1 Introduction
We consider a class of quasilinear elliptic equations of the form
(P)
{
-
div
(
A
(
x
,
u
)
∇
u
)
+
u
=
μ
in
Ω
,
u
=
0
on
∂
Ω
,
where Ω⊂ℝN is a bounded domain with smooth boundary ∂Ω (N≥3), μ∈ℳ(Ω) is a finite Radon measure, and A(x,u):Ω×ℝ→ℝN2 is a bounded symmetric Carathéodory matrix-valued function satisfying the following hypothesis, which will be always assumed to hold throughout the paper.
Hypothesis 1.1.
There exists α,β>0 and ϕ∈C1(ℝ), with ϕ(0)=0 and ϕ′(u)≥0 in ℝ, such that
A
(
x
,
u
)
ξ
⋅
ξ
≥
α
ϕ
′
(
u
)
|
ξ
|
2
,
|
A
(
x
,
u
)
ξ
|
≤
β
ϕ
′
(
u
)
|
ξ
|
for every ξ∈ℝN, u∈ℝ and a.e. x∈Ω.
Without loss of generality, we require ϕ′(u)≢0 in ℝ, and we set
(1.1)
ϕ
(
±
∞
)
=
lim
u
→
±
∞
ϕ
(
u
)
=
ϕ
±
∞
∈
ℝ
¯
,
with the possibilities ϕ+∞=+∞ and ϕ-∞=-∞ being both admissible. We explicitly observe that cases in which ϕ is bounded at +∞ and unbounded at -∞ (or vice versa) may also occur.
It is worth observing that in the model case A(x,u)=ϕ′(u)IN, where IN denotes the unit matrix of size N, problem (P) reduces to the equation
(1.2)
{
-
Δ
ϕ
(
u
)
+
u
=
μ
in
Ω
,
u
=
0
on
∂
Ω
,
hence, by the inversion g(v)=ϕ-1(v) (when possible),
(1.3)
{
-
Δ
v
+
g
(
v
)
=
μ
in
Ω
,
v
=
0
on
∂
Ω
.
The elliptic problem (1.3) has been intensively investigated in the last decades. In particular, when the nonlinearity g:ℝ→ℝ is a continuous function satisfying the sign condition sg(s)≥0, the solutions of (1.3) are meant in the following sense:
(1.4)
{
v
∈
L
1
(
Ω
)
,
g
(
v
)
∈
L
1
(
Ω
)
,
-
∫
Ω
v
Δ
ζ
+
∫
Ω
g
(
v
)
ζ
=
∫
Ω
ζ
𝑑
μ
for all
ζ
∈
C
2
(
Ω
¯
)
,
ζ
=
0
on
∂
Ω
and they always exist if the datum μ belongs to the Lebesgue space L1(Ω) or it is a diffuse measure with respect to the harmonic-capacity (see [7, 20, 17, 27, 30]; see also [18, Appendix 4B] for a survey on the subject). Let us explicitly remark that, by the requirement g(v)∈L1(Ω), the solutions of (1.3) in the sense of (1.4) formally translate back to solutions of (1.2) which are intrinsically function-valued (i.e., functions in L1(Ω)). On the other hand, the case where μ is any finite Radon measure turns out to be much more complicated. Indeed, it is well known that solutions of (1.3) in the sense of (1.4) may not exist for some choice of the datum μ if the nonlinear term g(s) grows too fast at infinity, e.g., as a power type function |s|p-1s with p≥NN-2 (see [6, 16, 3, 21, 29, 38]), formally corresponding to consider in problem (1.2) nonlinearities |ϕ(u)|≅|u|m as |u|→∞, 0<m≤N-2N. Analogous results have been proved in [4, 37] for an exponential nonlinearity (formally corresponding to a function ϕ in (1.2) with a logarithmic growth at infinity), and in [8, 24] when g:(-∞,s0)→ℝ has a vertical asymptote at some point s0 (formally corresponding to a bounded ϕ in (1.2)). Concerning the latter, it is significant to observe that cases in which problem (1.3) does not admit a solution in the sense of (1.4) may even occur if μ is a diffuse measure with respect to the harmonic-capacity (see [24, Theorem 1]; see also [19], where g is a maximal monotone graph with closed domain and the definition of solution is intrinsically measure-valued).
For the sake of simplicity, in problem (P) the coefficient of the lower order term u is equal to one. However, our results apply as well to the case
(1.5)
{
-
div
(
A
(
x
,
u
)
∇
u
)
+
c
u
=
μ
in
Ω
,
u
=
0
on
∂
Ω
,
with c>0. There are a lot of papers devoted to the study of (1.5) for c≥0, concerning in particular the existence of (function-valued) solutions and regularity properties (see [1, 9, 11, 10, 12] and references therein). In most of them, the matrix A(x,u) satisfies
α
|
ξ
|
2
(
1
+
|
s
|
)
γ
≤
〈
A
(
x
,
s
)
ξ
,
ξ
〉
≤
β
|
ξ
|
2
for some γ>0, and the datum μ belongs to some Lebesgue space Lm(Ω), with m≥1, or μ=div(g), with g∈Lr(Ω), for suitable choices of r (the case γ=0 has been largely studied, see [5, 13, 14, 15, 22] and references therein). Expectedly, for c=0, problem (1.5) is “strongly noncoercive” and a nonexistence result has been proved even for L∞-data if γ>1 (see [1]). In the spirit of [6, 3], when c>0, γ>1 and μ is a finite Radon measure, the nonexistence of L1-solutions of (1.5) has been proved in [11] in the case where the measure μ is concentrated on a set of zero harmonic-capacity.
To sum up, in much of the literature on the subject, although the datum μ is a measure, by definition, a solution of (1.5) belongs to some Lebesgue space. On the other hand, function-valued solutions may not exist (for specific choices of μ) if the diffusivity ϕ′(s) degenerates too fast at infinity. Hence, in these singular cases, it looks very natural to consider solutions which take values in the space of finite Radon measures (see [32] for preliminary results in this direction). In more detail, the present paper focus on the possibility of extending the concept of solution to problem (P) in the case where μ is any finite Radon measure and ϕ is any nondecreasing C1-function, in order to develop an existence theory which is consistent with respect to regularizing and smoothing approximations (solutions will be constructed by a standard regularization procedure; see problems (1) below). At the same time, it looks desirable to make the notion of solution independent of the level of degeneracy of diffusivity at infinity, including cases in which ϕ′ may vanish within one or more non-empty, possibly unbounded, intervals.
In the following, as already remarked, solutions are obtained via the approximating problems
P
n
{
-
div
(
A
(
x
,
u
n
)
∇
u
n
)
-
1
n
Δ
u
n
+
u
n
=
μ
n
in
Ω
,
u
n
=
0
on
∂
Ω
(see also (3.3)), and are meant as finite Radon measures u∈ℳ(Ω) which satisfy (P) in the following weak sense: ϕ(ur)∈W01,1(Ω), TK(ϕ(ur))∈W01,2(Ω) for all K>0 and
∫
{
ϕ
′
(
u
r
)
>
0
}
A
(
x
,
u
r
)
ϕ
′
(
u
r
)
∇
ϕ
(
u
r
)
⋅
∇
ρ
d
x
+
〈
u
,
ρ
〉
Ω
=
〈
μ
,
ρ
〉
Ω
for every ρ∈Cc1(Ω) (TK is the standard truncation function). Notice that, from this point of view, whether the (constructed) solutions are function-valued or not can be regarded as a regularity problem.
It is also worth observing that in the previous definition we have collected, roughly speaking, the minimal requirements which a solution should have – at least to solve problem (P) in the distributional sense – and the uniqueness of solutions in general fails within this class (see Remark 3.7). Hence, in order to look for uniqueness criteria, a natural question is whether some extra information can be draft from the above approximation procedure. More precisely, it is significant to investigate the qualitative properties of the singular part of the constructed solutions, by relying on the following decomposition of their positive and negative parts:
(1.6)
u
+
=
u
ac
+
+
[
u
s
+
]
d
,
2
+
u
c
,
2
+
,
u
-
=
u
ac
-
+
[
u
s
-
]
d
,
2
+
u
c
,
2
-
,
where
u
ac
±
and us± denote, respectively, the absolutely continuous and the singular part of u± with respect to the Lebesgue measure;
[
u
s
±
]
d
,
2
and uc,2± denote, respectively, the diffuse part and the concentrated part of us± with respect to the harmonic-capacity.
The main result of the present paper is Theorem 3.4, which focuses on the structure of the diffuse part of the constructed solutions with respect to the harmonic-capacity and contains, as a by-product, the following statements.
Theorem 1.2.
Let ϕ+∞=-ϕ-∞=∞ and let u be a solution of (P), obtained as a limiting point of a sequence {un} of solutions to problems (1). Then [us]d,2=0.
Theorem 1.3.
Let ϕ be bounded and let u be a solution of (P), obtained as a limiting point of a sequence {un} of solutions to problems (1). Then
(1.7)
[
u
s
±
]
d
,
2
≤
[
μ
s
±
]
d
,
2
⌞
{
x
∈
Ω
:
ϕ
(
u
r
(
x
)
)
=
ϕ
±
∞
}
.
Theorems 1.2 and 1.3 are specific for the constructed solutions and, in particular, imply that their singular parts need not be equal to the singular part of the datum μ (i.e., in general, us≠μs). Moreover, as a consequence of Theorem 3.4, in Corollary 3.6 we get the existence of L1-solutions for every diffuse measure μ with respect to the 2-capacity whenever ϕ+∞=-ϕ-∞=∞.
The most singular term in the right-hand side of equalities (1.6) – namely, the concentrated part of any constructed solution u with respect to the 2-capacity – satisfies, for every choice of ϕ, uc,2±≤μc,2± (this follows from (3.4) in Theorem 3.4), and is equal to the 2-concentrated part of μ when ϕ is bounded, i.e.,
(1.8)
u
c
,
2
=
μ
c
,
2
if
ϕ
∈
L
∞
(
ℝ
)
.
Analogous results when ϕ is unbounded are discussed in Propositions 4.1 and 4.3. These phenomena, which we call persistence (following the terminology adopted in the parabolic framework; see [33]), provide examples of nonexistence of function-valued solutions obtained by the approximating problems (1) and are in agreement with the heuristic idea that if ϕ′ vanishes too fast “when u is large”, then the diffusivity is not strong enough to provide smoothing of the concentrated part of solutions with respect to some specific capacity. On the other hand, sufficient conditions ensuring that us=0 – namely, that regularizing effects actually occur – are discussed in Proposition 4.5 when ϕ has a suitable polynomial growth at infinity (let us notice that our results are in the spirit of those in [6] for problem (1.3)), and in Proposition 4.6 in the case of a bounded ϕ (see [24, Theorem 13] for similar results for problem (1.3) when the nonlinearity g has a vertical asymptote).
Uniqueness is addressed in Section 5 under the more restrictive condition of Hypothesis 5.1, and with further requirements on the solutions in order to prevent nonuniqueness phenomena (in this regard, see [34, 35, 5, 22, 15, 31]). More precisely, when ϕ is bounded, uniqueness relies on the structural conditions in (1.7) and (1.8), which, in this sense, select a class of solutions where the problem is well-posed (see Theorem 5.2 below). On the other hand, if the assumption of the boundedness of ϕ is dropped, the situation is more subtle. In Definition 5.3 we introduce a new notion of weak entropy solution – which can be regarded as an “adjustment” of the concept of entropy solution given in [5] to the kind of equations considered in the present paper (see also [32] for preliminary developments in this direction) – and we prove existence and uniqueness in this class for every choice of the datum μ in the set of the diffuse measures with respect to the harmonic-capacity (see Theorems 5.4 and 5.6).
The content of the paper is distributed as follows. After some mathematical preliminaries in Section 2, we collect the main results in Sections 3, 4 and 5, and address the approximating problems in Section 6. Finally, in Section 7 we pass to the limit in the approximating problems, and Sections 8, 9 and 10 are devoted to the proofs of the main results.
2 Preliminary Notations
2.1 Radon Measures
In the following we denote by ℳ(Ω) the space of finite Radon measures on Ω, and by ℳ+(Ω)⊂ℳ(Ω) the cone of nonnegative (finite Radon) measures on Ω. For every μ∈ℳ(Ω), ∥μ∥ℳ(Ω):=|μ|(Ω) is the total variation of μ and, by abuse of notation, we extend the symbol of duality between the space ℳ(Ω) and Cc(Ω) – namely, 〈μ,ζ〉Ω=∫Ωζ(x)𝑑μ(x) – to μ-integrable functions ζ. Hereafter a Borel set E such that |E|=0 is called a null set, where |⋅| denotes the Lebesgue measure; moreover, the expression “almost everywhere”, or shortly a.e., means “up to null sets”.
For every μ∈ℳ(Ω) and every Borel set E⊆Ω, the restriction μ⌞E of μ to E is defined by setting (μ⌞E)(A):=μ(E∩A) for every Borel set A⊆Ω. In the following we denote by ℳac(Ω) (respectively, ℳs(Ω)) the set of measures which are absolutely continuous with respect to the Lebesgue measure (respectively, singular with respect to the Lebesgue measure) and, for every μ∈ℳ(Ω), we denote, respectively, by μac and μs its absolutely continuous part and its singular part with respect to the Lebesgue measure. By abuse of notation, we will often identify the absolutely continuous part μac of a measure μ with its density μr∈L1(Ω) (here μac(E)=∫Eμr(x)𝑑x for every Borel set E⊆Ω). Finally, every μ∈ℳ(Ω) can be uniquely decomposed into a difference μ=μ+-μ- of two nonnegative and mutually singular measures μ+ and μ-, called, respectively, the positive and the negative part of μ, and the equalities [μs]±=[μ±]s=:μs± follow by standard arguments.
2.2 Capacity
For every subset E⊆Ω, the p-capacity of E in Ω is defined as
c
1
,
p
(
E
)
=
c
1
,
p
(
E
;
Ω
)
:=
inf
v
∈
𝒰
Ω
E
∫
Ω
|
∇
v
|
p
d
x
(
p
∈
[
1
,
∞
)
)
,
where 𝒰ΩE denotes the set of functions v∈W01,p(Ω) such that 0≤v≤1 a.e. in Ω, and v=1 a.e. in a neighborhood of E (see [25] for basic definitions and properties).
For every p∈[1,∞), we denote by
ℳ
c
,
p
(
Ω
)
:=
{
μ
∈
ℳ
(
Ω
)
:
there exists a Borel set
E
⊆
Ω
such that
c
1
,
p
(
E
)
=
0
and
μ
=
μ
⌞
E
}
and
ℳ
d
,
p
(
Ω
)
:=
{
μ
∈
ℳ
(
Ω
)
:
|
μ
|
(
E
)
=
0
for every Borel set
E
⊆
Ω
such that
c
1
,
p
(
E
)
=
0
}
the sets of measures which are, respectively, concentrated and diffuse with respect to the p-capacity. For every μ∈ℳ(Ω), p∈[1,∞), there exists a unique couple (μc,p,μd,p) of (mutually singular) measures μc,p∈ℳc,p(Ω) and μd,p∈ℳd,p(Ω), called, respectively, the concentrated part and the diffuse part of μ with respect to the p-capacity, such that μ=μc,p+μd,p (see, e.g., [26]). Moreover, we have μc,p=[μs]c,p, μd,p=μac+[μs]d,p and
[
μ
s
±
]
c
,
p
=
[
μ
±
]
c
,
p
=
[
μ
c
,
p
]
±
=
:
μ
c
,
p
±
,
[
μ
±
]
d
,
p
=
[
μ
d
,
p
]
±
=
:
μ
d
,
p
±
.
For p∈(1,∞), it is known that μ∈ℳ(Ω) belongs to ℳd,p(Ω) if and only if μ∈L1(Ω)+W-1,p′(Ω) (where W-1,p′(Ω) denotes the dual space of the Sobolev space W01,p(Ω), see [15]). Then, if μ∈ℳd,p(Ω), every function v∈W01,p(Ω)∩L∞(Ω) also belongs to L∞(Ω,μ) and the duality symbol 〈μ,φ〉Ω can be extended to any φ∈W01,p(Ω)∩L∞(Ω).
A function f:Ω→ℝ is c1,p-quasi continuous in Ω if for every ε>0, there exists a set E⊆Ω, with c1,p(E)<ε, such that the restriction fΩ∖E is continuous in Ω∖E.
Every function u∈W1,p(Ω) has a c1,p-quasi continuous representative u~, defined c1,p-almost everywhere in Ω (see, e.g., [25]). Hence, in the following we will always identify every function u∈W1,p(Ω) with its (c1,p-essentially) unique c1,p-quasi continuous representative. Finally, for every E⊂Ω, we also have (see [22])
(2.1)
c
1
,
p
(
E
)
=
inf
v
∈
𝒲
Ω
E
∫
Ω
|
∇
v
|
p
d
x
,
where
𝒲
Ω
E
=
{
v
∈
W
0
1
,
p
(
Ω
)
:
v
=
1
c
1
,
p
-a.e. in
E
,
v
≥
0
c
1
,
p
-a.e. in
Ω
}
(
1
<
p
≤
N
)
.
2.3 Further Notations
For every s∈ℝ, we set s+=max{s,0}, s-=max{-s,0}. Moreover, for all K≥0, we set TK(s)=max{min{K,s},-K} and
(2.2)
H
K
+
(
s
)
:=
{
0
if
s
≤
K
,
s
-
K
if
K
<
s
<
K
+
1
,
1
if
s
≥
K
+
1
,
H
K
-
(
s
)
:=
{
0
if
s
≥
-
K
,
s
+
K
if
-
K
-
1
<
s
<
-
K
,
-
1
if
s
≤
-
K
-
1
.
In the sequel, we shall also use several technical results concerning the set 𝒴(Ω;ℝ) of Young measures on Ω×ℝ. For the precise definitions and statements, we refer to Appendix A.
6 Approximating Problems
For every μ∈ℳ(Ω), let {μn}⊆C∞(Ω¯) be the sequence defined in (3.3). Denoting by μ+ and μ- the positive and negative part of μ, respectively, we have
(6.1)
μ
n
=
[
μ
+
]
n
-
[
μ
-
]
n
,
with the sequences {[μ±]n}⊆C∞(Ω¯), [μ±]n≥0 in Ω, satisfying
(6.2)
[
μ
±
]
n
=
[
μ
r
±
]
n
+
[
μ
s
±
]
n
,
where
[
μ
r
±
]
n
(
x
)
=
∫
Ω
ρ
n
(
x
-
y
)
μ
r
±
(
y
)
𝑑
y
,
[
μ
s
±
]
n
(
x
)
=
∫
Ω
ρ
n
(
x
-
y
)
𝑑
μ
s
±
(
y
)
.
Then {[μr±]n}⊆C∞(Ω¯), {[μs±]n}⊆C∞(Ω¯), with [μr±]n≥0,[μs±]n≥0, and the following hold:
∥
[
μ
±
]
n
∥
L
1
(
Ω
)
≤
∥
μ
±
∥
ℳ
(
Ω
)
,
∥
[
μ
±
]
n
∥
L
∞
(
Ω
)
≤
C
n
,
[
μ
s
±
]
n
⇀
*
μ
s
±
in
ℳ
(
Ω
)
,
[
μ
r
±
]
n
→
μ
r
±
in
L
1
(
Ω
)
,
[
μ
±
]
n
(
x
)
→
μ
r
±
(
x
)
for a.e.
x
∈
Ω
.
For every n, there exists un∈W01,2(Ω)∩L∞(Ω) which solves problem (1) in the weak sense, i.e.,
(6.3)
∫
Ω
A
(
x
,
u
n
)
∇
u
n
⋅
∇
ρ
d
x
+
1
n
∫
Ω
∇
u
n
⋅
∇
ρ
d
x
+
∫
Ω
u
n
ρ
𝑑
x
=
∫
Ω
μ
n
(
x
)
ρ
(
x
)
𝑑
x
for every ρ∈W01,2(Ω) (this follows from the results in [28], by arguing as in [11, Section 2]).
The a priori estimates in Proposition 6.1 below are proved by standard techniques, most of them relying on a suitable choice of the test function ρ in (6.3) (see, e.g., [35, 20]).
Proposition 6.1.
For every
n
∈
ℕ
,
(6.4)
∥
u
n
∥
L
1
(
Ω
)
≤
∥
μ
∥
ℳ
(
Ω
)
,
and, for every
K
>
0
,
(6.5)
α
∫
{
K
<
|
ϕ
(
u
n
)
|
<
K
+
1
}
|
∇
(
ϕ
(
u
n
)
)
|
2
d
x
+
1
n
∫
{
K
<
|
ϕ
(
u
n
)
|
<
K
+
1
}
ϕ
′
(
u
n
)
|
∇
u
n
|
2
d
x
≤
∥
μ
∥
ℳ
(
Ω
)
,
(6.6)
α
∫
Ω
|
∇
T
K
(
ϕ
(
u
n
)
)
|
2
d
x
+
1
n
∫
{
|
ϕ
(
u
n
)
|
<
K
}
ϕ
′
(
u
n
)
|
∇
u
n
|
2
d
x
≤
K
∥
μ
∥
ℳ
(
Ω
)
.
For every
θ
>
0
, there exists a constant
C
θ
such that for every
n
∈
ℕ
, we have
(6.7)
α
∫
Ω
ϕ
′
(
u
n
)
|
∇
u
n
|
2
(
1
+
|
u
n
|
)
θ
+
1
𝑑
x
+
1
n
∫
Ω
|
∇
u
n
|
2
(
1
+
|
u
n
|
)
θ
+
1
𝑑
x
≤
C
θ
.
For every
1
≤
p
<
N
N
-
1
, there exists a constant
C
p
>
0
such that for every
n
, we have
(6.8)
∥
ϕ
(
u
n
)
∥
W
0
1
,
p
(
Ω
)
≤
C
p
.
For every
θ
∈
(
0
,
1
)
, there exists a constant
C
~
θ
>
0
such that for every
n
∈
ℕ
, we have
(6.9)
1
n
2
∫
Ω
|
∇
u
n
|
2
≤
C
~
θ
n
θ
-
1
2
.
Remark 6.2.
Since [TK(ϕ)]±=TK(ϕ±), we have
(6.10)
∥
ϕ
±
(
u
n
)
∥
W
0
1
,
2
(
Ω
)
≤
C
if
|
ϕ
±
∞
|
<
+
∞
,
∥
ϕ
(
u
n
)
∥
W
0
1
,
2
(
Ω
)
≤
C
if
ϕ
∈
L
∞
(
ℝ
)
,
where ϕ±∞ are the values in (1.1). Indeed, it suffices to choose in (6.6) K≥ϕ+∞ if ϕ+∞<+∞ and -K≤ϕ-∞ if ϕ-∞>-∞.
Proof of Proposition 6.1.
To prove (6.4) we consider in (6.3) the test function ρj=fj(un), where {fj} is a smooth approximation of the sign function (see [20]), and we take the limit as j→∞. Analogously, to prove (6.6) we use the test function ρ=TK(ϕ(un)), whereas (6.5) follows by choosing in (6.3) ρ=HK+(ϕ(un)) and ρ=HK-(ϕ(un)) (we omit the details). To prove (6.7), it suffices to choose in (6.3) the test function ρ=[1-(1+|un|)-θ]signun, with θ>0.
The proof of claim (iii) is analogous to that in [5, Lemmas 4.1 and 4.2] (see also [23, Appendix]).
Let us finally prove (iv). Since ∥μn∥L∞(Ω)≤Cn, with C>0 independent of n (see (3.3)), choosing in (6.3) the test functions ρ=(un-Cn)+ and ρ=(un+Cn)-, by Hypothesis 1.1 (i), we obtain ∥un∥L∞(Ω)≤Cn. Then, for every θ∈(0,1), by (6.7), it follows that
∫
Ω
|
∇
u
n
|
2
n
2
𝑑
x
=
∫
Ω
|
∇
u
n
|
2
(
1
+
|
u
n
|
)
θ
+
1
n
2
(
1
+
|
u
n
|
)
θ
+
1
𝑑
x
≤
(
1
+
C
n
)
θ
+
1
n
∫
Ω
|
∇
u
n
|
2
n
(
1
+
|
u
n
|
)
θ
+
1
𝑑
x
≤
C
~
θ
n
θ
-
1
2
.
∎
Proposition 6.3.
For every K≥0 and for every n∈N, we have
(6.11)
α
∫
{
u
n
>
K
}
|
∇
ϕ
(
u
n
)
|
2
d
x
d
t
≤
∥
μ
∥
ℳ
(
Ω
)
[
ϕ
+
∞
-
ϕ
(
K
)
]
if
ϕ
+
∞
<
+
∞
,
(6.12)
α
∫
{
u
n
<
-
K
}
|
∇
ϕ
(
u
n
)
|
2
d
x
d
t
≤
∥
μ
∥
ℳ
(
Ω
)
[
ϕ
(
-
K
)
-
ϕ
-
∞
]
if
ϕ
-
∞
>
-
∞
.
Proof.
Let us prove (6.11), the proof of (6.12) being analogous. For every K≥0, choosing in (6.3) the test function
f
(
u
n
)
=
∫
0
u
n
ϕ
′
(
s
)
χ
{
s
≥
K
}
(
s
)
𝑑
s
=
{
0
if
u
n
≤
K
,
ϕ
(
u
n
)
-
ϕ
(
K
)
if
u
n
>
K
,
by Hypothesis 1.1 (i), we obtain
α
∫
{
u
n
>
K
}
|
∇
ϕ
(
u
n
)
|
2
d
x
≤
α
∫
{
u
n
>
K
}
|
∇
ϕ
(
u
n
)
|
2
d
x
+
1
n
∫
{
u
n
>
K
}
ϕ
′
(
u
n
)
|
∇
u
n
|
2
+
∫
Ω
f
(
u
n
)
u
n
d
x
=
∫
Ω
μ
n
f
(
u
n
)
𝑑
x
≤
∥
μ
∥
ℳ
(
Ω
)
[
ϕ
+
∞
-
ϕ
(
K
)
]
.
∎
Proposition 6.4.
Let (4.3) hold with 0<γ≤1. Then (6.8) holds true for every 1≤p<22-γ.
Proof.
Fix 1≤p<22-γ and let θ≡θp:=2p-2+γ. Observe that γ≤1⇒22-γ≤2, and the range condition on p ensures that θ>0. By (4.3), we have
∫
Ω
|
∇
ϕ
(
u
n
)
|
p
d
x
=
∫
Ω
[
ϕ
′
(
u
n
)
]
p
2
|
∇
u
n
|
p
(
1
+
|
u
n
|
)
(
θ
+
1
)
p
2
(
1
+
|
u
n
|
)
(
θ
+
1
)
p
2
[
ϕ
′
(
u
n
)
]
p
2
d
x
≤
(
∫
Ω
ϕ
′
(
u
n
)
|
∇
u
n
|
2
(
1
+
|
u
n
|
)
θ
+
1
𝑑
x
)
p
2
(
∫
Ω
(
1
+
|
u
n
|
)
(
θ
+
1
)
p
2
-
p
[
ϕ
′
(
u
n
)
]
p
2
⋅
2
2
-
p
𝑑
x
)
2
-
p
2
≤
C
(
∫
Ω
ϕ
′
(
u
n
)
|
∇
u
n
|
2
(
1
+
|
u
n
|
)
θ
+
1
𝑑
x
)
p
2
(
∫
Ω
(
1
+
|
u
n
|
)
(
θ
+
1
)
p
2
-
p
-
γ
(
p
2
-
p
)
𝑑
x
)
2
-
p
2
=
C
(
∫
Ω
ϕ
′
(
u
n
)
|
∇
u
n
|
2
(
1
+
|
u
n
|
)
θ
+
1
𝑑
x
)
p
2
(
∫
Ω
(
1
+
|
u
n
|
)
𝑑
x
)
2
-
p
2
≤
C
θ
p
,
the latter inequality being a consequence of (6.4) and (6.7). ∎
7 Letting n→∞ in Problems (Pn)
The following proposition is a direct consequence of the estimates in (6.6) and (6.8).
Proposition 7.1.
There exists v∈W01,p(Ω) for every 1≤p<N-1N and TK(v)∈W01,2(Ω) for all K>0 such that, possibly up to a subsequence (denoted again {un} for simplicity), we have
(7.1)
ϕ
(
u
n
(
x
)
)
→
v
(
x
)
for a.e.
x
∈
Ω
,
ϕ
(
u
n
)
⇀
v
in
W
0
1
,
p
(
Ω
)
for every
1
≤
p
<
N
N
-
1
,
T
K
(
ϕ
(
u
n
)
)
⇀
T
K
(
v
)
in
W
0
1
,
2
(
Ω
)
for all
K
>
0
.
Obviously, (7.1) implies that ϕ-∞≤v(x)≤ϕ+∞ for a.e. x∈Ω.
Next, we study the limiting points of the sequence of Young measures associated with {un}.
Proposition 7.2.
Let {τn} be the sequence of Young measures associated with {un}.
There exist a Young measure
τ
∈
𝒴
(
Ω
;
ℝ
)
and a subsequence
{
u
n
j
}
⊆
{
u
n
}
such that
(7.2)
τ
n
j
→
τ
narrowly in
Ω
×
ℝ
.
Moreover, for a.e.
x
∈
Ω
,
(7.3)
supp
τ
x
⊆
{
ξ
∈
ℝ
:
ϕ
(
ξ
)
=
v
(
x
)
}
,
where
τ
x
is the disintegration of τ , and v is the function given in Proposition 7.1.
There exist
λ
1
,
λ
2
∈
ℳ
+
(
Ω
)
such that, possibly up to a subsequence denoted again
{
u
n
j
}
for simplicity, we have
(7.4)
u
n
j
+
⇀
*
u
b
,
1
+
λ
1
,
u
n
j
-
⇀
*
u
b
,
2
+
λ
2
in
ℳ
(
Ω
)
,
where
u
b
,
1
,
u
b
,
2
∈
L
1
(
Ω
)
, with
u
b
,
i
≥
0
in Ω , are defined by
(7.5)
u
b
,
1
(
x
)
=
∫
ℝ
ξ
+
d
τ
x
(
ξ
)
,
u
b
,
2
(
x
)
=
∫
ℝ
ξ
-
d
τ
x
(
ξ
)
(
for a.e.
x
∈
Ω
)
.
Remark 7.3.
Let us explicitly observe that if ϕ′>0 in ℝ, then, by (7.1), we get un(x)→ϕ-1(v(x)) for a.e. x∈Ω. From this convergence we infer that τx is the Dirac mass concentrated at the point ϕ-1(v(x)) (see, e.g., [36, Proposition 1]), hence (7.3) immediately follows.
Since ϕ is continuous and nondecreasing, the set Jx:={ξ∈ℝ:ϕ(ξ)=v(x)} is a closed (possibly degenerate or unbounded) interval for a.e. x∈Ω.
In view of Proposition 7.2 (ii), the function
(7.6)
u
b
(
x
)
:=
u
b
,
1
(
x
)
-
u
b
,
2
(
x
)
≡
∫
ℝ
ξ
d
τ
x
(
ξ
)
(
for a.e.
x
∈
Ω
)
belongs to the space L1(Ω). Moreover, setting λ:=λ1-λ2∈ℳ(Ω) and
(7.7)
u
:=
u
b
+
λ
∈
ℳ
(
Ω
)
,
by (7.4), we get
(7.8)
u
n
j
⇀
*
u
in
ℳ
(
Ω
)
.
Proof of Proposition 7.2.
(i) The convergence in (7.2) follows from the Prokhorov theorem, since the sequence {un} is uniformly bounded in L1(Ω) (see (6.4) and Proposition A.4).
It remains to prove (7.3). Without loss of generality, we may assume that all the convergences in Proposition 7.1 are satisfied along the subsequence {unj}. For every f∈Cc(ℝ), the sequence {f(ϕ(un))} is bounded in L∞(Ω), hence, by Proposition A.4, we have
f
(
ϕ
(
u
n
j
)
)
⇀
*
(
f
∘
ϕ
)
*
in
L
∞
(
Ω
)
,
where (f∘ϕ)*(x)=∫ℝf(ϕ(ξ))𝑑τx(ξ) for a.e. x∈Ω. On the other hand, by (7.1), we also get f(ϕ(unj))→f(v) in Lq(Ω) for every 1≤q<∞. Hence, there exists a null set N⊆Ω such that for every x∈Ω∖N and for every f as above, we have
(7.9)
f
(
v
(
x
)
)
=
(
f
∘
ϕ
)
*
(
x
)
=
∫
ℝ
f
(
ϕ
(
ξ
)
)
𝑑
τ
x
(
ξ
)
(the choice of the null set N can be made independent of f by separability arguments). Let x∈Ω∖N be fixed arbitrarily. For every K>0 such that -K<v(x)<K, let fK∈C0([-K,K]) satisfy fK(v(x))=0 and fK(s)>0 otherwise in (-K,K). Then (7.9) with f=fK gives ∫ℝfK(ϕ(ξ))𝑑τx(ξ)=0, whence (by the arbitrariness of K) τx({ξ∈ℝ:ϕ(ξ)≠v(x)})=0, and the conclusion follows.
(ii) Let us address the first convergence in (7.4), the proof of the second one being analogous. By (6.4) and Proposition A.4 (iii), there exists a subsequence (denoted again {unj} for simplicity) and a sequence of measurable sets Aj⊆Aj+1⊆Ω, with |Aj|→0, such that
(7.10)
u
n
j
+
χ
Ω
∖
A
j
⇀
u
b
,
1
in
L
1
(
Ω
)
,
where ub,1∈L1(Ω), ub,1≥0, is the barycenter of the limiting Young measure τ1 associated with the sequence {unj+}, i.e., ub,1(x)=∫ℝξ𝑑τx1(ξ) for a.e. x∈Ω. Since it can be easily seen that suppτx1⊆[0,∞) and τx1⌞(0,∞)=τx⌞(0,+∞), where τ is the Young measure given in step (i), the first equality in (7.5) follows.
By (6.4), the sequence {unj+χAj} is bounded in L1(Ω), whence there exists λ1∈ℳ+(Ω) such that (possibly up to a subsequence) unj+χAj⇀*λ1 in ℳ(Ω). By this convergence and (7.10), we get (7.4). ∎
Let us clarify the relation between the function v given in Proposition 7.1 and ub in (7.6).
Proposition 7.4.
Let {unj}, v and ub be, respectively, the sequence and the functions given in Propositions 7.1–7.2 and in (7.6). Then
(7.11)
v
(
x
)
=
ϕ
(
u
b
(
x
)
)
for a.e.
x
∈
Ω
.
Moreover,
(7.12)
u
n
j
→
u
b
a.e. in
Ω
′
=
{
x
∈
Ω
:
ϕ
′
(
u
b
(
x
)
)
>
0
}
.
Proof.
Let us firstly prove (7.11). By (7.3) and (7.6), for a.e. x∈Ω, we have ub(x)=∫Jxξ𝑑τx(ξ), where Jx={ξ∈ℝ:ϕ(ξ)=v(x)}. Since τx is a probability measure, by the previous equality we obtain that ub(x) belongs to Jx, and (7.11) follows.
Let us prove (7.12). By the above considerations, for a.e. x∈Ω, we have ub(x)∈Jx and
(7.13)
ϕ
(
u
n
j
(
x
)
)
→
ϕ
(
u
b
(
x
)
)
(see (7.1) and (7.11)). Fix any x as above such that, in addition, ϕ′(ub(x))>0. The latter implies that Jx={ub(x)} and, by the continuity of ϕ′, there exists r>0 such that
(7.14)
ϕ
′
(
ξ
)
>
0
for every
ξ
∈
[
u
b
(
x
)
-
r
,
u
b
(
x
)
+
r
]
.
Arguing by contradiction, let ε¯>0 be such that for every j0∈ℕ, there exists j¯≥j0 so that |unj¯(x)-ub(x)|>ε¯. In view of the monotonicity of ϕ, we have either
(7.15)
ϕ
(
u
n
j
¯
(
x
)
)
≥
ϕ
(
u
b
(
x
)
+
ε
¯
)
or
ϕ
(
u
n
j
¯
(
x
)
)
≤
ϕ
(
u
b
(
x
)
-
ε
¯
)
.
Since, by (7.14), ϕ(ub(x)+ε¯)>ϕ(ub(x)) and ϕ(ub(x)-ε¯)<ϕ(ub(x)), from (7.15) and (7.13) we get a contradiction and the conclusion follows. ∎
Let u be the limiting measure in (7.8). Then u=ub+λ in ℳ(Ω) (see (7.7)), and a natural question is whether the above equality coincides with the Lebesgue decomposition of u. The affirmative answer is the content of Proposition 7.6 below, which in turn relies on the following lemma, whose proof is postponed at the end of the section.
Lemma 7.5.
Let τ∈Y(Ω;R), {unj} and λi∈M(Ω) (i=1,2) be the Young measure, the subsequence and the measures given in Proposition 7.2, respectively. Then, for every f∈C(R), such that the following limits exist and are finite:
(7.16)
lim
s
→
+
∞
f
(
s
)
s
=
L
+
,
lim
s
→
-
∞
f
(
s
)
s
=
L
-
,
we have
f
(
u
n
j
)
⇀
*
f
*
+
L
+
λ
1
-
L
-
λ
2
in
ℳ
(
Ω
)
,
where f*∈L1(Ω) is defined by setting
(7.17)
f
*
(
x
)
=
∫
ℝ
f
(
ξ
)
𝑑
τ
x
(
ξ
)
for a.e.
x
∈
Ω
.
Proposition 7.6.
Let u∈M(Ω) and λi∈M+(Ω) be the measures in (7.7) and (7.4), respectively (i=1,2). Then
(7.18)
u
s
+
=
λ
1
≤
μ
s
+
,
u
s
-
=
λ
2
≤
μ
s
-
in
ℳ
(
Ω
)
,
(7.19)
u
b
(
x
)
=
u
r
(
x
)
for a.e.
x
∈
Ω
,
where ub∈L1(Ω) is the function in (7.6) and ur is the density of uac.
Remark 7.7.
By (7.11) and (7.19), we have v=ϕ(ur), and (7.3) can be rephrased as
(7.20)
supp
τ
x
⊆
J
x
=
{
ξ
∈
ℝ
:
ϕ
(
ξ
)
=
ϕ
(
u
r
(
x
)
)
}
for a.e.
x
∈
Ω
.
Moreover, the convergences in Propositions 7.1–7.4 read as follows:
(7.21)
ϕ
(
u
n
j
(
x
)
)
→
ϕ
(
u
r
(
x
)
)
for a.e.
x
∈
Ω
,
(7.22)
ϕ
(
u
n
j
)
⇀
ϕ
(
u
r
)
in
W
0
1
,
p
(
Ω
)
for every
1
≤
p
<
N
N
-
1
,
T
K
(
ϕ
(
u
n
j
)
)
⇀
T
K
(
ϕ
(
u
r
)
)
in
W
0
1
,
2
(
Ω
)
for all
K
>
0
,
(7.23)
u
n
j
→
u
r
a.e. in
Ω
′
=
{
x
∈
Ω
:
ϕ
′
(
u
r
(
x
)
)
>
0
}
,
u
n
j
+
⇀
*
u
b
,
1
+
u
s
+
,
u
n
j
-
⇀
*
u
b
,
2
+
u
s
-
in
ℳ
(
Ω
)
(see also (7.18)), where ub,i∈L1(Ω) are the functions in (7.5). Finally, for every f∈C(ℝ) satisfying (7.16), we have
(7.24)
f
(
u
n
j
)
⇀
*
f
*
+
L
+
u
s
+
-
L
-
u
s
-
in
ℳ
(
Ω
)
,
where f*∈L1(Ω) is the function in (7.17) and L± are the values in (7.16).
Proof of Proposition 7.6.
Let us prove the first claim in (7.18), the proof of the second one being analogous.
Fix any nonnegative ρ∈Cc1(Ω) and K∈ℕ. Multiplying the equation in (1) by ρHK+(un) and using Hypothesis 1.1(i), we have
(7.25)
∫
Ω
u
n
H
K
+
(
u
n
)
ρ
𝑑
x
≤
-
∫
Ω
{
A
(
x
,
u
n
)
∇
u
n
+
∇
u
n
n
}
⋅
∇
ρ
H
K
+
(
u
n
)
𝑑
x
+
∫
Ω
μ
n
H
K
+
(
u
n
)
ρ
𝑑
x
(here HK+ is the function in (2.2)). Let us consider separately the three terms in the right-hand side of the above inequality.
Since suppHK+⊆[K,+∞) and 0≤HK+(s)≤1 for all s∈ℝ, by Hypothesis 1.1 (ii), we have
|
∫
Ω
A
(
x
,
u
n
)
∇
u
n
⋅
∇
ρ
H
K
+
(
u
n
)
d
x
|
≤
β
∫
{
u
n
≥
K
}
|
∇
ϕ
(
u
n
)
|
|
∇
ρ
|
d
x
≤
β
∥
ρ
∥
C
1
(
Ω
¯
)
∥
ϕ
(
u
n
)
∥
W
0
1
,
p
(
Ω
)
|
{
u
n
≥
K
}
|
p
-
1
p
(7.26)
≤
C
1
K
p
-
1
p
(
K
>
0
)
,
with 1<p<NN-1 and C1 possibly depending on p and ρ (see (6.4) and (6.8)). Moreover, by (6.9), for every θ∈(0,1), we have
(7.27)
|
∫
Ω
∇
u
n
n
⋅
∇
ρ
H
K
+
(
u
n
)
𝑑
x
|
≤
∥
ρ
∥
C
1
(
Ω
¯
)
∫
Ω
|
∇
u
n
|
n
𝑑
x
≤
C
2
n
θ
-
1
4
for some constant C2>0 possibly depending on ρ and θ. Next, by (6.1)–(6.2), we get
(7.28)
∫
Ω
μ
n
H
K
+
(
u
n
)
ρ
𝑑
x
≤
∫
Ω
[
μ
+
]
n
H
K
+
(
u
n
)
ρ
𝑑
x
≤
∫
Ω
[
μ
r
+
]
n
H
K
+
(
u
n
)
ρ
𝑑
x
+
∫
Ω
[
μ
s
+
]
n
ρ
𝑑
x
,
where [μr+]n→μr in L1(Ω) and [μs+]n⇀*μs+ in ℳ(Ω). Let {unj} be any subsequence along which all the convergences in Propositions 7.1–7.4 and in Lemma 7.5 are satisfied. Since the sequence {HK+(unj)} is bounded in L∞(Ω), by Proposition A.4 (ii), we have
H
K
+
(
u
n
j
)
⇀
*
H
K
+
*
in
L
∞
(
Ω
)
as j→∞, where HK+*(x)=∫ℝHK+(ξ)𝑑τx(ξ) for a.e. x∈Ω. Concerning the left-hand side in (7.25), notice that applying Lemma 7.5, with fK(s)=sHK+(s), gives
(7.29)
lim
j
→
∞
∫
Ω
u
n
j
H
K
+
(
u
n
j
)
ρ
𝑑
x
=
∫
Ω
f
K
*
(
x
)
ρ
(
x
)
𝑑
x
+
〈
λ
1
,
ρ
〉
Ω
,
where fK*(x)=∫ℝξHK+(ξ)𝑑τx(ξ)∈L1(Ω). By (7.26)–(7.29), letting j→∞ in (7.25) gives
(7.30)
〈
λ
1
,
ρ
〉
Ω
≤
∫
Ω
f
K
*
(
x
)
ρ
(
x
)
𝑑
x
+
〈
λ
1
,
ρ
〉
Ω
≤
C
1
K
p
-
1
p
+
∫
Ω
H
K
+
*
(
x
)
μ
r
+
(
x
)
𝑑
x
+
〈
μ
s
+
,
ρ
〉
Ω
.
Since for a.e. x∈Ω, we have 0≤HK+*(x)≤∫ℝ𝑑τx(ξ)=1 and HK+*(x)≤∫{ξ≥K}𝑑τx(ξ)→0 as K→∞ (recall that τx is a probability measure), by the Dominated Convergence Theorem, we get μr+HK+*→0 in L1(Ω). Therefore, letting K→∞ in (7.30), we obtain
(7.31)
〈
λ
1
,
ρ
〉
Ω
≤
〈
μ
s
+
,
ρ
〉
Ω
for every nonnegative ρ∈Cc1(Ω) (hence, for every nonnegative ρ∈Cc(Ω), by standard approximation arguments). Since λ1 and μs+ are nonnegative Radon measures, from the previous inequality, we infer that λ1 is absolutely continuous with respect to μs+, hence singular with respect to the Lebesgue measure. Analogously, it can be checked that, for every ρ as above,
(7.32)
〈
λ
2
,
ρ
〉
Ω
≤
〈
μ
s
-
,
ρ
〉
Ω
.
Therefore, the measure λ=λ1-λ2 is singular with respect to the Lebesgue measure, (7.7) coincides with the Lebesgue decomposition of the measure u, and equality (7.19) follows at once. Moreover, since λ1 and λ2 are mutually singular (as they are, respectively, absolutely continuous with respect to μs+ and μs-), it follows that λ1=us+ and λ2=us-, whence (7.31)–(7.32) can be rephrased as 〈us±,ρ〉Ω≤〈μs±,ρ〉Ω. This concludes the proof. ∎
The following proposition will be used in the proof of Theorem 3.3.
Proposition 7.8.
Let {unj} be the sequence given in Proposition 7.2. Then
(7.33)
lim
j
→
∞
∫
{
x
∈
Ω
:
ϕ
′
(
u
r
(
x
)
)
=
0
}
|
∇
ϕ
n
j
(
u
n
j
)
|
d
x
=
0
,
where ϕnj(s):=ϕ(s)+snj (s∈R).
Proof.
We proceed in two steps. Step 1. Firstly, we prove that there exist two sequences {K1,m}m and {K2,m}m such that Ki,m>0, Ki,m→∞ as m→∞ (i=1,2), and
(7.34)
∫
{
x
∈
Ω
:
ϕ
′
(
u
r
(
x
)
)
=
0
}
(
∫
{
ξ
∈
ℝ
:
ϕ
(
ξ
)
=
ϕ
(
u
r
(
x
)
)
}
ϕ
′
(
g
m
(
ξ
)
)
d
τ
x
(
ξ
)
)
d
x
=
0
for all
m
∈
ℕ
,
where, for every ξ∈ℝ,
(7.35)
g
m
(
ξ
)
=
max
{
-
K
1
,
m
,
min
{
ξ
,
K
2
,
m
}
}
,
and the probability measure τx∈𝒫(ℝ) is the disintegration of the Young measure τ given in Proposition 7.2. To this end, for every K1,K2>0, let g(ξ)=max{-K1,min{ξ,K2}}. Then, for a.e. x∈Ω, we have
∫
{
ξ
∈
ℝ
:
ϕ
(
ξ
)
=
ϕ
(
u
r
(
x
)
)
}
ϕ
′
(
g
(
ξ
)
)
𝑑
τ
x
(
ξ
)
=
ϕ
′
(
-
K
1
)
∫
{
ξ
<
-
K
1
,
ϕ
(
ξ
)
=
ϕ
(
u
r
(
x
)
)
}
𝑑
τ
x
(
ξ
)
+
ϕ
′
(
K
2
)
∫
{
ξ
>
K
2
,
ϕ
(
ξ
)
=
ϕ
(
u
r
(
x
)
)
}
𝑑
τ
x
(
ξ
)
+
∫
{
ϕ
(
ξ
)
=
ϕ
(
u
r
(
x
)
)
,
-
K
1
≤
ξ
≤
K
2
}
ϕ
′
(
ξ
)
𝑑
τ
x
(
ξ
)
(some of the sets in the integrals in right-hand side of the above equality being possibly empty). In addition, if ϕ′(ur(x))=0, it follows that ϕ′(ξ)=0 for every ξ such that ϕ(ξ)=ϕ(ur(x)), whence
∫
{
-
K
1
≤
ξ
≤
K
2
,
ϕ
(
ξ
)
=
ϕ
(
u
r
(
x
)
)
}
ϕ
′
(
ξ
)
𝑑
τ
x
(
ξ
)
=
0
and
(7.36)
∫
{
ξ
∈
ℝ
:
ϕ
(
ξ
)
=
ϕ
(
u
r
(
x
)
)
}
ϕ
′
(
g
(
ξ
)
)
𝑑
τ
x
(
ξ
)
=
ϕ
′
(
-
K
1
)
∫
{
ξ
<
-
K
1
,
ϕ
(
ξ
)
=
ϕ
(
u
r
(
x
)
)
}
𝑑
τ
x
(
ξ
)
+
ϕ
′
(
K
2
)
∫
{
ξ
>
K
2
,
ϕ
(
ξ
)
=
ϕ
(
u
r
(
x
)
)
}
𝑑
τ
x
(
ξ
)
.
Setting Z0:={ξ∈ℝ:ϕ′(ξ)=0}, we distinguish the following cases.
(i) Let there exist M>0 such that Z0⊆[-M,M]. Then, for a.e. x such that ϕ′(ur(x))=0, we have {ξ∈ℝ:ϕ(ξ)=ϕ(ur(x))}⊆[-M,M] and, for every K1>M and K2>M, it follows that
{
ξ
<
-
K
1
,
ϕ
(
ξ
)
=
ϕ
(
u
r
(
x
)
)
}
=
∅
,
{
ξ
>
K
2
,
ϕ
(
ξ
)
=
ϕ
(
u
r
(
x
)
)
}
=
∅
.
Therefore, choosing in (7.36) sequences K1,m,K2,m→∞ with K1,m>M and K2,m>M, for every m, we have
(7.37)
∫
{
ξ
∈
ℝ
:
ϕ
(
ξ
)
=
ϕ
(
u
r
(
x
)
)
}
ϕ
′
(
g
m
(
ξ
)
)
𝑑
τ
x
(
ξ
)
=
0
for a.e.
x
such that
ϕ
′
(
u
r
(
x
)
)
=
0
(here gm is the function in (7.35)).
(ii) Let supZ0=+∞ and infZ0=-∞ (the cases supZ0<+∞ and infZ0=-∞ or supZ0=+∞ and infZ0>-∞ can be treated in a similar way). Then there exist two sequences {K1,m}m and {K2,m}m such that Ki,m→∞ (i=1,2) and ϕ′(-K1,m)=ϕ′(K2,m)=0 for all m∈ℕ. Thus, choosing in (7.36) K1=K1,m and K2=K2,m, equality (7.37) follows again for every m.
In all the above cases there exist {K1,m}m and {K2,m}m such that Ki,m→∞ as m→∞ (i=1,2) and (7.37) is satisfied for every m and a.e. x such that ϕ′(ur(x))=0. Hence, (7.34) follows at once.
Step 2. Letting {K1,m}m and {K2,m}m be the sequences given in the first part of the proof, and setting Km:=max{K1,m,K2,m}, from (6.7), for every θ>0, we have
∫
{
x
∈
Ω
:
ϕ
′
(
u
r
(
x
)
)
=
0
,
-
K
1
,
m
≤
u
n
j
(
x
)
≤
K
2
,
m
}
|
∇
ϕ
n
j
(
u
n
j
)
|
d
x
=
∫
{
x
∈
Ω
:
ϕ
′
(
u
r
(
x
)
)
=
0
,
-
K
1
,
m
≤
u
n
j
(
x
)
≤
K
2
,
m
}
ϕ
n
j
′
(
u
n
j
)
ϕ
n
j
′
(
u
n
j
)
|
∇
u
n
j
|
(
1
+
|
u
n
j
|
)
θ
+
1
2
(
1
+
|
u
n
j
|
)
θ
+
1
2
𝑑
x
≤
C
θ
(
1
+
K
m
)
θ
+
1
2
(
∫
{
x
∈
Ω
:
ϕ
′
(
u
r
(
x
)
)
=
0
,
-
K
1
,
m
≤
u
n
j
(
x
)
≤
K
2
,
m
}
[
ϕ
′
(
u
n
j
)
+
1
n
j
]
𝑑
x
)
1
2
≤
C
θ
,
K
m
{
(
∫
{
x
∈
Ω
:
ϕ
′
(
u
r
(
x
)
)
=
0
}
ϕ
′
(
g
m
(
u
n
j
)
)
𝑑
x
)
1
2
+
(
|
Ω
|
n
j
)
1
2
}
(here gm is the function in (7.35)). By (7.2), (7.20) and Proposition A.4 (ii), letting j→∞ in the previous inequality, we get
lim sup
j
→
∞
∫
{
x
∈
Ω
:
ϕ
′
(
u
r
(
x
)
)
=
0
,
-
K
1
,
m
≤
u
n
j
(
x
)
≤
K
2
,
m
}
|
∇
ϕ
n
j
(
u
n
j
)
|
d
x
(7.38)
≤
C
θ
,
K
m
(
∫
{
x
∈
Ω
:
ϕ
′
(
u
r
(
x
)
)
=
0
}
(
∫
{
ξ
∈
ℝ
:
ϕ
(
ξ
)
=
ϕ
(
u
r
(
x
)
)
}
ϕ
′
(
g
m
(
ξ
)
)
d
τ
x
(
ξ
)
)
d
x
)
1
2
=
0
,
the latter equality following by (7.34). Moreover, for every m, we have
∫
{
ϕ
′
(
u
r
)
=
0
}
|
∇
ϕ
n
j
(
u
n
j
)
|
d
x
=
∫
{
ϕ
′
(
u
r
)
=
0
,
K
1
,
m
≤
u
n
j
≤
K
2
,
m
}
|
∇
ϕ
n
j
(
u
n
j
)
|
d
x
(7.39)
+
∫
{
ϕ
′
(
u
r
)
=
0
,
u
n
j
>
K
2
,
m
}
|
∇
ϕ
n
j
(
u
n
j
)
|
d
x
+
∫
{
ϕ
′
(
u
r
)
=
0
,
u
n
j
<
-
K
1
,
m
}
|
∇
ϕ
n
j
(
u
n
j
)
|
d
x
,
and, for every 1<p<NN-1,
(7.40)
∫
{
ϕ
′
(
u
r
)
=
0
,
|
u
n
j
|
>
K
i
,
m
}
|
∇
ϕ
n
j
(
u
n
j
)
|
d
x
≤
∥
ϕ
n
j
(
u
n
j
)
∥
W
0
1
,
p
(
Ω
)
|
{
|
u
n
j
|
>
K
i
,
m
}
|
1
-
1
p
≤
C
¯
p
K
i
,
m
1
-
1
p
(i=1,2; see (6.4), (6.8) and (6.9)). Letting j→∞ in (7.39), by (7.38) and (7.40), we obtain
lim sup
j
→
∞
∫
{
x
∈
Ω
:
ϕ
′
(
u
r
(
x
)
)
=
0
}
|
∇
ϕ
n
j
(
u
n
j
)
|
d
x
≤
C
¯
p
(
1
K
1
,
m
1
-
1
p
+
1
K
2
,
m
1
-
1
p
)
,
and the claim follows by taking the limit as m→∞ in the above inequality. ∎
Proof of Lemma 7.5.
For every f∈C(ℝ) satisfying (7.16), there exist C0>0 such that
(7.41)
|
f
(
s
)
|
≤
C
0
(
1
+
|
s
|
)
for all
s
∈
ℝ
.
Moreover, by (7.16), for every ε>0, there exists Mε>0 such that
(7.42)
(
L
+
-
ε
)
s
≤
f
(
s
)
≤
(
L
+
+
ε
)
s
for
s
≥
M
ε
,
(7.43)
(
L
-
+
ε
)
s
≤
f
(
s
)
≤
(
L
-
-
ε
)
s
for
s
≤
-
M
ε
.
Fix any M>Mε and gi,M∈C(ℝ) (i=1,2,3), with 0≤gi,M≤1 and ∑i=13gi,M(s)=1 for all s∈ℝ, such that suppg1,M⊆[M,∞), suppg2,M⊆[-M-1,M+1] and suppg3,M⊆(-∞,-M]. Observe that the above conditions ensure that
(7.44)
g
1
,
M
(
s
)
=
1
for every
s
≥
M
+
1
and
g
3
,
M
(
s
)
=
1
for every
s
≤
-
M
-
1
.
Since
f
(
u
n
j
)
=
g
1
,
M
(
u
n
j
)
f
(
u
n
j
)
+
g
2
,
M
(
u
n
j
)
f
(
u
n
j
)
+
g
3
,
M
(
u
n
j
)
f
(
u
n
j
)
,
by (7.42)–(7.43), we have
(
L
+
-
ε
)
u
n
j
g
1
,
M
(
u
n
j
)
+
g
2
,
M
(
u
n
j
)
f
(
u
n
j
)
+
(
L
-
+
ε
)
u
n
j
g
3
,
M
(
u
n
j
)
≤
f
(
u
n
j
)
(7.45)
≤
(
L
+
+
ε
)
u
n
j
g
1
,
M
(
u
n
j
)
+
g
2
,
M
(
u
n
j
)
f
(
u
n
j
)
+
(
L
-
-
ε
)
u
n
j
g
3
,
M
(
u
n
j
)
.
Let us prove that, in the limit as j→∞,
(7.46)
u
n
j
g
1
,
M
(
u
n
j
)
⇀
*
G
1
,
M
*
+
λ
1
in
ℳ
(
Ω
)
,
(7.47)
u
n
j
g
3
,
M
(
u
n
j
)
⇀
*
G
3
,
M
*
-
λ
2
in
ℳ
(
Ω
)
,
(7.48)
f
(
u
n
j
)
g
2
,
M
(
u
n
j
)
⇀
*
G
2
,
M
*
in
L
∞
(
Ω
)
,
where G1,M*,G3,M*(x)∈L1(Ω) and G2,M*∈L∞(Ω) are defined by
G
1
,
M
*
(
x
)
=
∫
ℝ
ξ
g
1
,
M
(
ξ
)
𝑑
τ
x
(
ξ
)
,
G
3
,
M
*
(
x
)
=
∫
ℝ
ξ
g
3
,
M
(
ξ
)
𝑑
τ
x
(
ξ
)
and
G
2
,
M
*
(
x
)
=
∫
ℝ
f
(
ξ
)
g
2
,
M
(
ξ
)
𝑑
τ
x
(
ξ
)
for a.e. x∈Ω, respectively. To prove (7.46) (the proof of (7.47) being similar), observe that
(7.49)
u
n
j
g
1
,
M
(
u
n
j
)
=
[
u
n
j
g
1
,
M
(
u
n
j
)
-
u
n
j
+
]
+
u
n
j
+
,
and, by (7.44),
|
u
n
j
g
1
,
M
(
u
n
j
)
-
u
n
j
+
|
=
u
n
j
+
-
u
n
j
g
1
,
M
(
u
n
j
)
≤
M
+
1
a.e. in
Ω
.
Thus, from Proposition A.4 (ii) and the convergences in (7.2) and (7.4), letting j→∞ in (7.49) gives
u
n
j
g
1
,
M
(
u
n
j
)
=
[
u
n
j
g
1
,
M
(
u
n
j
)
-
u
n
j
+
]
+
u
n
j
+
⇀
*
h
1
,
M
*
+
u
b
,
1
+
λ
1
in
ℳ
(
Ω
)
,
where ub,1∈L1(Ω) is the function in (7.5) and h1,M*(x)=∫ℝ[ξg1,M(ξ)-ξ+]𝑑τx(ξ)∈L∞(Ω). Since for a.e. x∈Ω we have
h
1
,
M
*
(
x
)
+
u
b
,
1
(
x
)
=
∫
ℝ
[
(
ξ
g
1
,
M
(
ξ
)
-
ξ
+
)
+
ξ
+
]
𝑑
τ
x
(
ξ
)
=
G
1
,
M
*
(
x
)
,
then the convergence in (7.46) follows.
The sequence {f(unj)g2,M(unj)} is bounded in L∞(Ω) (recall that suppg2,M⊆[-M-1,M+1]), hence, by (7.2) and Proposition A.4 (ii), we get (7.48).
By (7.46)–(7.48), letting j→∞ in (7.45) gives
(
L
+
-
ε
)
∫
Ω
G
1
,
M
*
ρ
𝑑
x
+
(
L
+
-
ε
)
〈
λ
1
,
ρ
〉
Ω
+
(
L
-
+
ε
)
∫
Ω
G
3
,
M
*
ρ
𝑑
x
-
(
L
-
+
ε
)
〈
λ
2
,
ρ
〉
Ω
+
∫
Ω
G
2
,
M
*
ρ
𝑑
x
≤
lim inf
j
→
∞
∫
Ω
f
(
u
n
j
)
ρ
𝑑
x
≤
lim sup
j
→
∞
∫
Ω
f
(
u
n
j
)
ρ
𝑑
x
≤
∫
Ω
G
2
,
M
*
ρ
𝑑
x
+
(
L
+
+
ε
)
∫
Ω
G
1
,
M
*
ρ
𝑑
x
+
(
L
+
+
ε
)
〈
λ
1
,
ρ
〉
Ω
(7.50)
+
(
L
-
-
ε
)
∫
Ω
G
3
,
M
*
ρ
𝑑
x
-
(
L
-
-
ε
)
〈
λ
2
,
ρ
〉
Ω
for every nonnegative ρ∈Cc(Ω). Let us take the limit as M→∞ in the above inequalities. To this end, let us observe that for a.e. x∈Ω, we have
0
≤
G
1
,
M
*
(
x
)
≤
∫
ℝ
ξ
+
𝑑
τ
x
(
ξ
)
=
u
b
,
1
(
x
)
∈
L
1
(
Ω
)
,
G
1
,
M
*
(
x
)
=
∫
[
M
,
∞
)
ξ
g
1
,
M
(
ξ
)
𝑑
τ
x
(
ξ
)
→
0
as
M
→
∞
.
(Indeed, 0≤g1,M(ξ)≤1, g1,M(ξ)→0 for every ξ∈ℝ, and |ξg1,M(ξ)|≤ξ+∈L1(ℝ,τx).) Hence, by the Dominated Convergence Theorem, it follows that
(7.51)
G
1
,
M
*
→
0
in
L
1
(
Ω
)
as
M
→
∞
.
Similarly, it can be checked that
(7.52)
G
3
,
M
*
→
0
in
L
1
(
Ω
)
as
M
→
∞
.
Finally, for a.e. x∈Ω, we have
|
G
2
,
M
*
(
x
)
|
≤
C
0
∫
ℝ
(
1
+
|
ξ
|
)
𝑑
τ
x
(
ξ
)
=
C
0
(
1
+
u
b
,
1
(
x
)
+
u
b
,
2
(
x
)
)
∈
L
1
(
Ω
)
(see (7.41) and recall that τx is a probability measure) and G2,M*(x)→f*(x) as M→∞, where f*∈L1(Ω) is the function in (7.17). Hence, by the Dominated Convergence Theorem, it follows that
(7.53)
G
2
,
M
*
→
f
*
in
L
1
(
Ω
)
as
M
→
∞
.
In view of (7.51)–(7.53), letting M→∞ in (7.50), we obtain
(
L
+
-
ε
)
〈
λ
1
,
ρ
〉
Ω
-
(
L
-
+
ε
)
〈
λ
2
,
ρ
〉
Ω
+
∫
Ω
f
*
ρ
𝑑
x
≤
lim inf
j
→
∞
∫
Ω
f
(
u
n
j
)
ρ
𝑑
x
≤
lim sup
j
→
∞
∫
Ω
f
(
u
n
j
)
ρ
𝑑
x
≤
∫
Ω
f
*
ρ
𝑑
x
+
(
L
+
+
ε
)
〈
λ
1
,
ρ
〉
Ω
-
(
L
-
-
ε
)
〈
λ
2
,
ρ
〉
Ω
for all ρ∈Cc(Ω), ρ≥0, and the claim follows by the arbitrariness of ε and ρ. ∎
8 Proofs of Results in Section 3
Proof of Theorem 3.3.
Let u∈ℳ(Ω) be the limiting measure in (7.8). Then ϕ(ur)∈W01,p(Ω) for every 1≤p<NN-1, and for all K>0, we have TK(ϕ(ur))∈W01,2(Ω) (see Remark 7.7). Moreover, by (7.22) and (6.9), we get
(8.1)
ϕ
n
j
(
u
n
j
)
⇀
ϕ
(
u
r
)
in
W
0
1
,
p
(
Ω
)
for every
1
≤
p
<
N
N
-
1
,
where ϕnj is the function in (7.33). By the above remarks, in order to check that u satisfies the requirements in Definition 3.1, it only remains to prove the weak formulation (3.1). To this end, fix any ρ∈Cc1(Ω) and let {unj} be any sequence along which all the convergences in Section 7 are satisfied. Let us take the limit as j→∞ in the weak formulations of problems (Pnj), namely,
(8.2)
∫
Ω
A
(
x
,
u
n
j
)
∇
u
n
j
⋅
∇
ρ
d
x
+
1
n
j
∫
Ω
∇
u
n
j
⋅
∇
ρ
d
x
+
∫
Ω
u
n
j
ρ
𝑑
x
=
∫
Ω
μ
n
j
ρ
𝑑
x
.
By (7.8) and since μnj⇀*μ in ℳ(Ω), we have
lim
j
→
∞
∫
Ω
u
n
j
ρ
𝑑
x
=
〈
u
,
ρ
〉
Ω
,
lim
j
→
∞
∫
Ω
μ
n
j
ρ
𝑑
x
=
〈
μ
,
ρ
〉
Ω
.
Thus, (3.1) will follow by letting j→∞ in (8.2) if we prove that
(8.3)
lim
j
→
∞
∫
Ω
A
n
j
(
x
,
u
n
j
)
∇
u
n
j
⋅
∇
ρ
d
x
=
∫
{
x
∈
Ω
:
ϕ
′
(
u
r
(
x
)
)
>
0
}
A
(
x
,
u
r
)
ϕ
′
(
u
r
)
∇
ϕ
(
u
r
)
⋅
∇
ρ
d
x
,
where we have set
(8.4)
A
n
j
(
x
,
u
n
j
)
:=
A
(
x
,
u
n
j
)
+
1
n
j
I
N
(here IN denotes the unit matrix of size N). We have
∫
Ω
A
n
j
(
x
,
u
n
j
)
∇
u
n
j
⋅
∇
ρ
d
x
=
∫
Ω
A
n
j
(
x
,
u
n
j
)
ϕ
′
(
u
n
j
)
+
1
n
j
∇
ϕ
n
j
(
u
n
j
)
⋅
∇
ρ
d
x
=
∫
{
ϕ
′
(
u
r
)
>
0
}
A
n
j
(
x
,
u
n
j
)
ϕ
′
(
u
n
j
)
+
1
n
j
∇
ϕ
n
j
(
u
n
j
)
⋅
∇
ρ
d
x
(8.5)
+
∫
{
ϕ
′
(
u
r
)
=
0
}
A
n
j
(
x
,
u
n
j
)
ϕ
′
(
u
n
j
)
+
1
n
j
∇
ϕ
n
j
(
u
n
j
)
⋅
∇
ρ
d
x
.
Moreover, by Hypothesis 1.1 (ii), (7.23) and the Dominated Convergence Theorem, for all entries (ahk)nj of the matrix Anj(x,unj), we get
(8.6)
χ
{
x
∈
Ω
:
ϕ
′
(
u
r
(
x
)
)
>
0
}
(
a
h
k
)
n
j
(
x
,
u
n
j
(
x
)
)
ϕ
′
(
u
n
j
(
x
)
)
+
1
n
j
→
χ
{
x
∈
Ω
:
ϕ
′
(
u
r
(
x
)
)
>
0
}
a
h
k
(
x
,
u
r
(
x
)
)
ϕ
′
(
u
r
(
x
)
)
in Lp(Q) (p∈[1,∞)) and weakly* in L∞(Q) (h,k=1,…,N). Thus, (8.1) and (8.6) give
(8.7)
lim
j
→
∞
∫
{
ϕ
′
(
u
r
)
>
0
}
A
n
j
(
x
,
u
n
j
)
ϕ
′
(
u
n
j
)
+
1
n
j
∇
ϕ
n
j
(
u
n
j
)
⋅
∇
ρ
d
x
=
∫
{
ϕ
′
(
u
r
)
>
0
}
A
(
x
,
u
r
)
ϕ
′
(
u
r
)
∇
ϕ
(
u
r
)
⋅
∇
ρ
d
x
.
Moreover, by Hypothesis 1.1 (ii) and (7.33), we have
(8.8)
|
∫
{
ϕ
′
(
u
r
)
=
0
}
A
n
(
x
,
u
n
j
)
ϕ
′
(
u
n
j
)
+
1
n
j
∇
ϕ
n
j
(
u
n
j
)
⋅
∇
ρ
d
x
|
≤
C
~
∥
ρ
∥
C
1
(
Ω
¯
)
∫
{
ϕ
′
(
u
r
)
=
0
}
|
∇
ϕ
n
j
(
u
n
j
)
|
d
x
→
0
as j→∞. Combining (8.5) with (8.7) and (8.8), the convergence in (8.3) follows, and the proof of Theorem 3.3 is completed. ∎
Proof of Theorem 3.4.
The proof of (3.4) has been given in Proposition 7.6 (see (7.18)).
Let us address (3.5) and (3.7), the proof of (3.6) and (3.8) following in a similar way. By (3.3), we have [μ±]n=[μd,2±]n+[μc,2±]n, where μd,2± and μc,2± are, respectively, the diffuse part and the concentrated part of μ± with respect to the 2-capacity, and
(8.9)
[
μ
c
,
2
±
]
n
⇀
*
μ
c
,
2
±
in
ℳ
(
Ω
)
,
lim
n
→
∞
∫
Ω
[
μ
d
,
2
±
]
n
f
n
ρ
𝑑
x
=
〈
μ
d
,
2
±
,
f
ρ
〉
Ω
for every ρ∈Cc1(Ω) and fn,f∈W01,2(Ω)∩L∞(Ω) such that fn⇀f in W01,2(Ω), fn⇀*f in L∞(Ω) (the proof follows by standard properties of convolution).
Without loss of generality, we assume that ϕ+∞>0 (otherwise, ϕ+(s)=0 for all s∈ℝ, and by (7.18) the claim in (3.7) would be obviously satisfied). Let g∈Lip(ℝ) be any piecewise C1-function such that for some s0∈(0,ϕ+∞), g(s)=0 if s≤0, g(s)=1 if s≥s0, and g′≥0 a.e. in (0,s0). Notice that 0≤g(ϕ(ξ))≤1, g∘ϕ is nondecreasing, g(ϕ(ξ))=0 for all ξ≤0, and limξ→+∞g(ϕ(ξ))=1. For every nonnegative ρ∈Cc1(Ω), multiplying the equation in (1) by the test function ζ=ρg(ϕ(un))HK+(un) and using Hypothesis 1.1 (i), we obtain
∫
ℝ
{
α
[
(
g
∘
ϕ
)
H
K
+
]
′
(
u
n
)
ϕ
′
(
u
n
)
|
∇
u
n
|
2
+
1
n
[
(
g
∘
ϕ
)
H
K
+
]
′
(
u
n
)
|
∇
u
n
|
2
}
ρ
𝑑
x
+
∫
Ω
{
A
(
x
,
u
n
)
∇
u
n
⋅
∇
ρ
+
1
n
∇
u
n
⋅
∇
ρ
}
g
(
ϕ
(
u
n
)
)
H
K
+
(
u
n
)
𝑑
x
+
∫
Ω
g
(
ϕ
(
u
n
)
)
H
K
+
(
u
n
)
u
n
ρ
𝑑
x
≤
∫
Ω
μ
n
H
K
+
(
u
n
)
g
(
ϕ
(
u
n
)
)
ρ
𝑑
x
(here HK+ is the function in (2.2)). Since [(g∘ϕ)HK+]′≥0, 0≤HK,+(un)≤1 and g(ϕ(un))≥0 a.e. in Ω, by (6.1), we have
∫
Ω
A
n
(
x
,
u
n
)
∇
u
n
⋅
∇
ρ
g
(
ϕ
(
u
n
)
)
H
K
+
(
u
n
)
𝑑
x
+
∫
Ω
g
(
ϕ
(
u
n
)
)
H
K
+
(
u
n
)
u
n
ρ
𝑑
x
≤
∫
Ω
[
μ
+
]
n
g
(
ϕ
(
u
n
)
)
H
K
+
(
u
n
)
ρ
𝑑
x
(8.10)
≤
∫
Ω
[
μ
+
]
n
g
(
ϕ
(
u
n
)
)
ρ
𝑑
x
(here An is the matrix in (8.4)). Let {unj} be any subsequence along which all the convergences in Section 7 are satisfied. Since g′(s)=0 if s≤0 or s≥s0∈(0,ϕ+∞), the estimate in (6.6) ensures that the sequence {g(ϕ(unj))} is bounded in W01,2(Ω)∩L∞(Ω); moreover, 0≤g(ϕ(unj))≤1 a.e. in Ω for every j. Therefore, by (7.21), it follows that
(8.11)
g
(
ϕ
(
u
n
j
)
)
⇀
g
(
ϕ
(
u
r
)
)
in
W
0
1
,
2
(
Ω
)
,
g
(
ϕ
(
u
n
j
)
)
⇀
*
g
(
ϕ
(
u
r
)
)
in
L
∞
(
Ω
)
.
By (8.9) and (8.11), we have
lim sup
j
→
∞
∫
Ω
[
μ
+
]
n
j
g
(
ϕ
(
u
n
j
)
)
ρ
𝑑
x
≤
lim
j
→
∞
∫
Ω
[
μ
d
,
2
+
]
n
j
g
(
ϕ
(
u
n
j
)
)
ρ
𝑑
x
+
lim
j
→
∞
∫
Ω
[
μ
c
,
2
+
]
n
j
ρ
𝑑
x
(8.12)
=
〈
μ
d
,
2
+
,
g
(
ϕ
(
u
r
)
)
ρ
〉
Ω
+
〈
μ
c
,
2
+
,
ρ
〉
Ω
.
Let us address the remaining terms in the left-hand side of (8.10). Since the function ηK(s)=sg(ϕ(s))HK+(s) is continuous and satisfies (7.16) with L+=1 and L-=0, by (7.24), it follows that
u
n
j
g
(
ϕ
(
u
n
j
)
)
H
K
+
(
u
n
j
)
⇀
*
η
K
*
+
u
s
+
in
ℳ
(
Ω
)
,
where ηK*∈L1(Ω), ηK*≥0, is defined by ηK*(x)=∫ℝξg(ϕ(ξ))HK+(ξ)𝑑τx(ξ) for a.e. x∈Ω. Moreover, since ∥g(ϕ(unj))∥L∞(Ω)≤1, by arguing as in (7.26) and (7.27), we have
(8.13)
|
∫
Ω
A
n
j
(
x
,
u
n
j
)
∇
u
n
j
⋅
∇
ρ
g
(
ϕ
(
u
n
j
)
)
H
K
+
(
u
n
j
)
𝑑
x
|
≤
C
(
1
K
p
-
1
p
+
n
θ
-
1
4
)
,
where 1<p<NN-1, θ∈(0,1) and C possibly depends on ρ, p and θ. Letting j→∞ in (8.10), by (8.12)–(8.13), for every nonnegative ρ∈Cc1(Ω), we obtain
〈
u
s
+
,
ρ
〉
Ω
≤
〈
u
s
+
,
ρ
〉
Ω
+
∫
Ω
η
K
*
(
x
)
ρ
(
x
)
𝑑
x
≤
〈
μ
d
,
2
+
,
g
(
ϕ
(
u
r
)
)
ρ
〉
Ω
+
〈
μ
c
,
2
+
,
ρ
〉
Ω
+
C
K
p
-
1
p
.
Therefore, in the limit, as K→∞, we get
(8.14)
〈
u
s
+
,
ρ
〉
Ω
≤
〈
μ
d
,
2
+
,
g
(
ϕ
(
u
r
)
)
ρ
〉
Ω
+
〈
μ
c
,
2
+
,
ρ
〉
Ω
.
Let us distinguish two cases.
(i) If ϕ+∞=+∞, then, for every k∈ℕ, we choose in (8.14) g(s)=Hk+(s), where Hk+ is the function in (2.2) (with K=k). Since, by (6.5), for every k we have ∫Ω|∇Hk+(ϕ(unj))|2dx≤∥μ∥ℳ(Ω)α, in the limit as j→∞ (see (8.11)), we get
(8.15)
∫
Ω
|
∇
H
k
+
(
ϕ
(
u
r
)
)
|
2
d
x
≤
∥
μ
∥
ℳ
(
Ω
)
α
(
for all
k
)
and (8.14) reads
(8.16)
〈
u
s
+
,
ρ
〉
Ω
≤
〈
μ
d
,
2
+
,
H
k
+
(
ϕ
(
u
r
)
)
ρ
〉
Ω
+
〈
μ
c
,
2
+
,
ρ
〉
Ω
for every nonnegative ρ∈Cc1(Ω). Moreover, since Hk+(s)→0 as k→∞ for every s∈ℝ, by (8.15), we get
H
k
+
(
ϕ
(
u
r
)
)
⇀
0
in
W
0
1
,
2
(
Ω
)
,
H
k
+
(
ϕ
(
u
r
)
)
⇀
*
0
in
L
∞
(
Ω
)
,
whence limk→∞〈μd,2+,Hk+(ϕ(ur))ρ〉Ω=0 (see Section 2.2). As a consequence, letting k→∞ in (8.16), for every nonnegative ρ∈Cc1(Ω), we have 〈us+,ρ〉Ω≤〈μc,2+,ρ〉Ω. Plainly, this implies that us+=uc,2+, namely, ud,2+∈ℳac(Ω) and the proof of (3.5) is completed.
(ii) If ϕ+∞<+∞, we choose in (8.14) g(s)=gk(s), where for every k∈ℕ (so that ϕ+∞-k-1>0),
g
k
(
s
)
=
{
0
if
s
<
ϕ
+
∞
-
1
k
,
2
k
(
s
-
ϕ
+
∞
+
1
k
)
if
ϕ
+
∞
-
1
k
≤
s
≤
ϕ
+
∞
-
1
2
k
,
1
if
s
>
ϕ
+
∞
-
1
2
k
.
Hence, (8.14) gives
〈
u
s
+
,
ρ
〉
Ω
≤
〈
μ
d
,
2
+
,
g
k
(
ϕ
(
u
r
)
)
ρ
〉
Ω
+
〈
μ
c
,
2
+
,
ρ
〉
Ω
.
Taking into account the diffuse part with respect to the 2-capacity of the measures in the above inequality, we obtain
〈
[
u
s
+
]
d
,
2
,
ρ
〉
Ω
≤
〈
μ
d
,
2
+
,
g
k
(
ϕ
(
u
r
)
)
ρ
〉
Ω
≤
〈
μ
d
,
2
+
⌞
{
ϕ
+
(
u
r
)
≥
ϕ
+
∞
-
1
k
}
,
ρ
〉
Ω
,
where we identify the function ϕ+(ur)∈W01,2(Ω)∩L∞(Ω) (see Remark 3.2 (ii)) with its c1,2-quasi continuous representative v+, with 0≤v+≤ϕ+∞c1,2-almost everywhere in Ω. Therefore, (3.7) follows from the above inequality, by the arbitrarity of k. ∎
Proof of Corollary 3.6.
(i) If μs=0, inequalities (3.4) give us=0.
(ii) If ϕ+∞=+∞, by (3.5), we have ud,2+∈ℳac(Ω), whence us+=uc,2+. On the other hand, (3.4) gives uc,2+=us+≤μs+, whence uc,2+∈ℳd,2(Ω), since by assumption μ+∈ℳd,2(Ω). This implies that us+=uc,2+=0.
The proof of (iii) follows by arguing in a similar way. ∎
9 Proofs of Results in Section 4
Proof of Proposition 4.1.
Since, by assumption ϕ(ur)∈W01,p(Ω) for some 1<p<∞, from Hypothesis 1.1 (ii) we obtain χ{ϕ′(ur))>0}[ϕ′(ur)]-1A(x,ur)∇ϕ(ur)∈[Lp(Ω)]N. Therefore, (3.1) gives [u-μ]∈W-1,p(Ω), whence [u]c,p′=[μ]c,p′ (p′=pp-1) and the conclusion follows. ∎
Proof of Proposition 4.3.
The claim immediately follows by Proposition 6.4, since ϕ(unj)→ϕ(ur) a.e. in Ω (see (7.21)). ∎
Proof of Proposition 4.5.
Let u be any solution of problem (P) given by Theorem 3.3, and let {unj} be any sequence along which all the convergences in Section 7 are satisfied. In particular,
(9.1)
u
n
j
+
⇀
*
u
b
,
1
+
u
s
+
,
u
n
j
-
⇀
*
u
b
,
2
+
u
s
-
in
ℳ
(
Ω
)
,
where ub,i∈L1(Ω) are the functions in (7.5) and u=ub,1-ub,2+us∈ℳ(Ω).
Let us only prove claim (i), the proof of (ii) being similar. Let (4.4) hold with m>N-2N and fix p such that max{1,NmN+1}<p<NN-1 (observe that m>N-2N⇒NmN+1<NN-1). By (4.4), there exist c¯1,c¯2>0 such that
s
m
N
p
N
-
p
≤
c
¯
1
[
ϕ
(
s
)
]
N
p
N
-
p
+
c
¯
2
for all
s
≥
0
.
By the above estimate, the Sobolev inequality and (6.8), we have
∫
Ω
[
u
n
j
+
]
m
N
p
N
-
p
d
x
≤
c
¯
1
∫
Ω
|
ϕ
(
u
n
j
)
|
N
p
N
-
p
d
x
+
c
¯
2
|
Ω
|
≤
C
(
N
,
p
)
(
∫
Ω
|
∇
ϕ
(
u
n
j
)
|
p
d
x
)
N
N
-
p
+
c
¯
2
|
Ω
|
≤
C
¯
(
N
,
p
)
for C¯(N,p) independent of j, and the range conditions on p ensures that q:=mNpN-p>1. This implies that the sequence {unj+} is bounded in Lq(Ω) with q>1, hence it is weakly relatively compact in this space. As a consequence (see (9.1)), we get us+=0, whence u+∈ℳac(Ω). ∎
Proof of Proposition 4.6.
Let us only prove claim (i), the proof of (ii) being similar. Let {unj} be any sequence along which all the convergences in Section 7 are satisfied. By assumption (4.5) and since ϕ(s)≥0 for s≥0, we have 0≤ϕ(s)≤ϕ+∞-C(s+1)σ for all s≥0, where σ∈(0,1) and 0<C≤ϕ+∞. In particular, the previous inequality also gives 0≤ϕ(s)<ϕ+∞ for all s≥0. Let gδ∈Lip(ℝ) be defined as
(9.2)
g
δ
(
s
)
=
{
0
if
s
≤
δ
,
1
ϕ
+
∞
-
δ
(
s
-
δ
)
if
δ
<
s
≤
ϕ
+
∞
,
1
if
s
>
ϕ
+
∞
,
for
δ
∈
(
ϕ
+
∞
-
C
,
ϕ
+
∞
)
.
Notice that δ>ϕ+∞-C≥0, gδ(ϕ+∞)=1 for every δ, and for every p∈(1,2), we have
(9.3)
∫
Ω
|
∇
g
δ
(
ϕ
(
u
n
j
)
)
|
p
d
x
≤
1
[
ϕ
+
∞
-
δ
]
p
(
∫
{
ϕ
(
u
n
j
)
>
δ
}
|
∇
ϕ
(
u
n
j
)
|
2
)
p
2
|
{
ϕ
(
u
n
j
)
>
δ
}
|
1
-
p
2
.
In order to estimate the right-hand side of the above inequality, let kδ=min{ξ∈ℝ:ϕ(ξ)=δ}. Hence, ϕ(kδ)=δ, kδ>0, and by the nondecreasing character of ϕ, it follows that
(9.4)
{
x
∈
Ω
:
ϕ
(
u
n
j
(
x
)
)
>
δ
}
⊆
{
x
∈
Ω
:
u
n
j
(
x
)
>
k
δ
}
.
Moreover, by (4.5), we get (s+1)σ≥C[ϕ+∞-ϕ(s)]-1 if s≥0, whence, for every δ as in (9.2),
(9.5)
{
x
∈
Ω
:
ϕ
(
u
n
j
(
x
)
)
>
δ
}
⊆
{
x
∈
Ω
:
u
n
j
(
x
)
>
C
1
σ
[
ϕ
+
∞
-
δ
]
1
σ
-
1
>
0
}
(notice that C1σ[ϕ+∞-δ]-1σ-1>0, by the assumption ϕ+∞-C<δ<ϕ+∞). In view of (9.4), (9.5) and (6.11) (the latter with K=kδ), inequality (9.3) gives
∫
Ω
|
∇
g
δ
(
ϕ
(
u
n
j
)
)
|
p
d
x
≤
(
∫
{
u
n
j
>
k
δ
}
|
∇
ϕ
(
u
n
j
)
|
2
)
p
2
|
{
ϕ
(
u
n
j
)
>
δ
}
|
1
-
p
2
[
ϕ
+
∞
-
δ
]
p
≤
C
1
[
ϕ
+
∞
-
δ
]
p
[
ϕ
+
∞
-
ϕ
(
k
δ
)
]
p
2
|
{
ϕ
(
u
n
j
)
>
δ
}
|
1
-
p
2
≤
C
1
[
ϕ
+
∞
-
δ
]
p
2
[
ϕ
+
∞
-
δ
]
p
|
{
u
n
j
>
C
1
σ
[
ϕ
+
∞
-
δ
]
1
σ
-
1
}
|
1
-
p
2
(9.6)
≤
C
2
[
ϕ
+
∞
-
δ
]
1
σ
-
p
2
σ
-
p
2
{
C
1
σ
-
[
ϕ
+
∞
-
δ
]
1
σ
}
1
-
p
2
for some constant C2 independent of both j and δ (here we have also used (6.4)).
Since gδ(ϕ(unj))=gδ(ϕ+(unj)) and ϕ+(unj)⇀ϕ+(ur) in W01,2(Ω) (see (6.10) and (7.21)), letting j→∞ in (9.6), by the lower semicontinuity of the norm, we obtain
(9.7)
∫
Ω
|
∇
g
δ
(
ϕ
+
(
u
r
)
)
|
p
d
x
≤
C
2
[
ϕ
+
∞
-
δ
]
1
σ
-
p
2
σ
-
p
2
{
C
1
σ
-
[
ϕ
+
∞
-
δ
]
1
σ
}
1
-
p
2
.
Since limδ→ϕ+∞-{C1σ-[ϕ+∞-δ]1σ}=C1σ>0, taking the limit as δ→ϕ+∞- in the right-hand side of (9.7) gives, for all 1<p<2σ+1,
lim
δ
→
ϕ
+
∞
-
∫
Ω
|
∇
g
δ
(
ϕ
+
(
u
r
)
)
|
p
d
x
=
0
,
whence the set {x∈Ω:ϕ+(ur(x))=ϕ+∞} has zero p-capacity (see (2.1) and (9.2)).
Let μ∈ℳ(Ω) be any Radon measure such that μ+∈ℳd,p(Ω), with 1<p<2σ+1. Then μ+∈ℳd,2(Ω) (since p<2 and Ω is bounded) and, by (3.7),
[
u
s
+
]
d
,
2
≤
[
μ
s
+
]
d
,
2
⌞
{
x
∈
Ω
:
ϕ
+
(
u
r
(
x
)
)
=
ϕ
+
∞
}
≤
μ
+
⌞
{
x
∈
Ω
:
ϕ
+
(
u
r
(
x
)
)
=
ϕ
+
∞
}
=
0
,
since the set {x∈Ω:ϕ+(ur(x))=ϕ+∞} has zero p-capacity. This proves that [us+]d,2=0. Finally, since us+≤μs+ (see (3.4)), we also have uc,2+≤μc,2+=0, and the conclusion follows. ∎
10 Proof of Results in Section 5
Proof of Theorem 5.2.
The existence part follows from Theorems 3.3 and 3.4, hence it only remains to prove uniqueness.
Let u and w be two solutions of problem (P) satisfying (3.7)–(3.8). Then ϕ(ur) and ϕ(wr) belong to W01,2(Ω)∩L∞(Ω) (see (3.2)), and inequalities (3.7)–(3.8) give
(10.1)
[
u
s
±
]
d
,
2
=
[
u
s
±
]
d
,
2
⌞
{
ϕ
(
u
r
)
=
ϕ
±
∞
}
,
[
w
s
±
]
d
,
2
=
[
w
s
±
]
d
,
2
⌞
{
ϕ
(
w
r
)
=
ϕ
±
∞
}
;
here we identify ϕ(ur) and ϕ(wr) with their c1,2-quasi continuous representatives v(1),v(2)∈W01,2(Ω), which satisfy ϕ-∞≤v(i)≤ϕ+∞c1,2-almost everywhere in Ω.
By (4.2), we have uc,2=wc,2=μc,2, hence, for every ρ∈Cc1(Ω),
(10.2)
∫
Ω
{
B
(
x
,
u
r
)
∇
ϕ
(
u
r
)
-
B
(
x
,
w
r
)
∇
ϕ
(
w
r
)
}
⋅
∇
ρ
d
x
+
∫
Ω
(
u
r
-
w
r
)
ρ
𝑑
x
+
〈
[
u
s
]
d
,
2
-
[
w
s
]
d
,
2
,
ρ
〉
Ω
=
0
(see (3.1)); here we have set
(10.3)
B
(
x
,
s
)
=
1
ϕ
′
(
s
)
A
(
x
,
s
)
for a.e.
x
∈
Ω
,
s
∈
ℝ
(recall that ϕ′>0 by Hypothesis 5.1). Since, by Hypothesis 1.1 (ii), |B(x,ur)∇ϕ(ur)|≤β|∇ϕ(ur)|∈L2(Ω) and |B(x,wr)∇ϕ(wr)|≤β|∇ϕ(wr)|∈L2(Ω), equality (10.2) holds true for every ρ∈W01,2(Ω)∩L∞(Ω).
Let K>0 be fixed arbitrarily and let us choose ρ=TK(ϕ(ur)-ϕ(wr)) in (10.2). By (10.1) and since TK(s) is odd, we have
〈
[
u
s
]
d
,
2
-
[
w
s
]
d
,
2
,
T
K
(
ϕ
(
u
r
)
-
ϕ
(
w
r
)
)
〉
Ω
=
〈
[
u
s
]
d
,
2
,
T
K
(
ϕ
(
u
r
)
-
ϕ
(
w
r
)
)
〉
Ω
+
〈
[
w
s
]
d
,
2
,
T
K
(
ϕ
(
w
r
)
-
ϕ
(
u
r
)
)
〉
Ω
=
〈
[
u
s
+
]
d
,
2
,
T
K
(
ϕ
+
∞
-
ϕ
(
w
r
)
)
〉
Ω
+
〈
[
u
s
-
]
d
,
2
,
T
K
(
ϕ
(
w
r
)
-
ϕ
-
∞
)
〉
Ω
+
〈
[
w
s
+
]
d
,
2
,
T
K
(
ϕ
+
∞
-
ϕ
(
u
r
)
)
〉
Ω
+
〈
[
w
s
-
]
d
,
2
,
T
K
(
ϕ
(
u
r
)
-
ϕ
-
∞
)
〉
Ω
≥
0
,
since ϕ+∞-ϕ(wr)≥0, ϕ+∞-ϕ(ur)≥0, ϕ(wr)-ϕ-∞≥0 and ϕ(ur)-ϕ-∞≥0c1,2-almost everywhere in Ω. Therefore, 〈[us]d,2-[ws]d,2,TK(ϕ(ur)-ϕ(wr))〉Ω≥0 and (10.2) gives
∫
Ω
B
(
x
,
u
r
)
∇
(
ϕ
(
u
r
)
-
ϕ
(
w
r
)
)
⋅
∇
T
K
(
ϕ
(
u
r
)
-
ϕ
(
w
r
)
)
𝑑
x
+
∫
Ω
[
B
(
x
,
u
r
)
-
B
(
x
,
w
r
)
]
∇
ϕ
(
w
r
)
⋅
∇
T
K
(
ϕ
(
u
r
)
-
ϕ
(
w
r
)
)
𝑑
x
+
∫
Ω
(
u
r
-
w
r
)
T
K
(
ϕ
(
u
r
)
-
ϕ
(
w
r
)
)
𝑑
x
≤
0
.
Dividing the previous inequality by K and using Hypotheses 1.1 (i) and 5.1, we obtain
α
K
∫
Ω
|
∇
T
K
(
ϕ
(
u
r
)
-
ϕ
(
w
r
)
)
|
2
d
x
+
∫
Ω
(
u
r
-
w
r
)
T
K
(
ϕ
(
u
r
)
-
ϕ
(
w
r
)
)
K
d
x
≤
-
∫
Ω
[
B
(
x
,
u
r
)
-
B
(
x
,
w
r
)
]
∇
ϕ
(
w
r
)
⋅
∇
T
K
(
ϕ
(
u
r
)
-
ϕ
(
w
r
)
)
K
𝑑
x
≤
L
∫
{
|
ϕ
(
u
r
)
-
ϕ
(
w
r
)
|
≤
K
}
|
ϕ
(
u
r
)
-
ϕ
(
w
r
)
|
|
∇
ϕ
(
w
r
)
|
|
∇
T
K
(
ϕ
(
u
r
)
-
ϕ
(
w
r
)
)
|
K
d
x
≤
L
K
∫
Ω
|
∇
ϕ
(
w
r
)
|
|
∇
T
K
(
ϕ
(
u
r
)
-
ϕ
(
w
r
)
)
|
K
d
x
≤
α
2
K
∫
Ω
|
∇
T
K
(
ϕ
(
u
r
)
-
ϕ
(
w
r
)
)
|
2
d
x
+
L
2
K
2
α
∫
Ω
|
∇
ϕ
(
w
r
)
|
2
d
x
,
whence
∫
Ω
(
u
r
-
w
r
)
T
K
(
ϕ
(
u
r
)
-
ϕ
(
w
r
)
)
K
d
x
≤
L
2
K
2
α
∫
Ω
|
∇
ϕ
(
w
r
)
|
2
d
x
.
Letting K→0+ in the previous inequality, it follows that
∫
Ω
|
u
r
-
w
r
|
d
x
=
∫
Ω
(
u
r
-
w
r
)
sign
(
ϕ
(
u
r
)
-
ϕ
(
w
r
)
)
d
x
≤
0
(recall that ϕ is strictly increasing by Hypothesis 5.1), which plainly gives
(10.4)
u
r
=
w
r
a.e. in
Ω
.
Finally, by (10.4), equality (10.2) reads as 〈[us]d,2-[ws]d,2,ρ〉Ω=0 for every ρ∈W01,2(Ω)∩L∞(Ω), hence [us]d,2=[ws]d,2 in ℳ(Ω). Combining the latter equality with (10.4), we get u=w and the conclusion follows (recall that uc,2=wc,2=μc,2 by (4.2)). ∎
Proof of Theorem 5.4.
Let un∈W01,2(Ω)∩L∞(Ω) be a solution of problem (1). Fix any ψ∈W01,2(Ω)∩L∞(Ω) which satisfies the following conditions:
If ϕ+∞<+∞, there exists δ+∈(0,ϕ+∞) such that ψ≤ϕ+∞-δ+ a.e. in Ω,
If ϕ-∞>-∞, there exists δ-∈(0,-ϕ-∞) such that ψ≥ϕ-∞+δ- a.e. in Ω.
Observe that the requirements ϕ′>0 in ℝ and ϕ(0)=0 ensure that ϕ-∞<0<ϕ+∞, with both cases ϕ+∞=-ϕ-∞=∞ or ϕ+∞=∞ and ϕ-∞>-∞ (and vice versa) being admissible. Moreover, 0<ϕ+∞-δ+<ϕ+∞ and ϕ-∞<ϕ-∞+δ-<0. For every K>0, we have
∫
Ω
A
(
x
,
u
n
)
∇
u
n
⋅
∇
T
K
(
ϕ
(
u
n
)
-
ψ
)
𝑑
x
+
1
n
∫
Ω
∇
u
n
⋅
∇
T
K
(
ϕ
(
u
n
)
-
ψ
)
𝑑
x
+
∫
Ω
u
n
T
K
(
ϕ
(
u
n
)
-
ψ
)
𝑑
x
(10.5)
=
∫
Ω
μ
n
T
K
(
ϕ
(
u
n
)
-
ψ
)
𝑑
x
.
By Hypothesis 1.1 (i), we have
∫
Ω
A
(
x
,
u
n
)
∇
u
n
⋅
∇
T
K
(
ϕ
(
u
n
)
-
ψ
)
𝑑
x
=
∫
Ω
A
(
x
,
u
n
)
ϕ
′
(
u
n
)
∇
(
ϕ
(
u
n
)
-
ψ
)
⋅
∇
T
K
(
ϕ
(
u
n
)
-
ψ
)
𝑑
x
+
∫
Ω
A
(
x
,
u
n
)
ϕ
′
(
u
n
)
∇
ψ
⋅
∇
T
K
(
ϕ
(
u
n
)
-
ψ
)
𝑑
x
(10.6)
≥
α
∫
Ω
|
∇
T
K
(
ϕ
(
u
n
)
-
ψ
)
|
2
d
x
+
∫
Ω
A
(
x
,
u
n
)
ϕ
′
(
u
n
)
∇
ψ
⋅
∇
T
K
(
ϕ
(
u
n
)
-
ψ
)
d
x
.
Combining (10.6) and (10.5) gives
α
∫
Ω
|
∇
T
K
(
ϕ
(
u
n
)
-
ψ
)
|
2
d
x
+
∫
Ω
A
(
x
,
u
n
)
ϕ
′
(
u
n
)
∇
ψ
⋅
∇
T
K
(
ϕ
(
u
n
)
-
ψ
)
d
x
+
1
n
∫
Ω
∇
u
n
⋅
∇
T
K
(
ϕ
(
u
n
)
-
ψ
)
𝑑
x
+
∫
Ω
u
n
T
K
(
ϕ
(
u
n
)
-
ψ
)
𝑑
x
(10.7)
≤
∫
Ω
μ
n
T
K
(
ϕ
(
u
n
)
-
ψ
)
𝑑
x
.
Next, we set S+=ϕ-1(esssupx∈Ωψ+(x))≥0 and S-=ϕ-1(essinfx∈Ω(-ψ-(x)))≤0. Notice that the above quantities are well-defined by (C1) (if ϕ+∞<+∞) and (C2) (if ϕ-∞>-∞). For every couple of real numbers M-<S-≤0 and M+>S+≥0, we have
u
n
>
M
+
>
S
+
⇒
ϕ
(
u
n
)
>
ϕ
(
S
+
)
≥
ψ
a.e. in
Ω
,
u
n
<
M
-
<
S
-
⇒
ϕ
(
u
n
)
<
ϕ
(
S
-
)
≤
ψ
a.e. in
Ω
(see (C1)–C2), whence
∫
Ω
u
n
T
K
(
ϕ
(
u
n
)
-
ψ
)
𝑑
x
≥
∫
{
u
n
<
M
-
}
M
-
T
K
(
ϕ
(
u
n
)
-
ψ
)
𝑑
x
+
∫
{
u
n
>
M
+
}
M
+
T
K
(
ϕ
(
u
n
)
-
ψ
)
𝑑
x
+
∫
{
M
-
≤
u
n
≤
M
+
}
u
n
T
K
(
ϕ
(
u
n
)
-
ψ
)
𝑑
x
(10.8)
=
∫
Ω
g
M
(
u
n
)
T
K
(
ϕ
(
u
n
)
-
ψ
)
𝑑
x
,
where gM(s)=max{M-,min{s,M+}}. By (10.8), inequality (10.7) gives
α
∫
Ω
|
∇
T
K
(
ϕ
(
u
n
)
-
ψ
)
|
2
d
x
+
∫
Ω
A
(
x
,
u
n
)
ϕ
′
(
u
n
)
∇
ψ
⋅
∇
T
K
(
ϕ
(
u
n
)
-
ψ
)
d
x
+
1
n
∫
Ω
∇
u
n
⋅
∇
T
K
(
ϕ
(
u
n
)
-
ψ
)
𝑑
x
+
∫
Ω
g
M
(
u
n
)
T
K
(
ϕ
(
u
n
)
-
ψ
)
𝑑
x
(10.9)
≤
∫
Ω
μ
n
T
K
(
ϕ
(
u
n
)
-
ψ
)
𝑑
x
.
The rest of the proof is devoted to take the limit as n→∞ in (10.9). To this end, let {unj} be any subsequence along which all the convergences in Section 7 are satisfied. In particular, unj⇀*u in ℳ(Ω), where u is a solution of (P) given by Theorem 3.3. From the assumption μ∈ℳd,2(Ω) and (3.4) we infer that u∈ℳd,2(Ω), and, since ϕ′>0 in ℝ, by (7.23), we have
(10.10)
u
n
j
→
u
r
a.e. in
Ω
.
As a consequence, we obtain
(10.11)
lim
j
→
∞
∫
Ω
g
M
(
u
n
j
)
T
K
(
ϕ
(
u
n
j
)
-
ψ
)
𝑑
x
=
∫
Ω
g
M
(
u
r
)
T
K
(
ϕ
(
u
r
)
-
ψ
)
𝑑
x
and (see also Hypothesis 1.1 (ii))
(10.12)
A
(
x
,
u
n
j
)
ϕ
′
(
u
n
j
)
∇
ψ
→
A
(
x
,
u
r
)
ϕ
′
(
u
r
)
∇
ψ
in
[
L
2
(
Ω
)
]
N
.
In order to let j→∞ in the first two terms in the left-hand side of (10.9), let us preliminarily observe that there exists a constant C>0 (possibly depending on K and ∥ψ∥L∞(Ω)) such that for all j, we have
∫
Ω
|
∇
T
K
(
ϕ
(
u
n
j
)
-
ψ
)
|
2
d
x
=
∫
{
|
ϕ
(
u
n
j
)
-
ψ
|
≤
K
}
|
∇
(
ϕ
(
u
n
j
)
-
ψ
)
|
2
d
x
(10.13)
≤
2
∫
{
|
ϕ
(
u
n
j
)
|
≤
∥
ψ
∥
L
∞
(
Ω
)
+
K
}
|
∇
ϕ
(
u
n
j
)
|
2
d
x
+
2
∫
Ω
|
∇
ψ
|
2
d
x
≤
C
(here we have used (6.6)). Hence, by the above estimate and (10.10), we obtain
(10.14)
T
K
(
ϕ
(
u
n
j
)
-
ψ
)
⇀
T
K
(
ϕ
(
u
r
)
-
ψ
)
in
W
0
1
,
2
(
Ω
)
.
Combining (10.12) and (10.14), it can be easily seen that
(10.15)
lim
j
→
∞
∫
Ω
A
(
x
,
u
n
j
)
ϕ
′
(
u
n
j
)
∇
ψ
⋅
∇
T
K
(
ϕ
(
u
n
j
)
-
ψ
)
𝑑
x
=
∫
Ω
A
(
x
,
u
r
)
ϕ
′
(
u
r
)
∇
ψ
⋅
∇
T
K
(
ϕ
(
u
r
)
-
ψ
)
𝑑
x
,
whereas, by (6.9) and (10.13),
(10.16)
lim
j
→
∞
1
n
j
∫
Ω
∇
u
n
j
⋅
∇
T
K
(
ϕ
(
u
n
j
)
-
ψ
)
𝑑
x
=
0
.
Moreover, since μ∈ℳd,2(Ω), by (3.3) and (10.14), we get
(10.17)
lim
j
→
∞
∫
Ω
T
K
(
ϕ
(
u
n
j
)
-
ψ
)
μ
n
j
𝑑
x
=
〈
μ
,
T
K
(
ϕ
(
u
r
)
-
ψ
)
〉
Ω
.
In view of (10.11) and (10.14)–(10.17), letting j→∞ in (10.9) gives
α
∫
Ω
|
∇
T
K
(
ϕ
(
u
r
)
-
ψ
)
|
2
d
x
+
∫
Ω
A
(
x
,
u
r
)
ϕ
′
(
u
r
)
∇
ψ
⋅
∇
T
K
(
ϕ
(
u
r
)
-
ψ
)
d
x
+
∫
Ω
g
M
(
u
r
)
T
K
(
ϕ
(
u
r
)
-
ψ
)
d
x
≤
〈
μ
,
T
K
(
ϕ
(
u
r
)
-
ψ
)
〉
Ω
.
Finally, letting M+→∞ and M-→-∞, we obtain the entropy inequality (5.1), for every ψ∈W01,2(Ω)∩L∞(Ω), satisfying (C1) and (C2). By the arbitrariness of δ+∈(0,ϕ+∞) in (C1) and δ-∈(0,-ϕ-∞) in (C2), it can be checked that (5.1) holds true for all ψ∈W01,2(Ω)∩L∞(Ω), as in (5.2)–(5.3). Therefore, the conclusion follows. ∎
Proof of Proposition 5.5.
(i) Let μ∈ℳd,2 and let u be any solution of problem (P) given in Theorem 3.3. Then u∈ℳd,2(Ω), by (3.4).
(ii) Let ϕ∈L∞(ℝ), ϕ′>0 in ℝ, μ∈ℳd,2(Ω) and let u be a solution of problem (P) satisfying (3.7)–(3.8). By (3.2), we have ϕ(ur)∈W01,2(Ω) and, since μ∈ℳd,2(Ω) (namely, μc,2=0), by (4.2), it follows that u∈ℳd,2(Ω). Hence, for every ψ∈W01,2(Ω)∩L∞(Ω), we can choose in equality (3.1) the test function ρ=TK(ϕ(ur)-ψ) and, by Hypothesis 1.1 (i), we get
α
∫
Ω
|
∇
T
K
(
ϕ
(
u
r
)
-
ψ
)
|
2
d
x
+
∫
Ω
A
(
x
,
u
r
)
ϕ
′
(
u
r
)
∇
ψ
⋅
∇
T
K
(
ϕ
(
u
r
)
-
ψ
)
d
x
+
∫
Ω
u
r
T
K
(
ϕ
(
u
r
)
-
ψ
)
𝑑
x
+
〈
[
u
s
]
d
,
2
,
T
K
(
ϕ
(
u
r
)
-
ψ
)
〉
Ω
≤
〈
μ
,
T
K
(
ϕ
(
u
r
)
-
ψ
)
〉
Ω
.
Therefore, the entropy inequalities (5.1) will follow if we prove that 〈[us]d,2,TK(ϕ(ur)-ψ)〉Ω≥0 for all ψ as above, satisfying in addition (5.2)–(5.3). To this end, it suffices to observe that by (3.7)–(3.8) (equivalently, see (10.1)), we have
〈
[
u
s
]
d
,
2
,
T
K
(
ϕ
(
u
r
)
-
ψ
)
〉
Ω
=
〈
[
u
s
+
]
d
,
2
,
T
K
(
ϕ
+
∞
-
ψ
)
〉
Ω
+
〈
[
u
s
-
]
d
,
2
,
T
K
(
ψ
-
ϕ
-
∞
)
〉
Ω
≥
0
,
since (5.2)–(5.3) ensure that ϕ+∞-ψ≥0 and ψ-ϕ-∞≥0 a.e. in Ω (hence c1,2-almost everywhere, as both ϕ(ur),ψ∈W01,2(Ω) are identified with their c1,2-quasi continuous representatives). ∎
The proof of Theorem 5.6 relies on the following lemma.
Lemma 10.1.
Let ϕ′>0 and let μ∈M(Ω) be such that μ=f-div(G), where f∈L2(Ω) and G∈[L2(Ω)]N. Then every weak entropy solution u of problem (P) satisfies ϕ(ur)∈W01,2(Ω).
Proof.
Choosing ψ=0 in the entropy inequalities (5.1), for every K>0, we get
α
∫
Ω
|
∇
T
K
(
ϕ
(
u
r
)
)
|
2
d
x
≤
α
∫
Ω
|
∇
T
K
(
ϕ
(
u
r
)
)
|
2
d
x
+
∫
Ω
u
r
T
K
(
ϕ
(
u
r
)
)
d
x
≤
∫
Ω
T
K
(
ϕ
(
u
r
)
)
f
(
x
)
𝑑
x
+
∫
Ω
∇
T
K
(
ϕ
(
u
r
)
)
⋅
G
(
x
)
𝑑
x
≤
(
∥
f
|
L
2
(
Ω
)
+
∥
|
G
|
∥
L
2
(
Ω
)
)
∥
T
K
(
ϕ
(
u
r
)
)
∥
W
0
1
,
2
(
Ω
)
,
whence ∥TK(ϕ(ur))∥W01,2(Ω)≤Cα for some C>0 independent of K, and the conclusion follows by letting K→∞. ∎
Proof of Theorem 5.6.
Let u be a weak entropy solution of problem (P). Since μ∈ℳd,2(Ω), there exist f∈L1(Ω) and G∈[L2(Ω)]N such that μ=f-div(G) (see [15]), and (5.1) reads as
∫
Ω
{
α
|
∇
T
K
(
ϕ
(
u
r
)
-
ψ
)
|
2
+
u
r
T
K
(
ϕ
(
u
r
)
-
ψ
)
+
A
(
x
,
u
r
)
ϕ
′
(
u
r
)
∇
ψ
⋅
∇
T
K
(
ϕ
(
u
r
)
-
ψ
)
}
𝑑
x
(10.18)
≤
∫
Ω
{
f
T
K
(
ϕ
(
u
r
)
-
ψ
)
+
G
⋅
∇
T
K
(
ϕ
(
u
r
)
-
ψ
)
}
𝑑
x
for every K>0 and ψ∈W01,2(Ω)∩L∞(Ω) satisfying (5.2)–(5.3). For every j∈ℕ, let μj∈ℳd,2(Ω) be defined as
μ
j
=
f
j
-
div
(
G
)
,
where {fj}⊆L∞(Ω) is any sequence such that
(10.19)
f
j
→
f
in
L
1
(
Ω
)
.
For every j, let uj be any weak entropy solution of problem (P) with datum μj, given by Theorem 5.4. By Lemma 10.1, we have ϕ(ujr)∈W01,2(Ω), hence uj∈ℳd,2(Ω) (indeed, [uj]c,2=[μj]c,2=0, by Proposition 4.1). Moreover, by arguing as in the proof of Theorem 5.2, it can be checked that every uj satisfies the weak formulation
(10.20)
∫
Ω
A
(
x
,
u
j
r
)
ϕ
′
(
u
j
r
)
∇
ϕ
(
u
j
r
)
⋅
∇
ρ
d
x
+
〈
u
j
,
ρ
〉
Ω
=
〈
μ
j
,
ρ
〉
Ω
,
with test functions ρ∈W01,2(Ω)∩L∞(Ω). We proceed in three steps.
Step 1. Let us prove that for every K>0 and j∈ℕ, we have
T
K
(
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
)
∈
W
0
1
,
2
(
Ω
)
and
α
∫
Ω
|
∇
T
K
(
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
)
|
2
d
x
+
∫
Ω
u
r
T
K
(
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
)
d
x
+
∫
Ω
A
(
x
,
u
r
)
ϕ
′
(
u
r
)
∇
ϕ
(
u
j
r
)
⋅
∇
T
K
(
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
)
d
x
(10.21)
≤
∫
Ω
f
T
K
(
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
)
𝑑
x
+
∫
Ω
G
⋅
∇
T
K
(
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
)
𝑑
x
.
Indeed, for every h∈ℕ, choosing ψ=Th(ϕ(ujr)) in (10.18) gives
α
∫
Ω
|
∇
T
K
(
ϕ
(
u
r
)
-
T
h
(
ϕ
(
u
j
r
)
)
)
|
2
d
x
+
∫
Ω
u
r
T
K
(
ϕ
(
u
r
)
-
T
h
(
ϕ
(
u
j
r
)
)
)
d
x
+
∫
Ω
A
(
x
,
u
r
)
ϕ
′
(
u
r
)
∇
T
h
(
ϕ
(
u
j
r
)
)
⋅
∇
T
K
(
ϕ
(
u
r
)
-
T
h
(
ϕ
(
u
j
r
)
)
)
𝑑
x
(10.22)
≤
∫
Ω
f
T
K
(
ϕ
(
u
r
)
-
T
h
(
ϕ
(
u
j
r
)
)
)
𝑑
x
+
∫
Ω
G
⋅
∇
T
K
(
ϕ
(
u
r
)
-
T
h
(
ϕ
(
u
j
r
)
)
)
𝑑
x
.
From the previous inequality and Hypothesis 1.1 (ii), since ϕ(ujr)∈W01,2(Ω), we obtain
α
∫
Ω
|
∇
T
K
(
ϕ
(
u
r
)
-
T
h
(
ϕ
(
u
j
r
)
)
)
|
2
d
x
≤
K
(
∥
u
r
∥
L
1
(
Ω
)
+
∥
f
∥
L
1
(
Ω
)
)
+
(
β
∥
ϕ
(
u
j
r
)
∥
W
0
1
,
2
(
Ω
)
+
∥
|
G
|
∥
L
2
(
Ω
)
)
(
∫
Ω
|
∇
T
K
(
ϕ
(
u
r
)
-
T
h
(
ϕ
(
u
j
r
)
)
)
|
2
d
x
)
1
2
,
whence
(10.23)
∥
T
K
(
ϕ
(
u
r
)
-
T
h
(
ϕ
(
u
j
r
)
)
)
∥
W
0
1
,
2
(
Ω
)
≤
C
for some C=C(K,j)>0 independent of h. Since, for every fixed K and j, we have
(10.24)
T
K
(
ϕ
(
u
r
)
-
T
h
(
ϕ
(
u
j
r
)
)
)
→
T
K
(
ϕ
(
u
r
)
-
(
ϕ
(
u
j
r
)
)
a.e. in
Ω
as
h
→
∞
,
from (10.23), it follows that TK(ϕ(ur)-ϕ(ujr))∈W01,2(Ω) and
(10.25)
T
K
(
ϕ
(
u
r
)
-
T
h
(
ϕ
(
u
j
r
)
)
)
⇀
T
K
(
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
)
in
W
0
1
,
2
(
Ω
)
as
h
→
∞
.
In view of (10.24)–(10.25), and (since ϕ(ujr)∈W01,2(Ω))
∫
Ω
|
∇
ϕ
(
u
j
r
)
-
∇
T
h
(
ϕ
(
u
j
r
)
)
|
2
d
x
=
∫
{
|
ϕ
(
u
j
r
)
|
>
h
}
|
∇
ϕ
(
u
j
r
)
|
2
d
x
→
0
as
h
→
∞
,
letting h→∞ in (10.22), by the lower semicontinuity of the W01,2-norm, we get (10.21).
Step 2. Let us prove that for every K>0 and j∈ℕ, we have
∫
Ω
{
A
(
x
,
u
j
r
)
ϕ
′
(
u
j
r
)
∇
ϕ
(
u
j
r
)
⋅
∇
T
K
(
ϕ
(
u
j
r
)
-
ϕ
(
u
r
)
)
+
u
j
r
T
K
(
ϕ
(
u
j
r
)
-
ϕ
(
u
r
)
)
}
𝑑
x
(10.26)
≤
∫
Ω
f
j
T
K
(
ϕ
(
u
j
r
)
-
ϕ
(
u
r
)
)
𝑑
x
+
∫
Ω
G
⋅
∇
T
K
(
ϕ
(
u
j
r
)
-
ϕ
(
u
r
)
)
𝑑
x
.
Choosing ρ=TK(ϕ(ujr)-ϕ(ur))∈W01,2(Ω)∩L∞(Ω) in (10.20), we obtain
∫
Ω
A
(
x
,
u
j
r
)
ϕ
′
(
u
j
r
)
∇
ϕ
(
u
j
r
)
⋅
∇
T
K
(
ϕ
(
u
j
r
)
-
ϕ
(
u
r
)
)
𝑑
x
+
∫
Ω
u
j
r
T
K
(
ϕ
(
u
j
r
)
-
ϕ
(
u
r
)
)
𝑑
x
+
〈
u
j
s
,
T
K
(
ϕ
(
u
j
r
)
-
ϕ
(
u
r
)
)
〉
Ω
=
〈
μ
j
,
T
K
(
ϕ
(
u
j
r
)
-
ϕ
(
u
r
)
)
〉
Ω
(10.27)
=
∫
Ω
f
j
T
K
(
ϕ
(
u
j
r
)
-
ϕ
(
u
r
)
)
𝑑
x
+
∫
Ω
G
⋅
∇
T
K
(
ϕ
(
u
j
r
)
-
ϕ
(
u
r
)
)
𝑑
x
.
Concerning the above equality, since ujs∈ℳd,2(Ω), by Theorem 3.4 (recall that uj is a solution of problem (P) given by Theorem 5.4, hence by Theorem 3.3), the following hold:
Either ujs+=0 if ϕ+∞=+∞ (see (3.5)–(3.6)), or if ϕ+∞<+∞, then, by (3.7)–(3.8), we get
(10.28)
〈
u
j
s
+
,
T
K
(
ϕ
(
u
j
r
)
-
ϕ
(
u
r
)
)
〉
Ω
≥
〈
u
j
s
+
,
T
K
(
ϕ
+
∞
-
ϕ
+
(
u
r
)
)
〉
Ω
≥
0
.
Either ujs-=0 if ϕ-∞=-∞ (see (3.5)–(3.6)), or if ϕ-∞>-∞, then, by (3.7)–(3.8), we get
(10.29)
-
〈
u
j
s
-
,
T
K
(
ϕ
(
u
j
r
)
-
ϕ
(
u
r
)
)
〉
Ω
≥
〈
u
j
s
-
,
T
K
(
-
ϕ
-
(
u
r
)
-
ϕ
-
∞
)
〉
Ω
≥
0
.
Combining (10.28)–(10.29) with (10), inequality (10.26) follows at once. Step 3. Let us conclude the proof. By (10.21) and (10.26), we obtain
α
∫
Ω
|
∇
T
K
(
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
)
|
2
d
x
+
∫
Ω
(
u
r
-
u
j
r
)
T
K
(
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
)
d
x
+
∫
Ω
[
B
(
x
,
u
r
)
-
B
(
x
,
u
j
r
)
]
∇
ϕ
(
u
j
r
)
⋅
∇
T
K
(
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
)
𝑑
x
(10.30)
≤
∫
Ω
(
f
-
f
j
)
T
K
(
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
)
𝑑
x
,
where, for the sake of brevity, B(x,s) is the matrix defined in (10.3), namely, B(x,s)=1ϕ′(s)A(x,s). By (10.30), and Hypotheses 1.1 (ii) and 5.1, we get
α
K
∫
Ω
|
∇
T
K
(
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
)
|
2
d
x
+
∫
Ω
(
u
r
-
u
j
r
)
T
K
(
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
)
K
d
x
≤
L
∫
{
|
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
|
≤
K
}
|
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
|
|
∇
ϕ
(
u
j
r
)
|
|
∇
T
K
(
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
)
|
K
d
x
+
∫
Ω
|
f
-
f
j
|
d
x
≤
L
K
∫
Ω
|
∇
ϕ
(
u
j
r
)
|
|
∇
T
K
(
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
)
|
K
d
x
+
∫
Ω
|
f
-
f
j
|
d
x
≤
L
2
K
2
α
∫
Ω
|
∇
ϕ
(
u
j
r
)
|
2
d
x
+
α
2
K
∫
Ω
|
∇
T
K
(
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
)
|
2
d
x
+
∫
Ω
|
f
-
f
j
|
d
x
,
whence
∫
Ω
(
u
r
-
u
j
r
)
T
K
(
ϕ
(
u
r
)
-
ϕ
(
u
j
r
)
)
K
d
x
≤
L
2
K
2
α
∫
Ω
|
∇
ϕ
(
u
j
r
)
|
2
d
x
+
∫
Ω
|
f
-
f
j
|
d
x
.
Taking the limit as K→0+ in the previous inequality gives ∫Ω|ur-ujr|dx≤∫Ω|f-fj|dx, thus from (10.19) it follows ujr→ur in L1(Ω) as j→∞.
On the other hand, if w is another weak entropy solution of problem (P) with datum μ, it can be easily checked that the first part of the proof holds true as well with u replaced by w, whence ujr→wr in L1(Ω) as j→∞. By these considerations, it follows that ur=wr a.e. in Ω. By this equality, the weak formulation (3.1) (for u and w, respectively) gives 〈us,ρ〉Ω=〈ws,ρ〉Ω for every ρ∈Cc1(Ω). Therefore, u=w in ℳ(Ω) and this shows that for every μ∈ℳd,2(Ω), there exists at most one weak entropy solution of problem (P).
The existence of the (unique) weak entropy solution u of problem (P) for every μ∈ℳd,2(Ω) follows from Theorem 5.4, which in turn ensures that u belongs to ℳd,2(Ω) (see Proposition 5.5 (i)) and satisfies the properties stated in Theorem 3.4. ∎
Remark 10.2.
Let μ∈ℳd,2(Ω), ϕ′>0 in ℝ and ϕ+∞=-ϕ-∞=∞. Concerning the model problem (1.2), it is informative to observe that in this case uniqueness immediately follows by the structural conditions (3.5)–(3.6). More precisely, for every μ∈ℳd,2(Ω), there exists a unique solution u of (1.2), in the sense of Definition 3.1, which belongs to the Lebesgue space L1(Ω).
Indeed, it is a consequence of the more general results in [18, Corollary 4.B.1] (it suffices to set v=ϕ(u), g(v)=u and rephrase (1.2) as (1.3)).
A natural question is whether similar considerations continue to hold if one removes the requirement ϕ+∞=-ϕ-∞=∞. When ϕ is bounded, the affirmative answer is the content of Theorem 5.2, where the proof of uniqueness only relies on (3.7)–(3.8) (similar results for the model equation (1.3) have been previously obtained in [24, Proposition 3] and [19, Theorem 1]). Hence, it only remains to address the case in which ϕ+∞<∞ and ϕ-∞=-∞ (the case ϕ+∞=∞ and ϕ-∞>-∞ can be treated analogously).
Proposition 10.3.
Let ϕ be as in Hypothesis 1.1, satisfying ϕ′>0 in R, ϕ+∞<∞ and ϕ-∞=-∞. Then, for every μ∈Md,2(Ω), there exists at most one solution u∈Md,2(Ω) of (1.2) which satisfies (3.5)–(3.8).
Proof.
We follow the proof of [18, Corollary 4.B.1]. Let u1,u2∈ℳd,2(Ω) be two solutions of (1.2) (in the sense of Definition 3.1) satisfying (3.5)–(3.8), whence
(10.31)
u
i
s
-
=
0
,
u
i
s
+
=
u
i
s
+
⌞
{
x
∈
Ω
:
ϕ
+
(
u
i
r
(
x
)
)
=
ϕ
+
∞
}
(
i
=
1
,
2
)
.
Setting σ=u1-u2∈ℳd,2(Ω), it can be seen that w:=ϕ(u1r)-ϕ(u2r) is the unique solution of the problem -∫ΩwΔζ=-〈σ,ζ〉Ω for every ζ∈C02(Ω¯), w∈L1(Ω) (see [35]). Consider any sequence {σn}⊆W-1,2(Ω)∩ℳ(Ω), with σn⇀*σ in ℳ(Ω) and ∥σn∥ℳ(Ω)≤C for some C independent of n, and let wn∈H01(Ω) be the unique solution of
-
Δ
w
n
=
-
σ
n
in
Ω
,
w
n
=
0
on
∂
Ω
.
It is well known that wn→w in L1(Ω) and, for any K>0,
(10.32)
〈
σ
n
,
T
K
(
w
n
)
〉
Ω
=
-
∫
Ω
|
∇
T
K
(
w
n
)
|
2
d
x
.
Since σ∈ℳd,2(Ω), by a proper choice of {σn} (see [22, Section 3]), for every K>0, there exists CK>0 such that ∥TK(wn)∥W01,2(Ω)≤CK, whence TK(wn)⇀TK(w) in W01,2(Ω) and 〈σnTK(wn)〉Ω→〈σ,TK(w)〉Ω as n→∞. Therefore, letting n→∞ in (10.32) gives
0
≥
〈
σ
,
T
K
(
w
)
〉
Ω
=
∫
Ω
(
u
1
r
-
u
2
r
)
T
K
(
ϕ
(
u
1
r
)
-
ϕ
(
u
2
r
)
)
𝑑
x
+
〈
u
1
s
+
-
u
2
s
+
,
T
K
(
ϕ
(
u
1
r
)
-
ϕ
(
u
2
r
)
)
〉
Ω
≥
∫
Ω
(
u
1
r
-
u
2
r
)
T
K
(
ϕ
(
u
1
r
)
-
ϕ
(
u
2
r
)
)
𝑑
x
(here we have used (10.31)). By the arbitrariness of K, this proves that u1r=u2r a.e. in Ω and the conclusion follows by arguing as in the last part of the proof of Theorem 5.6. ∎
