=0
on ∂Ω,
where µ ∈ M(Ω) and the absorption term is linear.
Before proceeding, we need to define the test function we are going to use
(
(|v| − k) sign(v) if |v| ≥ k,
Gk (v) =
0
otherwise.
Now we introduce the following problem
(
−∆v + v = φ in Ω,
v=0
on ∂Ω,
where φ ∈ Lq (Ω), q >
(2.4)
N
.
2
Thus we have
Z
Z
∇v · ∇Gk (v) +
Ω
Z
vGk (v) =
Ω
φGk (v).
Ω
25
2.2 A linear case proved via duality
2. The problem in some regular cases
We recall that ∇v = ∇Gk (v) where Gk (v) =
6 0 and we state that the second
term is positive, so that we have
Z
Z
2
|∇Gk (v)| ≤
φGk (v).
Ω
Ω
Now we apply Stampacchia’s regularity (see Theorem 1.6.1) in order to obtain
that v ∈ L∞ (Ω).
We use v as a test function in (2.3) and we use u as a test function in (2.4)
Z
Z
Z
∇u · ∇v + uv =
µv,
Ω
Ω
Ω
||
Z
Z
Z
∇v · ∇u +
Ω
vu =
Ω
φu.
Ω
This takes us to give the following definition.
Definition 2.3. Let µ ∈ M(Ω). A function u ∈ L1 (Ω) is a duality solution
to (2.3) if for every φ ∈ L∞ (Ω) then
Z
Z
µv =
φu,
Ω
Ω
where v is the solution to (2.4) with datum φ.
We present an existence result.
Theorem 2.2.1. Let µ ∈ M(Ω). Then there exists a unique solution u to
(2.3) via duality. Moreover u ∈ Lq (Ω), for every p < NN−2 .
Proof. We have already stated that φ ∈ Lq (Ω), q >
Stampacchia’s regularity.
N
2
so that we can apply
Let define the linear functional T : Lq (Ω) → R by
Z
hT, φi =
vdµ.
Ω
We prove that T is continuous
Z
|hT, φi| ≤
|v|d|µ| ≤ ||µ||M(Ω) ||v||L∞ (Ω) ,
Ω
so that we can apply the Riesz rappresentation theorem (see Theorem 1.3.7).
0
Thus, there exists a unique u ∈ Lq (Ω) such that
Z
hT, φi =
uφ, ∀φ ∈ Lq (Ω).
Ω
26
2. The problem in some regular cases
2.2 A linear case proved via duality
Since L∞ (Ω) ⊂ Lq (Ω), we clearly keep the last achievement for every φ ∈
L∞ (Ω) and say that u is the duality solution to (2.3).
We now want to show that the solution u does not depend on q; infact if
we have p > q > N2 we can make the same argument for φp , φq so that
Z
Z
Z
up φ =
µv =
uq φ, ∀g ∈ L∞ (Ω)
Ω
Ω
Ω
and so up = uq a.e.
Then, for every > 0, we can take the regularity of the solution u up to
N 0
L( 2 ) − (Ω), where ( N2 )0 = NN−2 .
Remark 22. In the same way, it can be proved some regularity for the gradient
of the duality solution to (2.3). Indeed, u ∈ W01,q (Ω), for every q < NN−1 .
At this point, we can improve our picture and observe that we studied the
existence of the solution when the data is a general measure but we had to
increase the regularity of the absorption term in order to obtain the existence.
Figure 2.1: We have given an example to understand the duality solution.
27
2.2 A linear case proved via duality
2. The problem in some regular cases
28
Chapter 3
Measure data
In this chapter we will approach the semilinear Dirichlet problem where
the datum is either a general measure or an absolutely continuous measure.
Before proceeding this way, we have to study what we missed with the variational method; since we know that (2∗ )0 > 1 (N ≥ 3), we have no possibility
to reach the existence of the solution through the variational setting when the
datum is in L1 (Ω).
So we are going to approach the problem in another way.
Let us consider the problem
(
−∆u + g(u) = µ in Ω,
u=0
on ∂Ω,
(3.1)
where g : R → R is a continuous function that satisfies the sign condition (2.2).
At the beginning µ will be an L1 (Ω) function and later a general measure.
3.1
L1(Ω) and measure data
In the linear Dirichlet problem there is, in some sense, no difference between
L (Ω) or general measure data; the solution always exists and the estimates
are essentially the same (see Section 1.6).
This is no more true in the semilinear problem due to the absorption role
played by g.
We are going to show that the solution always exists when the datum is in
L1 (Ω).
Meanwhile, we will need to strength the assumptions on the measure data in
order to obtain the existence of the solution.
1
29
3.1 L1 (Ω) and measure data
3. Measure data
Let us provide the following.
Definition 3.1. Let g : R → R be a continuous function and let µ ∈ Lp (Ω).
A function u : Ω → R is a solution to (3.1) if:
ˆ u ∈ L1 (Ω),
ˆ g(u) ∈ L1 (Ω),
Z
Z
Z
g(u)ϕ =
µϕ,
ˆ − u∆ϕ +
Ω
Ω
∀ϕ ∈ C0∞ (Ω).
Ω
Remark 23. We have denoted by C0∞ (Ω) the set of functions that belong to
C ∞ (Ω) and are zero on the boundary of Ω.
Therefore, we are ready to present the existence result when the datum is
in L1 (Ω).
To briefly explain what we are going to do, we intend to approximate both
the datum and the absorption term in order to obtain a problem that we are
already able to solve.
At that point, we get the solution to the problem where the term g and the
datum are approximated and then we prove that the limit of this solution
solves (3.1).
Theorem 3.1.1. Let g : R → R be a continuous function satisfying the sign
condition (2.2) and let µ ∈ L1 (Ω). Then there exists a solution u to (3.1).
Proof. As already stated, we prove the existence of the solution via approximation.
We take µn as the truncation of µ at level n and we take gn as the truncation
of g at level n.
The problem becames something we are able to solve through the variational
setting (see Section 2.1)
(
−∆un + gn (un ) = µn in Ω,
un = 0
on ∂Ω.
We define


1




 |u| − k sign(u)
Bk, (u) =


−1




0
if u ≥ k + ,
if − k − ≤ u < −k or k ≤ u < k + ,
if u ≤ −k − ,
otherwise.
30
3.1 L1 (Ω) and measure data
3. Measure data
Using Bk, (un ) as a test function we obtain
Z
Z
Z
gn (un )Bk, (un ) ≤
µn Bk, (un ) ≤
Ω
|µn |.
{|un |≥k}
Ω
Letting tend to zero, by the dominated convergence theorem (see Theorem
1.5.6)
Z
Z
gn (un )Bk, (un )→
|gn (un )|,
{|un |≥k}
Ω
so that
Z
Z
|gn (un )| ≤
{|un |≥k}
In particular taking k = 0
Z
Z
Z
|µ|.
|µn | ≤
|gn (un )| ≤
Ω
|µn |.
{|un |≥k}
Ω
Ω
We can bring us back to the following linear problem
(
−∆un = µn − g(un ) in Ω,
un = 0
on ∂Ω,
where we have just proved that the datum is bounded in L1 (Ω).
Therefore, by Stampacchia’s regularity (see Theorem 1.6.5), we can state that
un is bounded in W01,q (Ω) for every q < NN−1 .
Moreover, by the Rellich-Kondrachov theorem (see Theorem 1.3.5), we know
that
W01,q (Ω) ,→,→ Lp (Ω) with 1 ≤ p < q ∗ ,
so that we can say, up to subsequences, that un converges to some function u
in L1 (Ω), and un converges to u a.e.
31
3.1 L1 (Ω) and measure data
3. Measure data
Thus, we have started from
Z
Z
Z
µn v,
− un ∆v + gn (un )v =
Ω
Ω
∀v ∈ C0∞ (Ω),
Ω
and we do not have any problems in the first and the third term when we let
n tend to infinity.
Instead, we can not take the second term to the limit yet; we need some extra
work. We want to apply the Vitali convergence theorem (see Theorem 1.5.7)
but we still miss the equi-integrability condition (1.3).
Thus we have
Z
Z
|gn (un )| =
E
Z
Z
|gn (un )| +
E∩{|un |≤k}
|gn (un )| ≤
E∩{|un |>k}
Z
E∩{|un |≤k̃}
|gn (un )|+
Z
+
{|un |>k̃}
|µn | ≤
|gn (un )| + ,
2
E∩{|un |≤k̃}
where in the last inequality we used that µn satisfies the equi-integrability
condition. We also used that, by Stampacchia’s regularity, we have that un is
N
bounded in M N −2 (Ω) (see Theorem 1.6.6) so that
C
n
o
|un | > k̃ ≤
N
≤ δ̃.
k̃ N −2
Since E was arbitrary, we have
Z
Z
|gn (un )| ≤ max |gn | dx = max |gn ||E| ≤ .
2
E∩{|un |<k̃}
E
[−k̃,k̃]
[−k̃,k̃]
Therefore, the function g satisfies the equi-integrability condition and it can
be applied the Vitali convergence theorem that guarantees us that
Z
Z
L1
gn (un ) → g(u), so that
gn (un )v →
g(u)v.
Ω
Ω
The conclusion holds since we proved that the limit of un satisfies
Z
Z
Z
− u∆v + g(u)v =
µv, ∀v ∈ C0∞ (Ω).
Ω
Ω
Ω
Thus, we have proved we have solution to (3.1) when the datum is in L1 (Ω).
32
3.1 L1 (Ω) and measure data
3. Measure data
We can improve our picture that explains what we have done so far.
Now we show the first important difference between the linear and the
semilinear problem. In the linear case we have the existence of the solution
whatever measure we are treating. In the semilinear problem, the fact that
g(u) must belong to L1 (Ω) forces the solution to the problem to be more regular, instead, less regularity on the datum translates in less regularity for the
solution. Thus, we have a sort of balance where the regularity of the measure
is explained by the concentrated part. If a measure is concentrated we could
not have solution. This can be observed with an example that deals with the
most common concentrated measure: Dirac delta measure.
Before presenting the example, we need to establish what is a solution when
the datum is a general measure.
Let us provide the following.
Definition 3.2. Let g : R → R be a continuous function and let µ ∈ M(Ω).
A function u : Ω → R is a solution to (3.1) if
ˆ u ∈ L1 (Ω),
ˆ g(u) ∈ L1 (Ω),
Z
Z
Z
ˆ − u∆ϕ +
g(u)ϕ =
ϕdµ,
Ω
Ω
∀ϕ ∈ C0∞ (Ω).
Ω
33
3.1 L1 (Ω) and measure data
3. Measure data
Now we present the example.
Example 3.1. We consider the following problem
(
−∆u + |u|p−1 u = δ0 in B1 ,
u=0
on ∂B1 ,
(3.2)
where B1 is the unit ball centered at 0. We are going to prove that, if p ≥
then the problem (3.2) has no solution.
N
,
N −2
We start assuming, by contradiction, that the problem has a solution.
Let ϕ ∈ Cc∞ (RN ) such that supp(ϕ) ⊂ B1 , and ϕ(0) 6= 0. For every n ∈ N∗ let
ϕn : B1 → R,
be the function defined by
ϕn (x) = ϕ(nx).
This function is regular enough to be used as a test function in the weak
formulation.
We have
Z
Z
Z
p−1
−
u∆ϕn +
|u| uϕn =
δ0 ϕn = ϕ(0).
(3.3)
B1
B1
B1
The previous equation holds for every n. We let n tend to infinity.
We have no problem to apply the dominated convergence theorem to a bounded
function and we know ϕn → 0 so that
Z
lim
|u|p−1 uϕn = 0.
n→∞
B1
For what concerns the first term on the left hand side we have
Z
Z
Z
2
u∆ϕn = n
u∆ϕ(nx)dx,
u∆ϕn =
B1
B1
B1
n
n
where in the first equality we used that supp(ϕn ) ⊂ B 1 . Now we can say
n
Z
B1
Hölder
u∆ϕn ≤ n2
! p1
Z
p
=n
p
p0
|u|
|∆ϕ(nx)| dx
B1
B1
n
2− pN0
! 10
Z
n
! p1
Z
|u|p
! 10
p
Z
p0
|∆ϕ| dy
,
B1
B1
n
n
where the last equality holds by the change of variable y = nx.
34
=
3. Measure data
3.2 Polynomial growth
Letting n tend to infinity we note that the right hand side vanishes only if
≤ 0, but this is equally to say that p ≥ NN−2 .
2− N
p0
By the dominated convergence theorem, we have
Z
lim
u∆ϕn = 0,
n→∞
B1
and it follows from (3.3) that ϕ(0) = 0, a contradiction, as ϕ has been chosen
such that ϕ(0) 6= 0.
Thus, we get the following picture.
3.2
Polynomial growth
We focus our attention to the existence of the solution when considering
the problem (3.1) with measure data. Let us assume some conditions on the
growth of the term g, then we will be able to prove that either the problem
(3.1) has a solution regardless of what measure we are treating or, in view
of the Example 3.1, we could need to restrict the set of measures in order to
obtain the existence.
In this section we are going to treat the problem (3.1) where the nonlinearity
g has polynomial growth at most
|g(t)| ≤ C(|t|p + 1),
where C, p > 0.
35
∀t ∈ R,
3.2 Polynomial growth
3. Measure data
We are going to show there is a critical value p for the growth of g:
ˆ if p <
(3.1);
N
,
N −2
we have no problems to prove existence of the solution to
ˆ if p ≥ NN−2 , then this would mean further assumptions on the measure
data in order to guarantee the existence of the solution.
3.2.1
Subcritical growth when p <
the measure
N
N −2
approximating
Before presenting and proving the existence theorem, we need some tools
that are usefuls in the proof of the theorem.
The first result concerns the possibility of approximating a general measure
with a sequence of functions which guarantees some regularity of the norm
convergence.
Lemma 3.2.1. For every µ ∈ M(Ω) we can find a sequence (µn )n∈N ∈ C ∞ (Ω)
converging weakly to µ in the sense of measures such that
lim ||µn ||L1 (Ω) = ||µ||M(Ω)
n→∞
Proof. We construct the sequence (µn )n∈N through convolution with a mollifier.
Let (ρn )n∈N be a sequence of mollifiers and µn : Ω → R be a function defined by
Z
µn (x) =
ρn (x − y)dµ(y),
Ω
∞
so that (µn )n∈N ∈ C (Ω).
If ρn is an even function and for every ψ ∈ C0 (Ω) we have
Z
Z
Z Z
F ubini
ψµn =
ρn (x − y)ψ(x)dx dµ(y) = (ρn ∗ ψ)dµ,
Ω
Ω
Ω
Ω
by the regularity of the convolution we have that (ρn )n∈N converges uniformly
to ψ in Ω so that we deduce the weak convergence in the sense of measures
Z
Z
ψµn −→
ψdµ, ∀ψ ∈ C0 (Ω).
Ω
Ω
In particular, by the lower semicontinuity of the norm under weak convergence
we have
||µ||M(Ω) ≤ lim ||µn ||L1 (Ω) .
n→∞
36
3. Measure data
3.2 Polynomial growth
Since we have
Z
ρn (x − y)dx ≤ 1,
Ω
we obtain
||µn ||L1 (Ω) ≤ ||µ||M(Ω) ,
and the conclusion holds.
This result is useful in the proof of the lemma we are going to establish
now.
Lemma 3.2.2. Let g : R → R be a continuous function satisfying the sign
condition (2.2) and let µ ∈ M(Ω). If u is a solution to (3.1) then we have the
following absorption estimate
||g(u)||L1 (Ω) ≤ ||µ||M(Ω) .
(3.4)
Proof. Let (fn )n∈N ⊂ L∞ (Ω) be a sequence converging to g(u) in L1 (Ω) and let
(µn )n∈N ⊂ L∞ (Ω) be a sequence converging to µ in the sense of the measures
and such that (µn )n∈N is bounded in L1 (Ω).
Let us consider the problem (3.1) as a linear problem
(
−∆un = µn − fn
u=0
in Ω,
on ∂Ω.
By the linear theory, we know that the solution to the linear problem where
the datum is in L∞ (Ω) satisfies
un ∈ W01,2 (Ω) ∩ L∞ (Ω).
Let define


1
if t > ,


 t S (t) =
if − ≤ t ≤ ,



−1
if t < −.
37
3.2 Polynomial growth
3. Measure data
Using S (un ) as a test function we obtain
Z
Z
Z
2 0
|∇un | S (un ) + fn S (un ) =
S (un )µn .
Ω
Ω
Ω
Since S (un ) is nondecreasing and |S (un )| ≤ 1, we have
Z
Z
|µn | = ||µn ||L1 (Ω) .
fn S (un ) ≤
Ω
Ω
We can apply Stampacchia’s regularity theory (see Theorem 1.6.5) and say
that un is bounded in W01,q (Ω), for every q < NN−1 . By Rellich-Kondrachov
compactness theorem, we deduce that there exists a subsequence (unk )k∈N
converging to some function v in L1 (Ω). Since u and v satisfies the same
problem, by uniqueness of the linear Dirichlet problem, we have u = v. This
implies that (un )n∈N converges to u in L1 (Ω). By the dominated convergence
theorem, we know
L1
fn S (un ) → g(u)S (u),
and we obtain
Z
g(u)S (un ) ≤ lim ||µn ||L1 (Ω) .
n→∞
Ω
By Lemma 3.2.1, we can choose a sequence (µn )n∈N such that
lim ||µn ||L1 (Ω) = ||µ||M(Ω) .
n→∞
Thus we deduce
Z
g(u)S (un ) ≤ ||µ||M(Ω) ,
Ω
since this is true for every > 0 then we have
Z
g(u) sign(u) ≤ ||µ||M(Ω) ,
Ω
and the conclusion holds by the sign condition.
38
3. Measure data
3.2 Polynomial growth
We are now ready to state and to prove the existence theorem of this section.
Theorem 3.2.3. Let g : R → R be a continuous function satisfying the sign
condition (2.2) and such that
|g(t)| ≤ C(|t|p + 1),
where C, p > 0. If p <
solution.
N
N −2
∀t ∈ R,
(3.5)
then for every µ ∈ M(Ω) the problem (3.1) has a
Proof. Let (µn )n∈N be a sequence in L∞ (Ω), bounded in L1 (Ω) and that converges weakly to µ in the sense of measures in Ω.
Recalling Remark 21, we get the existence of the variational solution (see
Section 2.1) to the problem
(
−∆un + g(un ) = µn in Ω,
un = 0
on ∂Ω,
where the datum is in L∞ (Ω).
Thus, for every ϕ ∈ C0∞ (Ω) we obtain
Z
Z
Z
ϕµn .
− un ∆ϕ + g(un )ϕ =
Ω
Ω
Ω
Since (µn )n∈N is bounded in L1 (Ω), we are regular enough to apply (3.4), and
say that the sequence ((µn ) − (g(un ))n∈N is bounded in L1 (Ω).
Applying Stampacchia’s regularity theory to the following problem
(
−∆un = µn − g(un ) in Ω,
un = 0
on ∂Ω,
then we can say that there exists a subsequence (unk )k∈N that converges to
some function u in Lr (Ω), for 1 ≤ r < NN−2 .
Since p < NN−2 , then it follows from a generalized version of the dominated
convergence theorem that g(unk )k∈N converges to g(u) in L1 (Ω).
Letting k tend to infinity we have that for every ϕ ∈ C0∞ (Ω)
Z
Z
Z
− u∆ϕ + g(u)ϕ =
ϕ dµ.
Ω
Ω
Ω
The conclusion holds since we proved that the limit of the solution to the
approximated problem is the solution to the problem (3.1).
39
3.2 Polynomial growth
3. Measure data
Thus, we can improve our picture in the following way.
3.2.2
Method of sub and supersolutions
We want to explain the method of sub and supersolutions in order to find
a solution to the problem (3.1) when the nonlinearity g has a critical or supercritical polynomial growth.
We will start clarifying what we mean by “sub-supersolutions”.
Definition 3.3. Let g : R → R be a continuous function and let µ ∈ M(Ω).
A function u : Ω → R is a subsolution to (3.1) if:
ˆ u ∈ L1 (Ω),
ˆ g(u) ∈ L1 (Ω),
Z
Z
Z
ˆ − u∆ϕ +
g(u)ϕ ≤
ϕdµ,
Ω
Ω
Ω
40
∀ϕ ∈ C0∞ (Ω), ϕ ≥ 0.
3. Measure data
3.2 Polynomial growth
In the same way we provide the following.
Definition 3.4. Let g : R → R be a continuous function and let µ ∈ M(Ω).
A function u : Ω → R is a supersolution to (3.1) if:
ˆ u ∈ L1 (Ω),
ˆ g(u) ∈ L1 (Ω),
Z
Z
Z
g(u)ϕ ≥
ϕdµ,
ˆ − u∆ϕ +
Ω
Ω
∀ϕ ∈ C0∞ (Ω), ϕ ≥ 0.
Ω
This method relies on the idea that between a subsolution and a supersolution to (3.1) there should be a solution. From now on, we could require the
nonlinearity g to satisfy the following integrability condition.
Definition 3.5. Let g : R → R be a continuous function. We say that g
satisfies the integrability condition if for every w, w, w ∈ L1 (Ω) such that
w ≤ w ≤ w in Ω, if g(w) ∈ L1 (Ω) and if g(w) ∈ L1 (Ω), then g(w) ∈ L1 (Ω).
Remark 24. If g(s) = sq , then, for every q ≥ 0, the function g satisfies the
integrability condition. We can also present the following function g(s) =
s2 | sin(s)| that does not satisfy the integrability condition.
In order to prove the main theorem for the sub and supersolution method,
we should ask for an equivalent formulation of the integrability condition.
Lemma 3.2.4. Let g : R → R be a continuous function. Then, g satisfies the
integrability condition if and only if for every w, w ∈ L1 (Ω) such that w ≤ w
in Ω and g(w), g(w) ∈ L1 (Ω), there exists h ∈ L1 (Ω) such that, for every
w ∈ L1 (Ω), if w ≤ w ≤ w in Ω, then |g(w)| ≤ h in Ω.
[see proof in [14], Lemma 6.9].
We present an important corollary of the previous lemma.
Corollary 3.2.5. Let g : R → R be a continuous function satisfying the
integrability condition (3.5). If (un )n∈N is a monotone sequence converging
to u in L1 (Ω) and if for every n ∈ N, (g(un ))n∈N ∈ L1 (Ω), then (g(un ))n∈N
converges to g(u) in L1 (Ω) if and only if g(u) ∈ L1 (Ω)
[see proof in [14], Corollary 6.10].
The method relies on the following theorem, that is proved via Schauder fixed
point theorem (see Theorem 1.5.10).
Theorem 3.2.6. Let g : R → R be a continuous function satisfying the integrability condition (3.5) and let µ ∈ M(Ω). If the problem (3.1) has a subsolution u and supersolution u and if u ≤ u in Ω, then there exists a solution u
to (3.1) such that u ≤ u ≤ u.
41
3.2 Polynomial growth
3. Measure data
Proof. We first modify g so that the solution of the modified problem is itself
solution to the problem (3.1).
We define g̃ : Ω × R → R as


g(u(x)) if t < u(x),
g̃(x, t) = g(t)
if u(x) ≤ t ≤ u(x),


g(u(x)) if t > u(x).
As defined, g̃ is a Carathéodory function (see Definition 1.28) and, by integrability condition for g, for every v ∈ L1 (Ω), we have g̃(·, v) ∈ L1 (Ω).
From now on, we will proceed step by step.
i) At first we prove that if u satisfies
(
−∆u = µ − g̃(·, u) in Ω,
u=0
on ∂Ω,
(3.6)
then u ≤ u ≤ u. Thus g̃(·, u) = g(u) and u is a solution to (3.1).
We only prove that u ≤ u in Ω. The other inequality u ≤ u is similar.
We observe that u is a solution to (ii), and u is a supersolution to (3.1).
Taking the difference of the two formulations, we have, for every ϕ ∈
C0∞ (Ω) such that ϕ ≥ 0,
Z
Z
Z
(u − u)∆ϕ ≥ [g̃(·, u) − g(u)]ϕ =
χ{u≤u} [g̃(·, u) − g(u)]ϕ.
Ω
Ω
Ω
We apply Kato’s inequality up to the boundary (see Theorem 1.6.10) to
the function u − u
Z
Z
+
(u − u) ∆ϕ ≥
χ{u>u} χ{u≤u} [g̃(·, u) − g(u) ϕ = 0,
Ω
Ω
then we have
Z
(u − u)+ ≤ 0.
Ω
+
Thus, we obtain (u − u) = 0 in Ω, that is u ≤ u in Ω.
This implies, by contruction, g̃(·, u) = g(u). Thus u is solution to (3.1).
ii) Let define
G : L1 (Ω) → L1 (Ω),
v → u.
42
3. Measure data
3.2 Polynomial growth
This map assigns to every v ∈ L1 (Ω) the solution u to the following
linear problem
(
−∆u = µ − g̃(·, v) in Ω,
u=0
on ∂Ω.
We want to prove that this map is continuous in L1 (Ω).
By the integrability condition we can apply Lemma 3.2.4.
Therefore, we have that, for every x ∈ Ω and for every v ∈ L1 (Ω), there
exists h(x) ∈ L1 (Ω) such that
|g̃(x, v(x))| ≤ h(x).
We choose a sequence (vn )n∈N converging to some function v in L1 (Ω).
We can still say
|g̃(x, vn (x))| ≤ h(x).
By the dominated convergence theorem, we have that (g̃(·, vn (x)))n∈N converges to g̃(·, v(x)) in L1 (Ω). Thus, by the uniqueness of the solution to
the linear problem, we can say that un converges to u in L1 (Ω). Therefore,
we proved that G is continuous.
iii) In this step we intend to prove that the set G(L1 (Ω)) is bounded and
relatively compact in L1 (Ω).
For every v ∈ L1 (Ω) we have
||µ − g̃(·, v)||M(Ω) ≤ ||µ||M(Ω) + ||g̃(·, v)||M(Ω) ≤ ||µ||M(Ω) + ||h||L1 (Ω) .
By the linear theory we know
||G(v)||L1 (Ω) = ||u||L1 (Ω) ≤ C ||µ||M(Ω) + ||h||L1 (Ω) .
Thus, the set G(L1 (Ω)) is bounded in L1 (Ω).
By Stampacchia’s regularity theory we have that, for every q < NN−1 , G(v)
is bounded W01,q (Ω). Thus, by the Rellich-Kondrachov theorem, we have
that the set G(L1 (Ω)) is relatively compact in L1 (Ω).
iv) Now we can finally apply Schauder fixed point theorem so that G has a
fixed point u ∈ L1 (Ω). By the first step, u is a solution to (3.1) and the
conclusion holds.
3.2.3
Critical and supercritical growth when p ≥
through sub and supersolutions method
N
N −2
We showed in Example 3.1 that a critical or a supercritical growth for the
term g means that the existence of the solution to (3.1) is not guaranteed.
43
3.2 Polynomial growth
3. Measure data
Therefore, our goal is to understand what assumptions on the measure we
should ask in order to obtain the existence of the solution to the problem. We
will formally restrict the set of the measures we should consider as data.
First of all, we characterize the measures which are absolutely continuous with
0
respect to the W 2,p capacity (see Definition 1.17). We recall that we denote
this property by “µ capW 2,p0 ”. Now, we give a compactness result concerning this set of measure.
The proofs of the two results that we are going to state are present in [14].
Proposition 3.2.7. Let 1 < p < ∞ and let µ ∈ M(Ω) be a nonnegative
0
measure. Then, let Ω ⊂ RN and let V ⊂ W 2,p (Ω) be a closed vector subspace
containing Cc∞ (Ω). If µ capW 2,p0 , then there exists a sequence (µn )n∈N in
M(Ω) of nonnegative measure so that:
ˆ µn ∈ V 0 ,
∀n ∈ N ,
ˆ the sequence (µn )n∈N is nondecreasing and converges strongly to µ in
M(Ω).
We have just presented a way of approximating nonnegative absolutely continuous measures. We give a final ingredient that will be useful in the proof of
the next theorem.
Lemma 3.2.8. Let µ ∈ M(Ω), and let v be the solution to the following linear
Dirichlet problem
(
−∆v = µ in Ω,
v=0
on ∂Ω.
We have, for every 1 < p < +∞, v ∈ Lp (Ω) if and only if there exists a
constant C ≥ 0 such that
Z
ϕdµ ≤ C||ϕ|| 2,p0 , ∀ϕ ∈ C0∞ (Ω).
W
(Ω)
Ω
The proof of the above lemma relies on the Riesz representation theorem.
Now, we are ready to present and to prove the main theorem of this section.
Theorem 3.2.9. Let g : R → R be a continuous function satisfying the sign
condition (2.2), integrability condition (3.5), and such that
|g(t)| ≤ C(|t|p + 1),
∀t ∈ R,
where C, p > 0. If p ≥ NN−2 then, for every µ ∈ M(Ω) such that µ capW 2,p0 ,
the problem (3.1) has a solution.
44
3. Measure data
3.2 Polynomial growth
Proof. In order to apply the method of sub and supersolutions, our approach
will be to find a supersolution to the problem (3.1) with datum max {µ, 0}.
We will reason in the same way in order to find a subsolution related to the
problem (3.1) with datum min {µ, 0}.
We note that max {µ, 0} capW 2,p0 . Thus, we can apply Proposition 3.2.7
where we assume
0
0
V = W 2,p (Ω) ∩ W01,p (Ω).
Therefore, we have that there exists a nondecreasing sequence (µn )n∈N of nonnegative measure such that
0
0
0
µn ∈ W 2,p (Ω) ∩ W01,p (Ω) , ∀n ∈ N.
We also have that (µn )n∈N converges strongly to max {µ, 0} in M(Ω).
Now we have to prove an existence result concerning the datum we have just
introduced.
Lemma 3.2.10. There exists a nondecreasing sequence (un )n∈N in L1 (Ω) such
that, for every n ∈ N, we have that un is a solution to the problem (3.1) with
datum µn .
Proof. Let us consider the following linear problem
(
−∆v n = µn in Ω,
vn = 0
on ∂Ω.
Since µn ≥ 0, by the weak maximum principle (see Theorem 1.6.7), v n ≥ 0.
By the sign condition (2.2), for every ϕ ∈ C0∞ (Ω) such that ϕ ≥ 0 in Ω, we
can say
Z
g(v n )ϕ ≥ 0.
Ω
We have stated that
µn ∈ W
2,p0
(Ω) ∩
0
0
W01,p (Ω)
,
∀n ∈ N,
so that we can apply Lemma 3.2.8 having, for every 1 < p < ∞, v n ∈ Lp (Ω).
Recalling that |g(t)| ≤ C(|t|p + 1), we have that g(v n ) ∈ L1 (Ω), and v n is a
supersolution to (3.1).
We proceed by induction. Since v 0 is a supersolution and 0 is a subsolution
to (3.1) with datum µ0 , then it follows from Theorem 3.2.6 that there exists a
solution u0 to to the problem such that 0 ≤ u0 ≤ v 0 .
Let us assume we have defined un−1 ≤ v n−1 . By comparison, we have un−1 ≤
v n . We also have that un−1 is a subsolution, and v n is a supersolution to (3.1).
As before, there exists a solution un to (3.1) such that un−1 ≤ un ≤ v n .
The sequence (un )n∈N just defined satisfies the properties we need.
45
3.2 Polynomial growth
3. Measure data
Assumed the above lemma, we can continue to prove Theorem 3.2.9.
By comparison, we have that un is bounded by the solution to the linear
problem with datum max {µ, 0}.
We have that (un )n∈N converges pointwise to some function u.
By the monotone convergence theorem (see Theorem 1.5.5), (un )n∈N converges
to u in L1 (Ω).
By the absorption estimate (3.4), then it follows from Corollary 3.2.5 that
(g(un ))n∈N converges to g(u) in L1 (Ω).
Thus, we have started from
Z
Z
Z
ϕ dµn , ∀ϕ ∈ C0∞ (Ω).
− un ∆ϕ + g(un )ϕ =
Ω
Ω
Ω
Letting n tend to infinity we are regular enough to say
Z
Z
Z
− u∆ϕ + g(u)ϕ =
ϕ d(max {µ, 0}), ∀ϕ ∈ C0∞ (Ω).
Ω
Ω
Ω
We can reason in the same way approaching the datum min {µ, 0} to obtain a
solution u.
We have u ≤ 0 ≤ u. Therefore, by Theorem 3.2.6, we deduce that there exists
a solution u to (3.1) with datum µ and the conclusion holds.
Thus, we have the following.
46
3. Measure data
3.3
3.3 Exponential growth when N = 2
Exponential growth when N = 2
Let us describe, in the special case N = 2, the mechanism behind the
existence of the solution to (3.1) when the datum is a general measure and the
nonlinearity g has exponential growth at most
|g(t)| ≤ C(et + 1),
∀t ∈ R,
where C > 0.
As in the previous section, we want to investigate the way we need to proceed
in order to obtain the existence. We will need to restrict the set of measures.
We will show how much a concentrated measure can be tall in order to obtain
the existence of the solution.
In this section we will prove the main existence theorem by the method of
sub and supersolutions.
Before proving the main theorem, we need a pair of tools to show that the
exponential of the solution to the linear problem is bounded in L1 (Ω) such
that we can assume g(u) ∈ L1 (Ω) in the proof of the main theorem.
We begin with an L1 -estimate concerning the exponential of the Newtonian
potential.
Lemma 3.3.1. Let N = 2 and let µ ∈ M(Ω) be nonnegative. Let v : R2 → R
be the Newtonian potential generated by µ, that is
Z
d
1
log
dµ(y),
v(x) =
2π Ω
|x − y|
where d ≥ diam Ω. If ||µ||M(Ω) < 4π, then ev ∈ L1 (Ω) and the following
estimate holds
||ev ||L1 (Ω) ≤ C,
for some C[||µ||M(Ω) , d] > 0.
[see proof in [7], Theorem 1].
Since it is quite technical, we do not present the proof of the lemma just
stated. Briefly, it is proved with the help of the Jensen inequality and, later,
with the Fubini theorem. We now state the last tool before proving the existence theorem.
Lemma 3.3.2. Let N = 2 and let v be the solution of the following linear
problem
(
−∆v = µ in Ω,
v=0
on ∂Ω.
where µ ∈ M(Ω) such that, for every x ∈ Ω, µ({x}) < 4π, then we have
ev ∈ L1 (Ω).
47
3.3 Exponential growth when N = 2
3. Measure data
Proof. Let µ+ = max {µ, 0}. We can say that there exists r > 0 such that for
every a ∈ Ω
µ+ (Br (a) ∩ Ω) < 4π.
For a ∈ Ω, let v1 and v2 , respectively, be the Newtonian potentials generate
by µ+ bBr (a) and µ+ bΩ\Br (a) where d ≥ diam Ω.
We will prove a sublemma that will be useful in order to complete the proof.
Sublemma 3.3.3. The solution v to the linear problem satisfies v ≤ v1 + v2 .
Proof. We know that
(
−∆(v1 + v2 ) = µ+ ,
−∆v = µ.
Thus,
∆(v − v1 − v2 ) ≥ 0,
in the sense of distributions in Ω.
By construction, v1 + v2 is positive in Ω. Therefore, we have
(v − v1 − v2 )+ ≤ v + ≤ |v|.
So we have,
1
Z
1
(v − v1 − v2 ) ≤
(x∈Ω:d(x,Ω)<)
+
Z
|v|.
(x∈Ω:d(x,Ω)<)
It can be proved that the term on the right hand side converges to zero as tends to zero (see [14], Proposition 3.5). Therefore, the term on the left hand
side converges to zero. Applying the Theorem 1.6.8, we have
∆(v − v1 − v2 ) ≥ 0,
in the sense of (C0∞ (Ω))0 . By the weak maximum principle (see Theorem 1.6.7)
the conclusion holds.
We assume the sublemma and we conclude the proof.
We note that v2 is harmonic in Br (a). Thus, we can say that, for 0 < θ < 1,
v2 is bounded in Brθ (a). Therefore,
ev ≤ Cev1 in Brθ (a) ∩ Ω.
The measure µ+ bBr (a) satisfies conditions on Lemma 3.3.2 so that we have
ev1 ∈ L1 (Ω). Therefore,
ev ∈ L1 (Brθ (a) ∩ Ω).
The conclusion holds when we cover Ω with finitely many balls of radius rθ so
that to obtain ev ∈ L1 (Ω).
48
3. Measure data
3.3 Exponential growth when N = 2
Now we are ready to present and to prove the existence theorem.
Theorem 3.3.4. Let g : R → R be a continuous function satisfying the sign
condition (2.2) and such that
|g(t)| ≤ C(|e|t + 1),
∀t ∈ R,
where C > 0. If N = 2 and if µ ∈ M(Ω) is such that, for every x ∈ Ω,
µ({x}) < 4π, then the problem (3.1) has a solution.
Proof. We consider the following linear problem
(
−∆v = max {µ, 0} in Ω,
v=0
on ∂Ω.
By Lemma 3.3.2, ev ∈ L1 (Ω), so we deduce g(v) ∈ L1 (Ω).
By the weak maximum principle, v ≥ 0. By the sign condition, we have
g(v) ≥ 0.
Thus, we can say v is a supersolution to (3.1).
Now we consider
(
−∆v = min {µ, 0} in Ω,
v=0
on ∂Ω.
By the weak maximum principle, v ≤ 0, whence ev ∈ L∞ (Ω). Thus, again, we
deduce g(v) ∈ L1 (Ω).
As before, we can reason in the same way to say v is a subsolution to (3.1).
It follows from the nondecrease of the exponential function that, for every
v ∈ L1 (Ω) such that v ≤ v ≤ v, we have g(v) ∈ L1 (Ω) .
The conclusion holds since we have all the tools to apply Theorem 3.2.6 in
order to say that there exists a solution u to (3.1) such that v ≤ u ≤ v .
Remark 25. We state that, if µ({x}) = 4π for every x ∈ Ω, then the problem (3.1) has solution if and only if the term g satisfies also the integrability
condition.
Remark 26. In the case N ≥ 3 the problem (3.1) admits solution for every
µ ∈ M(Ω) such that µ ≤ 4πHN −2 . This case was studied in [3], Theorem 1.
Again, we provide the improved picture in the next page.
49
3.4 Diffuse measure data
3.4
3. Measure data
Diffuse measure data
We showed that the problem (3.1) where µ is a general measure or an
0
absolutely continuous measure with respect to the W 2,p capacity needs some
conditions on the growth of the term g in order to obtain the existence of the
solution.
In this section we want to investigate for which set of measures the problem (3.1) admits solution regardless of the growth of g; we will only require
g : R → R to be a continuous function satisfying sign condition (2.2).
This set of measures exists and it is the set of absolutely continuous measures
with respect to the W 1,2 capacity (µ capW 1,2 ).
Remark 27. We will refer to the set of absolutely continuous measures with
respect to the W 1,2 capacity as the set of diffuse measures.
On the other hand we will also prove that, if the problem (3.1) admits solution
for every nondecreasing continuous g, then the measure must be diffuse. Thus,
µ capW 1,2 is, in some sense, “optimal” in order to obtain the existence of
the solution to the problem.
First of all, we provide some definitions and tools whose will be helpful in
the proof of the main proposition.
50
3. Measure data
3.4 Diffuse measure data
We start by defining what we mean by quasicontinuous function and quasicontinuous representatives.
Definition 3.6. A function u : Ω → R is quasicontinuous with respect to
the W 1,2 capacity if for every > 0 there exists a Borel set E ⊂ Ω such that
capW 1,2 (E) ≤ and u is continuous in Ω \ E.
We give the following proposition to establish the existence of quasicontinuous
representatives.
Proposition 3.4.1. For every u ∈ W 1,2 (Ω) there exists a quasicontinuous
function û : Ω → R with respect to the W 1,2 capacity such that u = û a.e. in
Ω.
[see proof in [14], Proposition A.8.]
Remark 28. The previous proposition can be seen as the analog of the Egorov
theorem (see Theorem 1.5.3) for Sobolev functions.
We need another result about quasicontinuous functions which will be useful
in proving the equi-integrability condition on the approximated gn (un ) in the
main proposition below.
Corollary 3.4.2. Let µ ∈ M(Ω) be a diffuse measure and let u be the solution
to the linear Dirichlet problem (1.4) with datum µ. Then, for every s ≥ 0
Z
sign(û)dµ ≥ 0.
{|û|>s}
[see proof in [14], Corollary 9.3.]
The following lemma clarifies what type of estimates are satisfied by the quasicontinuous representative of the solution to the linear Dirichlet problem.
Lemma 3.4.3. Let µ ∈ M(Ω) be the datum to the linear Dirichlet problem
(
−∆u = µ in Ω,
u=0
on ∂Ω,
then for every s > 0 we have
capW 1,2 ({|û| > s}) ≤
for some constant C[Ω] > 0.
51
C
||µ||M(Ω) ,
s
3.4 Diffuse measure data
3. Measure data
Proof. By Proposition 1.4.2, we have for every s > 0
capW 1,2 ({|v̂| > s}) ≤
1
||v||2W 1,2 (RN ) ,
s2
∀v ∈ W01,2 (Ω).
It can be proved (see [14], Lemma 4.7), for k > 0, we have Tk (u) ∈ W01,2 (Ω)
(we have denoted by Tk (u) as the truncation of u at level k) and
1
1
2
≥ C1 ||DTk (u)||L2 (Ω)
C1 k 2 ||µ||M(Ω)
P oincare
≥
||Tk (u)||W 1,2 (Ω) .
Therefore,
capW 1,2 ({|Tk (û)| > s}) ≤
C2 k
||µ||M(Ω) .
s2
If k ≥ s
{|Tk (û)| > s} = {|û| > s}.
So we take k = s and deduce that
capW 1,2 ({|û| > s}) ≤
C2
||µ||M(Ω) ,
s
ant the conclusion holds.
Now we want to provide an equivalent characterization of diffuse measures.
Lemma 3.4.4. Let µ ∈ M(Ω), then µ is diffuse if and only if for every > 0
there exists δ > 0 such that if A ⊂ Ω is a Borel set such that capW 1,2 (A) ≤ δ,
then |µ|(A) ≤ .
Proof. We start with necessary implication. If A ⊂ Ω is a Borel set such that
capW 1,2 (A) = 0, then for every > 0, |µ|(A) ≤ . This implies |µ|(A) = 0,
and, by definition, µ is a diffuse measure.
We prove the sufficient implication by contradiction. Let (αn )n∈N be a sequence
of positive numbers converging to zero. Suppose, by contradiction, that there
exist > 0 and a sequence (An )n∈N of Borel subsets of Ω such that for every
n∈N
capW 1,2 (An ) ≤ αn and µ(An ) ≥ .
Let define
A=
∞ [
∞
\
An .
j=0 n=j
By the monotonicity and the by subadditivity of the capacity, we have that
for every j ∈ N
capW 1,2 (A) ≤ capW 1,2 (
∞
[
An ) ≤
n=j
∞
X
n=j
capW 1,2 (An ) ≤
∞
X
αn .
n=j
P
Therefore, if ∞
n=o αn converges, then capW 1,2 (A) = 0 and µ(A) ≥ .
Thus we have a contradiction and the conclusion holds.
52
3. Measure data
3.4 Diffuse measure data
Thus, we are now ready to prove that problem (3.1) with diffuse measure
as datum always admits solution.
Proposition 3.4.5. Let g : R → R be a continuous function satisfying sign
condition (2.2) and let µ ∈ M(Ω) be a diffuse measure, then the problem (3.1)
has a solution.
Proof. The result is proved by approximating g with bounded functions and
by keeping µ fixed.
Let gn : R → R be a continuous bounded function satisfying sign condition
and such that (gn )n∈N converges uniformly to g in bounded subsets of R. For
istance we can take the truncation of g at level n.
Let us consider the following problem
(
−∆un + gn (un ) = µ in Ω,
un = 0
on ∂Ω.
By gn boundedness, we have already established the existence of the solution
in this case in Theorem 3.2.3.
Before proceeding we need an L1 -estimate for gn (un ).
Subproposition 3.4.6. For every s ≥ 0, there exists C1 such that, for every
Borel set E ⊂ Ω and for every n ∈ N, we have
Z
|gn (un )| ≤ C1 |E| + |µ|({|ûn | > s}).
E
Proof. We have, for every Borel set E ⊂ Ω and for every s ≥ 0
Z
Z
Z
|gn (un )| ≤
|gn (un )| +
|gn (un )|.
E∩{|un |≤s}
E
E∩{|un |>s}
By gn boundedness in bounded sets, we can control the first term on the right
hand side
Z
|gn (un )| ≤ C1 |E ∩ {|un | ≤ s}| ≤ C1 |E|.
E∩{|un |≤s}
By Corollary 3.4.2 applied to the linear problem with datum µ − gn (un ),
Z
Z
sign(un )gn (un ) ≤
sign(ûn )dµ.
{|un |>s}
{|ûn |>s}
By the sign condition we obtain
Z
Z
|gn (un )| ≤
{|un |>s}
sign(ûn )dµ ≤ |µ|({|ûn | > s}),
{|ûn |>s}
and the conclusion holds.
53
3.4 Diffuse measure data
3. Measure data
Using the Subproposition 3.4.6 we can complete the proof.
By Lemma 3.4.3, we have
capW 1,2 ({|ûn | > s}) ≤
C2
||µ − gn (un )||M(Ω) .
s
Thus, by triangle inequality and by absorption estimate 3.4, we can say
capW 1,2 ({|ûn | > s}) ≤
C3
||µ||M(Ω) .
s
Recalling that µ is diffuse we can apply Lemma 3.4.4.
Thus, by the above subproposition, we have that the sequence (gn )n∈N is equiintegrable.
Again, by Stampacchia’s regularity theory, there exists a sub-sequence (unk )k∈N
converging to some function u in L1 (Ω) and a.e. in Ω.
Since (gn )n∈N is equi-integrable, we can apply the Vitali convergence theorem
and say that (gn (unk ))n∈N converges to g(u) in L1 (Ω).
The conclusion holds since we proved that u is a solution to (3.1).
Remark 29. In the previous section we proved that treating problem where the
0
datum is an absolutely continuous measure with respect to the W 2,p capacity
implies some regularity on the term g. Instead, by the above proposition, we
have formally shown that we need less regularity when measure is diffuse. This
fact agrees with capacity theory since the second one is a subset of the first
one.
Thus, we can complete our scheme and we get the final picture of what we
did throughout this work.
54
3. Measure data
3.4 Diffuse measure data
At this point we want to show that the existence of the solution to the
problem (3.1) where the datum is a diffuse measure is a necessary and sufficient implication. We mean that if we prove the problem (3.1) admits solution
for every nondecreasing continuous g then the measure data must be diffuse.
The proof of the proposition relies on a lemma regarding L1 -function properties and on a lemma regarding properties of the Legendre transform.
Lemma 3.4.7 (de la Vallée Poussin). Let f ∈ L1 (Ω) then there exists a nonnegative nondecreasing function h : [0, ∞] → R such that
ˆ h(0) = 0,
h(t)
= +∞,
t→+∞ t
ˆ lim
ˆ h(f ) ∈ L1 (Ω).
[see [9], Remark 23.]
We also need the following property.
Lemma 3.4.8. Let h : [0, ∞] → R be a nonnegative continuous function such
that h(0) = 0. If
h(t)
= +∞,
lim
t→+∞ t
then the Legendre trasform of h defined by
h∗ (t) = sup{st − h(s)},
∀t ∈ [0, +∞],
s≥0
is a nonnegative nondecreasing continuous function such that h∗ (0) = 0.
[see proof in [14], Lemma 9.8.]
Now, we introduce the last tool we need to prove our result. This tool is
an identity of capacities.
Lemma 3.4.9. We have, ∀K ⊂⊂ Ω
Z
inf{ |∆ϕ| : ϕ ∈ C0∞ (Ω) and ϕ ≥ 1 on K}
Ω
Z
= 2 inf{
|∇ϕ|2 : ϕ ∈ C0∞ (Ω) and ϕ ≥ 1 on K}.
Ω
[see proof in [14], Lemma 9.9.]
55
3.4 Diffuse measure data
3. Measure data
Remark 30. We note that the capacities defined on the above proposition have
the same sets of zero capacity.
We provide the last tool before presenting the last result of this section.
Lemma 3.4.10. Let g : R → R be a continuous function satisfying sign
condition (2.2). Given µ ∈ M(Ω), let u be a solution to (3.1). Thus,
ˆ if µ ≤ 0, then u ≤ 0 in Ω,
ˆ if µ ≥ 0, then u ≥ 0 in Ω.
Proof. We prove the first point. We have
∆u = g(u) − µ ≥ g(u),
in the sense of (C0∞ (Ω))0 . Thus, by Kato’s inequality up to the boundary and
by sign condition
∆u+ ≥ χ{u>0} g(u) ≥ 0,
in the sense of (C0∞ (Ω))0 . The conclusion follows from the weak maximum
principle.
We are ready to present and to prove the sufficient implication we have already
asserted.
Proposition 3.4.11. Let µ ∈ M(Ω) be a nonnegative measure. If the problem
(3.1) has solution for every continuous nondecreasing function g : R → R, then
the measure must be diffuse.
Proof. For simplicity we assume g(0) = 0.
Assuming that g is a continuous nondecreasing function that we will fix later,
we have g satisfies the sign condition.
Since µ ≥ 0, then it follows from Lemma 3.4.10 that u ≥ 0.
Let K ⊂⊂ Ω and let ϕ ∈ C0∞ (Ω) be a nonnegative function such that ϕ ≥ 1
on K. Thus, we can use ϕ as a test function in the weak formulation to the
problem (3.1) and obtain
Z
Z
Z
− u∆ϕ + g(u)ϕ =
ϕdµ ≥ µ(K),
Ω
Ω
Ω
where the last inequality holds since µ is nonnegative.
We suppose capW 1,2 (K) = 0.
By Lemma 3.4.9, we can choose a sequence, by truncation, (ϕn )n∈N of nonnegative functions in C0∞ (Ω) such that
ˆ ϕn ≥ 1,
∀n ∈ N,
ˆ (∆ϕn )n∈N converges to 0 in L1 (Ω),
56
3. Measure data
3.4 Diffuse measure data
ˆ by the linear theory (ϕn )n∈N converges to 0 in L1 (Ω),
ˆ by a truncation argument 0 ≤ ϕn ≤ 1 in Ω,
∀n ∈ N.
Therefore we can say
Z
Z
−
g(u)ϕn ≥ µ(K).
u∆ϕn +
Ω
(3.7)
Ω
At first we want to apply the dominated convergence theorem to the first term
on the left hand side in (3.7). Since (∆ϕn )n∈N converges to 0 in L1 (Ω), we
know that there exists a subsequence (∆ϕnk )k∈N converging to 0 a.e. in Ω. We
can also find a function f ∈ L1 (Ω) such that
|∆ϕnk | ≤ f,
∀k ∈ N.
Let us fix g = h∗ , whose h∗ is the Legendre trasform of a function h : [0, +∞] →
R defined in Lemma 3.4.8. As defined, we must have
st ≤ h∗ (t) + h(s),
∀s, t ≥ 0 ∈ N.
So this can be applied to our case obtaining
|u∆ϕnk | ≤ |uf | ≤ h∗ (u) + h(f ).
By Lemma 3.4.8, we have h(f ) ∈ L1 (Ω). Recalling that (∆ϕnk )k∈N converges
to 0 a.e. we can apply the dominated convergence theorem and say
Z
lim
u∆ϕn = 0.
n→∞
Ω
Again, we want to apply the dominated convergence theorem to the second
term on the left hand side in (3.7).
Since (ϕn )n∈N is bounded in L∞ (Ω) and converges to 0 in L1 (Ω). Again, by
the dominated convergence theorem, we have
Z
h∗ (u)ϕn = 0.
lim
n→∞
Ω
Recalling inequality (3.7), we have µ(K) ≤ 0. The conclusion holds since
µ is nonnegative, so we must have µ(K) = 0 for every K ⊂⊂ Ω such that
capW 1,2 (K) = 0.
Remark 31. We proved that the set of diffuse measures is the largest one for
which we get the existence of the solution to (3.1), regardless of the growth of
term g.
Remark 32. If µ is a signed measure for which the problem (3.1) has a solution
for every nondecreasing continuous function g, we can prove the same result
above with the help of some tools arising from the next chapter. We will show
that max{µ, 0}, min{µ, 0} admits solution in the same cases, so that they are
diffuse with respect to the W 1,2 capacity and the conclusion holds.
57
3.4 Diffuse measure data
3. Measure data
58
Chapter 4
Good measures and reduced
measures
Let us consider the problem
(
−∆u + g(u) = µ in Ω,
u=0
on ∂Ω,
(4.1)
where µ ∈ M(Ω) is a general measure.
For simplicity, in all this chapter we assume g : R → R to be a continuous,
nondecreasing function such that
g(t) = 0,
∀t ≤ 0.
(4.2)
Remark 33. Condition (4.2) is only needed to consider general signed measures.
From now on, unless explicitly stated, we denote by (gn )n∈N a sequence of
bounded functions gn : R → R which are continuous, nondecreasing, and such
that:
ˆ 0 ≤ g1 (t) ≤ g2 (t) ≤ ... ≤ g(t),
ˆ gn (t) converges to g(t),
∀t ∈ R,
∀t ∈ R.
Remark 34. For istance, we can take gn as the truncation of g at level n.
In the first part of this chapter we want to investigate properties of measures
such that problem (4.1) admits a solution.
In the last part we will consider problems that do not admit solutions and
we will introduce an approximation scheme to get, in some sense, the “nearest
problem” that admits solution.
59
4.1 Good measures
4.1
4. Good measures and reduced measures
Good measures
At first we want to clarify what we mean with “good measures” for a fixed
g.
Definition 4.1. We say that µ ∈ M(Ω) is a good measure if the problem (4.1)
admits solution in the sense of the Definition 3.2. The set of good measures is
denoted by G.
We start with an observation arising from what we saw in the last chapter.
Remark 35. At the end of the previous chapter we proved that diffuse measures
are good measures for any g.
The converse of this statement is not true. In Section 3.3, we showed that
measures such that 0 < µ({x}) < 4π are good. But these measures are not
diffuse; we have cap({x}) = 0 while 0 < µ({x}) < 4π.
Now we present some properties of good measures.
We start with a lemma that will be useful in the proof of an important property of good measures: a measure that stays below a good measure is itself
good.
Lemma 4.1.1. Let µ ∈ M(Ω) be a good measure to the problem (4.1). Let us
consider the following problem
(
−∆un + gn (un ) = µ in Ω,
un = 0
on ∂Ω.
Then we have:
ˆ un converges to u in W01,1 (Ω),
ˆ gn (un ) converges to g(u) in L1 (Ω),
where u is the solution to (4.1).
Proof. We have
Z
Z
Z
− un ∆ϕ + gn (un )ϕ =
ϕdµ,
Ω
and
Ω
Z
−
Ω
Z
Z
u∆ϕ +
Ω
∀ϕ ∈ C0∞ (Ω),
g(u)ϕ =
Ω
ϕdµ,
∀ϕ ∈ C0∞ (Ω).
Ω
Combining the previous equalities.
Z
Z
− (un − u)∆ϕ + (gn (un ) − g(u))ϕ = 0,
Ω
Ω
60
∀ϕ ∈ C0∞ (Ω).
4. Good measures and reduced measures
4.1 Good measures
Therefore, we can say
Z
Z
Z
− (un − u)∆ϕ + (gn (un ) − gn (u))ϕ = (g(u) − gn (u))ϕ,
Ω
Ω
∀ϕ ∈ C0∞ (Ω).
Ω
By uniqueness, we have
Z
Z
|gn (un ) − gn (u)| ≤
|g(u) − gn (u)| → 0.
Ω
Ω
Thus, by triangle inequality
Z
Z
|gn (un ) − g(u)| ≤ 2 |g(u) − gn (u)| → 0.
Ω
Ω
In this way we have just proved that gn (un ) converges to g(u) in L1 (Ω). This
implies, by linear theory stability, that ∆un converges to ∆u in L1 (Ω) and
then un converges to u in W01,1 (Ω), and so the conclusion holds.
Proposition 4.1.2 (Property 1). Let µ1 be a good measure, then any other
measure µ2 ≤ µ1 is a good measure.
Proof. Let consider the following problems with µ1 , µ2 data
(
−∆u1,n + gn (u1,n ) = µ1 in Ω,
u1,n = 0
on ∂Ω,
(
−∆u2,n + gn (u2,n ) = µ2
u2,n = 0
in Ω,
on ∂Ω.
Since µ2 ≤ µ1 , by comparison, we have u2,n ≤ u1,n a.e.
Recalling that g is nondecreasing we have gn (u2,n ) ≤ gn (u1,n ).
By Lemma 4.1.1, we have that gn (u1,n ) converges to g(u∗1 ) in L1 (Ω).
Thus, by the generalized version of the dominated convergence theorem, we
have that gn (u2,n ) converges to g(u∗2 ) in L1 (Ω), where u∗i is the solution to
(
−∆u∗i + g(u∗i ) = µi in Ω,
u∗i = 0
on ∂Ω.
Thus we have
Z
−
u∗2 ∆ϕ
Z
+
Ω
Ω
g(u∗2 )ϕ
Z
=
ϕdµ2 ,
∀ϕ ∈ C0∞ (Ω),
Ω
and the conclusion holds since we proved that µ2 is a good measure.
We deduce a consequence.
61
4.1 Good measures
4. Good measures and reduced measures
Corollary 4.1.3. Let µ ∈ M(Ω) such that µ+ is diffuse then µ is a good
measure.
Proof. We have already noted that diffuse measures are good. Thus, µ+ is a
good measure. By Proposition 4.1.2 we have that µ ≤ µ+ is a good measure.
Now we state that the supremum of two good measures is itself a good
measure.
Proposition 4.1.4 (Property 2). Let µ1 , µ2 be good measures then sup{µ1 , µ2 }
is a good measure.
Since we need some tools arising from the theory of reduced measure, the
proof is postponed to the next section.
We present a consequence of the previous proposition.
Corollary 4.1.5. The set G of good measures is convex.
Proof. Let µ1 , µ2 ∈ G, we want to show that, for every 0 ≤ t ≤ 1, tµ1 + (1 −
t)µ2 ∈ G.
We note
tµ1 + (1 − t)µ2 ≤ sup{µ1 , µ2 },
∀t ∈ [0, 1].
By Proposition 4.1.4 we know that sup{µ1 , µ2 } is a good measure. Then it
follows from Proposition 4.1.2 that tµ1 + (1 − t)µ2 is a good measure and the
conclusion holds.
The next property we present concerns the topological structure of the set
G of good measures.
Proposition 4.1.6 (Property 3). The set G of good measures is closed with
respect to strong convergence in M(Ω).
Proof. Let (µk )k∈N ⊂ G such that µk converges strongly to µ in M(Ω). We
want to show that µ ∈ G.
Let us consider
(
−∆uk + g(uk ) = µk in Ω,
uk = 0
on ∂Ω.
By uniqueness, we have
Z
|g(uk1 ) − g(uk2 )| ≤ ||µk1 − µk2 ||M(Ω) ,
Ω
and we deduce
Z
|uk1 − uk2 | ≤ 2||µk1 − µk2 ||M(Ω) .
Ω
62
4. Good measures and reduced measures
4.1 Good measures
Since (µk )k∈N is a Cauchy sequence in M(Ω), then (uk )k∈N and (g(uk ))k∈N are
Cauchy sequence in L1 (Ω).
Thus, there exist u, v ∈ L1 (Ω) such that:
ˆ uk converges to u in L1 (Ω),
ˆ g(uk ) converges to v in L1 (Ω).
Therefore we deduce v = g(u) a.e.
Since we have shown
Z
Z
Z
− u∆ϕ + g(u)ϕ =
ϕdµ,
Ω
Ω
∀ϕ ∈ C0∞ (Ω),
Ω
then µ is a good measure and the conclusion holds.
We continue by stating another property of G.
Proposition 4.1.7 (Property 4). We have
\
G(g) = {µ ∈ M(Ω) such that µ+ is a diffuse measure},
g
where we take the intersection over all continuous, nondecreasing functions g
such that g(0) = 0.
Remark 36. We have already proved this property in Proposition 3.4.11.
The next property gives us a tool to understand if a measure is good.
Proposition 4.1.8 (Property 5). Let µ ∈ M(Ω). The following conditions
are equivalent:
i) µ is a good measure,
ii) µ+ is a good measure,
iii) µc is a good measure.
Proof. i) ⇒ ii)
We have that µ, 0 ∈ G. By Proposition 4.1.4, we obtain that µ+ = sup{µ, 0} ∈
G.
ii) ⇒ i)
By Proposition 4.1.2, since µ ≤ µ+ , we have that µ is a good measure.
ii) ⇒ iii)
+
+
We have (µ+ − µc )d = (µ+ )d = µ+
d ≥ 0, and (µ − µc )c = µc − µc ≥ 0.
Thus, combining the previous inequalities,
+
+
µ+
d + µc = µ ≥ µc ,
63
4.2 Reduced Measures
4. Good measures and reduced measures
and the conclusion holds from Proposition 4.1.2.
iii) ⇒ ii)
We note that for every measure we can say µ+ = sup{µd , µc } by decomposition. We stated that diffuse measure are good measure. By Proposition 4.1.4,
µ+ is a good measure.
The conclusion holds since we proved all implications.
Previous properties takes us to another consequence regarding the set of
good measures. This set is closed with respect to the sum with diffuse measures.
Corollary 4.1.9. Let Md (Ω) be the space of diffuse measure, then we have
G + Md (Ω) ⊂ G.
Proof. Let µ ∈ G. By Proposition 4.1.8, µc ∈ G. Let λ ∈ Md (Ω), then we can
say that (µ + λ)c = µc . Thus, (µ + λ)c ∈ G, and, by Proposition 4.1.8, the
conclusion holds since µ + λ ∈ G.
Remark 37. We note that G + G 6⊆ G. For istance, in case N = 2, if we take
µ = 2πδ0 , λ = (2π + )δ0 , then µ + λ is not a good measure for a given g that
grows as an exponential function. This was proved in Section 3.3.
These tools prepares us to the next section, where we will introduce the
concept of “reduced measures”.
4.2
Reduced Measures
We want to investigate problems similar to (4.1) that does not admit solution, that is when µ is not a good measure. In some sense, we want a measure
that best approximate the starting measure so that the solution we get is a sort
of approximation of the “solution” to the starting problem. For this purpose,
we want to introduce the concept of “reduced measures”; at first, to understand what we mean by “reduced measures”, we need a convergence result.
Thus, we introduce the first approximation method keeping µ fixed, and by
approximating g.
Proposition 4.2.1. Let µ ∈ M(Ω), and let un be the solution to
(
−∆un + gn (un ) = µ in Ω,
un = 0
on ∂Ω.
Thus, we have that the nonincreasing un converges to u∗ in L1 (Ω), where u∗
is the largest subsolution to (4.1).
Moreover
Z
u∗ ∆ϕ ≤ 2||µ||M(Ω) ||ϕ||L∞ (Ω) , ∀ϕ ∈ C0∞ (Ω).
(4.3)
Ω
64
4. Good measures and reduced measures
4.2 Reduced Measures
Proof. By the comparison, we have that un is nonincreasing.
By absorption estimate (3.4)
||gn (un )||L1 (Ω) ≤ ||µ||M(Ω) .
Therefore, we deduce that
||∆un ||M(Ω) ≤ 2||µ||M(Ω) ,
and, by Stampacchia’s regularity theory (see Theorem 1.6.5), we have that
(un )n∈N is bounded in W01,q (Ω), for every q < NN−1 . Then it follows from
the Rellich-Kondrachov compactness theorem (see Theorem 1.3.5) that, up
to subsequences, un converges to some function u∗ in L1 (Ω). Therefore un
converges to u∗ a.e. We also have that gn (un ) converges to g(u) a.e.
Then, by Fatou’s lemma (see Lemma 1.5.4), we have
||g(u∗ )||L1 (Ω) ≤ ||µ||M(Ω) ,
and also we obtain (4.3).
Again, by Fatou’s lemma, we have
Z
Z
Z
∗
∗
− u ∆ϕ + g(u )ϕ ≤
ϕdµ,
Ω
Ω
∀ϕ ∈ C0∞ (Ω), ϕ ≥ 0 in Ω.
Ω
We have proved that u∗ is a subsolution to (4.1), but we still have to prove
that u∗ is the largest subsolution. Let v be any subsolution to (4.1), then
−∆v + gn (v) ≤ −∆v + g(v) ≤ µ, in (C0∞ (Ω))0 .
Then, by comparison, v ≤ un a.e. and letting n tend to infinity we have
v ≤ u∗ . The conclusion holds since we proved that u∗ is the largest subsolution
to (4.1).
Remark 38. We note that u∗ does not depend on the sequence (gn )n∈N . Moreover, we can deduce that, if µ is a good measure, then u∗ coincides with the
solution to the problem.
Now we are ready to give the following definition.
Definition 4.2. Given µ ∈ M(Ω). Let µ∗ ∈ M(Ω) be the unique measure
such that
Z
Z
Z
∗
∗
− u ∆ϕ + g(u )ϕ =
ϕdµ∗ , ∀ϕ ∈ C0∞ (Ω),
(4.4)
Ω
Ω
Ω
where u∗ is defined in Proposition 4.2.1. The measure µ∗ is called the reduced
measure of µ.
65
4.2 Reduced Measures
4. Good measures and reduced measures
Remark 39. Since the terms on the left hand side of (4.4) are bounded by
Proposition 4.2.1 and by absorption estimate, we can apply the Riesz rappresentation theorem and say that there exists a unique measure that satisfies
(4.4).
Remark 40. Obviously, we remark that the measure µ∗ does depend on g.
In order to highlight its importance, we want to follow another approximation scheme to get the largest subsolution u∗ . We want to keep the term g
fixed and we approximate the measure via convolution.
Let (ρn )n∈N be a sequence of mollifiers in RN such that, for every n ≥ 1,
supp(ρn ) ⊂ B 1 . Let µ ∈ M(Ω), we define
n
Z
µn (x) = ρn ∗ µ(x) =
ρn (x − y)dµ(y), ∀x ∈ RN .
Ω
We provide the following proposition.
Proposition 4.2.2. Let us consider
(
−∆un + g(un ) = µn
un = 0
in Ω,
on ∂Ω.
Thus, if we assume that g is also convex, then un converges to u∗ in L1 (Ω),
where u∗ is defined in Proposition 4.2.1.
Proof. Let µ be a good measure, then u = u∗ satisfies
(
−∆u + g(u) = µ in Ω,
u=0
on ∂Ω,
and the conclusion holds by uniqueness.
Let ω ⊂⊂ Ω. If n ≥ 1 is great enough, we have, by convolution properties,
∆(ρn ∗ u) + ρn ∗ g(u) = µn in ω.
Then, by the convexity of g,
∆(ρn ∗ u − un ) = ρn ∗ g(u) − g(un )
Jensen
≥
g(ρn ∗ u) − g(un ) in (C0∞ (ω))0 .
By Kato’s inequality up to the boundary (see Theorem 1.6.10),
∆(ρn ∗ u − un )+ ≥ [g(ρn ∗ g(u)) − un ]+ in (C0∞ (ω))0 .
By the estimate obtained in Proposition 4.2.1, we have
||∆un ||M(Ω) ≤ 2||µ||M(Ω) ,
66
∀n ≥ 1,
(4.5)
4. Good measures and reduced measures
4.2 Reduced Measures
so that, by Stampacchia’s regularity theory and by the Rellich-Kondrachov
compactness theorem, we can say that there exists a subsequence (unk )k∈N
converging to some function v in L1 (Ω).
Letting n tend to infinity in (4.5),
−∆(u − v)+ ≤ 0 in (C0∞ (ω))0 ,
and by arbitrarity of ω ⊂⊂ Ω
−∆(u − v)+ ≤ 0 in (C0∞ (Ω))0 .
Since a general version of the Riesz representation theorem guarantees us that
a positive or negative distribution is a measure, then it follows from Stampacchia’s regularity that (u − v)+ ∈ W01,1 (Ω) .
Thus, by the weak maximum principle (see Theorem 1.6.7), we have that
(u − v)+ ≤ 0 a.e.
Thus, we can say
u ≤ v a.e.
It follows from Fatou’s lemma that v is a subsolution to (4.1) and by comparison we have
u ≥ v a.e.
Thus, we have
u = v a.e.
The conclusion holds observing that v is indipendent of the subsequence (unk )k∈N
so that we must have that un converges to u = u∗ in L1 (Ω).
Now let µ be a general measure. We can assume that un converges to some
function v in L1 (Ω).
It follows from Fatou’s lemma that v is a subsolution to the problem. By
Proposition 4.2.1 we have that v ≤ u∗ .
Let us consider the problem
(
−∆u∗n + g(u∗n ) = ρn ∗ µ∗ in Ω,
u∗n = 0
on ∂Ω,
Since the datum is good, we have already proved that u∗n converges to u∗ in
L1 (Ω). By comparison we have u∗n ≤ un a.e. and letting n tend to infinity we
have u∗ ≤ v. Thus, the conclusion holds.
4.2.1
Properties of reduced measures
In this section we present some properties for the reduced measures.
At first, we need a technical tool.
67
4.2 Reduced Measures
4. Good measures and reduced measures
Lemma 4.2.3. µ∗ ≥ µd − µ−
c in Ω.
[see proof in [6], Lemma 1.]
We are ready to present the main property concerning reduced measure.
Proposition 4.2.4 (Property 1). The measure µ∗ is the largest good measure
≤ µ.
Proof. Let λ ≤ µ be a good measure, then we must prove that λ ≤ µ∗ .
Let us consider
(
−∆v + g(v) = λ in Ω,
v=0
on ∂Ω,
By Lemma 4.2.3,
∗
µd − µ−
c ≤ µ ≤ µ,
and taking diffuse parts, we deduce
(µ∗ )d = (µ)d .
Therefore, we have
λd ≤ µd = (µ∗ )d .
By definition, u∗ is the largest subsolution to (4.1). Thus, we have
v ≤ u∗ a.e.
By inverse maximum principle (see Theorem 1.6.11),
λc = (−∆v)c ≤ (−∆u∗ )c = (µ∗ )c .
The conclusion holds since we proved that λ ≤ µ∗ .
The next theorem is an approximation result concerning reduced measures.
It assures that a measure is well approximated by the reduced measure.
Proposition 4.2.5 (Property 2). There exists a Borel set Σ ⊂ Ω with capW 1,2 (Σ) =
0 such that
(µ − µ∗ )(Ω \ Σ) = 0.
Proof. As we have seen in the previous proof,
(µ∗ )d = (µ)d .
Thus, we have
(µ − µ∗ ) = (µ − µ∗ )c ,
and the conclusion holds recalling the definition of the concentrated part of a
measure.
68
4. Good measures and reduced measures
4.2 Reduced Measures
We continue with a corollary concerning a further characterization of the
reduced measures.
Corollary 4.2.6. Let µ ∈ M(Ω), then we have
||µ − µ∗ ||M(Ω) = min ||µ − λ||M(Ω) .
λ∈G
Proof. For every λ ∈ G we have
|µ − λ| = (µ − λ)+ + (µ − λ)− ≥ (µ − λ)+ = µ − inf{µ, λ}.
It follows from Proposition 4.1.4 that inf{µ, λ} ∈ G. We also proved in Proposition 4.2.4 that µ∗ is the largest good measure lower than µ, then we can
say
inf{µ, λ} ≤ µ∗ .
Therefore, we have
|µ − λ| ≥ µ − inf{µ, λ} ≥ µ − µ∗ ≥ 0.
The conclusion holds since we proved
||µ − λ||M(Ω) ≥ ||µ − µ∗ ||M(Ω) .
Remark 41. With Proposition 4.2.5 and with Corollary 4.2.6 we have shown
that µ∗ is the best approximation of µ in G. Therefore, it makes sense looking
for the solution to the problem associated to the measure data µ∗ when the
problem with measure data µ does not admit solution.
We show that the reduced measures are order preserving.
Proposition 4.2.7 (Property 3). Let µ, λ ∈ M(Ω). If µ ≤ λ, then µ∗ ≤ λ∗ .
Proof. By Proposition 4.2.4, µ∗ is the largest good measure lower than µ, but
λ∗ is also a good measure. Then we must have
µ∗ ≤ µ ≤ λ∗ ≤ λ,
and the conclusion holds.
Proposition 4.2.8 (Property 4). If µ1 , µ2 ∈ G are mutually singular, then
(µ1 + µ2 )∗ = (µ1 )∗ + (µ2 )∗ .
69
4.2 Reduced Measures
4. Good measures and reduced measures
Remark 42. We recall that measures are said to be mutually singular if they
are concentrated on disjoint sets.
Proof. To prove the proposition we show two inequalities.
i) (µ1 )∗ + (µ2 )∗ ≤ (µ1 + µ2 )∗ .
We have
(µ1 )∗ + (µ2 )∗ ≤ [(µ1 )∗ + (µ2 )∗ ]+ = sup{(µ1 )∗ , (µ2 )∗ }.
By Proposition 4.1.4, sup{(µ1 )∗ , (µ2 )∗ } is a good measure. By Proposition
4.1.2, (µ1 )∗ + (µ2 )∗ is a good measure. By construction,
(µ1 )∗ + (µ2 )∗ ≤ µ1 + µ2 ,
but, by Proposition 4.2.4, we know which is the largest good measure
lower than µ1 + µ2 . So we must have
(µ1 )∗ + (µ2 )∗ ≤ (µ1 + µ2 )∗ .
ii) (µ1 )∗ + (µ2 )∗ ≥ (µ1 + µ2 )∗ .
Let λ ≤ µ1 +µ2 be a good measure. By the Radon-Nikodyn decomposition,
we have that there exist λ0 , λ1 , λ2 ∈ M(Ω) such that
λ = λ0 + λ1 + λ2 ,
where λ0 is singular with respect to |µ1 | + |µ2 |, λ1 (λ2 ) is absolutely continuous with respect to |µ1 | (|µ2 |).
Since λ0 , λ1 , λ2 ≤ λ, it follows from Proposition 4.1.2 that all three measures are good.
We also have
λ0 ≤ 0, λ1 ≤ µ1 , λ2 ≤ µ2 .
Therefore,
(µ1 )∗ + (µ2 )∗ ≥ λ0 + λ1 + λ2 = λ,
The conclusion holds recalling that the measure λ was arbitrary so that
(µ1 )∗ + (µ2 )∗ ≥ (µ1 + µ2 )∗ .
From the previous proposition we can deduce some consequences that we
state for completeness.
Corollary 4.2.9. Let µ ∈ M(Ω), then we have:
ˆ (µ∗ )d = (µd )∗ = µd ,
70
4. Good measures and reduced measures
4.2 Reduced Measures
ˆ (µ∗ )c = (µc )∗ ,
ˆ (µ∗ )+ = (µ+ )∗ ,
ˆ (µ∗ )− = µ− .
[see proof in [6], Corollary 10.]
The next result arises from the previous proposition, and it explains us what
happens to the reduced measure when we sum a general measure and a diffuse
measure.
Corollary 4.2.10. Let µ ∈ M(Ω), λ ∈ Md (Ω), then
(µ + λ)∗ = µ∗ + λ.
Proof. By Proposition 4.2.8 we have
(µ + λ)∗ = µ∗c + (µd + λ)∗ = µ∗c + µd + λ = (µ∗c + µ∗d ) + λ = µ∗ + λ.
We give a final property for reduced measures.
Proposition 4.2.11 (Property 5). Let µ, λ ∈ M(Ω), then we have
|µ∗ − λ∗ | ≤ |µ − λ|.
Moreover, we can say
(µ∗ − λ∗ )+ ≤ (µ − λ)+ .
[see proof in [6], Theorem 10.]
Remark 43. In the previous proposition we have stated that the map µ → µ∗
is a sort of contraction. The distance between the reduced elements is lower
(or equal) than the distance between the starting elements.
4.2.2
General nonlinearities g
In this section we briefly consider the case where g : R → R is a continuous, nondecreasing function such that g(0) = 0. Therefore, we will not require
condition (4.2). We will not prove these results but we intend to understand
if it is possible to approach the problem in this case.
At first, we present a proposition similar to Proposition 4.2.1.
71
4.2 Reduced Measures
4. Good measures and reduced measures
Proposition 4.2.12. Let µ ∈ M(Ω), and let un be the solution to
(
−∆un + gn (un ) = µ in Ω,
un = 0
on ∂Ω,
then there exists u∗ ∈ L1 (Ω) such that un converges to u∗ in L1 (Ω). If µ ≥ 0,
then u∗ ≥ 0 is the largest subsolution to (4.1). If µ ≤ 0, then u∗ ≤ 0 is the
smallest supersolution to (4.1). Moreover, we have
Z
u∗ ∆ϕ ≤ 2||µ||M(Ω) ||ϕ||L∞ (Ω) , ∀ϕ ∈ C0∞ (Ω).
(4.6)
Ω
[see proof in [6], Lemma 4.]
Thus, we are ready to show which problem u∗ satisfies. This is explained
in the following theorem.
Theorem 4.2.13. Let (un )n∈N be the sequence defined in Proposition 4.2.12.
Then un converges to u∗ in L1 (Ω) where u∗ satisfies
(
−∆u∗ + g(u∗ ) = (µ+ )∗ + (−µ− )∗ in Ω,
u∗ = 0
on ∂Ω.
[see proof in [6], Theorem 15.]
Remark 44. We have stated that if we consider a more general function g then
we can speak of u∗ but we have to consider a slightly different reduced measure.
72
Appendix A
Uniqueness
In this section we want to prove that if g is nondecreasing and a solution
to the semilinear Dirichlet problem does exist then it is unique.
Proposition A.0.14. Let g : R → R be a continuous, nondecreasing function,
such that g(0) = 0. Let µ ∈ M(Ω), then the following semilinear Dirichlet
problem
(
−∆u + g(u) = µ in Ω,
(A.1)
u=0
on ∂Ω,
has at most one solution u ∈ L1 (Ω).
Proof. Let (ρn )n∈N be a sequence of mollifiers in RN such that, for every n ≥ 1,
supp(ρn ) ⊂ B 1 . Thus, we define
n
Z
µn (x) = ρn ∗ µ(x) =
ρn (x − y)dµ(y),
∀x ∈ RN .
Ω
We provide a result that is useful in order to prove the proposition.
Subproposition A.0.15. Let p : R → R be a continuous, bounded, nondecreasing function, such that p(0) = 0. Let f ∈ L1 (Ω) and let v ∈ L1 (Ω) be the
unique solution to
(
−∆v = f in Ω,
v=0
on ∂Ω,
then
Z
Z
−
P (v)∆ϕ ≤
Ω
Z
f p(v)ϕ,
Ω
t
p(s)ds, ∀t ∈ R.
where P (t) =
0
[see proof in [6], Lemma B.1.]
73
∀ϕ ∈ C ∞ (Ω), ϕ ≥ 0 in Ω,
A. Uniqueness
Let us consider the following linear problem
(
−∆un = µn − g(u) in Ω,
un = 0
on ∂Ω.
(A.2)
Applying the Subproposition A.0.15 we obtain
Z
Z
(µn − g(u)) p(un )ϕ, ∀ϕ ∈ C ∞ (Ω), ϕ ≥ 0 in Ω,
− P (un )∆ϕ ≤
Ω
Ω
where p is a function satisfying the assumptions of the subproposition. Thus,
if we take 0 ≤ p(t) ≤ 1, ∀t ∈ R, then we have
Z
Z
Z
Z
− P (un )∆ϕ+ g(u)p(un )ϕ ≤
µn p(un )ϕ ≤ ||ϕ||L∞ (Ω) (µn )+ ≤ ||ϕ||L∞ (Ω) ||µ+ ||M(Ω) .
Ω
Ω
Ω
Ω
We let n tend to infinity
Z
Z
− P (u)∆ϕ + g(u)p(u)ϕ ≤ ||ϕ||L∞ (Ω) ||µ+ ||M(Ω) .
Ω
Ω
We apply last inequality to a sequence of nondecreasing continuous functions
(pn )n∈N such that pn (t) = 0 if t ≤ 0 and pn (t) ≥ 1 if t ≥ n1 .
We let n tend to infinity in order to obtain
Z
Z
+
− u ∆ϕ +
g(u)ϕ ≤ ||ϕ||L∞ (Ω) ||µ+ ||M(Ω) ,
Ω
∀ϕ ∈ C ∞ (Ω), ϕ ≥ 0 in Ω.
[u>0]
Clearly, we can also obtain
Z
Z
−
− u ∆ϕ +
g(u)ϕ ≤ ||ϕ||L∞ (Ω) ||µ− ||M(Ω) ,
Ω
∀ϕ ∈ C ∞ (Ω), ϕ ≥ 0 in Ω.
[u<0]
and conclude that
Z
Z
− |u|∆ϕ+ g(u) sign(u)ϕ ≤ ||ϕ||L∞ (Ω) ||µ||M(Ω) , ∀ϕ ∈ C ∞ (Ω), ϕ ≥ 0 in Ω.
Ω
Ω
Therefore, if u, v are solutions to (A.1) with datums µ1 , µ2 , then, for every
ϕ ∈ C ∞ (Ω) such that ϕ ≥ 0 in Ω, we have
Z
Z
− |u − v|∆ϕ + (g(u) − g(v)) sign(u)ϕ ≤ ||ϕ||L∞ (Ω) ||µ1 − µ2 ||M(Ω) .
Ω
Ω
Recalling that g is notdecreasing and choosing ϕ = 1, then we obtain
Z
|g(u) − g(v)| ≤ ||µ1 − µ2 ||M(Ω) .
Ω
This implies that, if u,v are solutions to the same problem, it must be u =
v.
74
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