Amin Farjudian,∗School of Computer Science, The University of
Nottingham Ningbo China, 199 Taikang East Road, Ningbo, 315100, China. email: [email protected]
Behrouz Emamizadeh, School of Mathematical Sciences, The University of Nottingham Ningbo China, 199 Taikang East Road, Ningbo, 315100, China. email: [email protected]
ABSOLUTE CONTINUITY IN PARTIAL
DIFFERENTIAL EQUATIONS
Abstract
In this note we study a function which frequently appears in partial diff eren-tial equations. We prove that this function is absolutely continuous, hence it can be written as a definite integral. As a result we obtain some estimates regarding solutions of the Hamilton-Jacobi systems.
1
Introduction
LetH be a differential operator of orderm ∈ Nand let f ∈ Lp(D) be a positive
function, wherep ∈(1,∞) andDis a smooth bounded domain inRn. Consider the
equation:
H(u)= f, inD (1)
A functionu∈Wm,p(D)∩C(D) is called a strong solution of (1) provided thatH(u)= f almost everywhere (a. e.) inD. We assume the operatorHsatisfies the following condition:
For anyu∈Wm(D) andγ∈R: H(u)=0 a. e. inEγ B{x∈D|u(x)=γ} (P)
Mathematical Reviews subject classification: Primary: 28D05, 35F21
Key words: Absolute continuity, Weak convergence, Rearrangements of functions, Measure preserv-ing maps, Hamilton-Jacobi systems
∗
Amin Farjudian’s work on this article has been partially supported by the Natural Science Foundation of China (Grant No. 61070023) and Ningbo Natural Science Programme by Ningbo S&T bureau (Grant No. 2010A610104).
For a measurable function h : D → R, the distribution function ofh, denoted λh(α), is defined as follows:
λh(α)B| {x∈D|h(x)≥α} | ≡ | {h≥α} |, (∀α∈R)
where| · |denotes then-dimensional Lebesgue measure. Clearlyλhis decreasing, and
ifhis continuous, thenλhwill be strictly decreasing. Moreover, in case the graph
ofh has no significant flat sections (i. e.∀γ ∈ R : | {h=γ} | = 0) then λh will be
continuous. The decreasing rearrangement ofh, denotedh∗(s), is defined as follows:
h∗: [0,|D|]→
R
h∗(s)=inf{α|λh(α)≤s}
Note that whenhis continuous and its graph has no significant flat sections then: λh◦h∗(s)=s and h∗◦λh(α)=α.
We also need to recall some background from rearrangements of functions. Given g0:D⊆Rn →R, the rearrangement class generated byg0, denotedR(g0), is the set
of functionsg:D→Rsuch thatλg(α)=λg0(α), for every realα. In caseg0∈L
p(D)
thenR(g0) ⊆ Lp(D), and∀g ∈ R(g0) : kgkp = kg0kp. The weak closure ofR(g0)
inLp(D) is denoted asR(g0) which, unlikeR(g0), enjoys some nice properties and
characterizations that are stated in the following lemma. For the proof and further reading see [3, 4, 5, 9]:
Lemma 1. Let g0 ∈Lp(D)be a non-negative function, andR(g0)be the
rearrange-ment class generated by g0. Then:
(1) R(g0)is convex, and weakly compact in Lp(D).
(2) R(g0)=co(R(g0)), the closed convex hull ofR(g0).
(3) The following characterization stands:
R(g0)=
(
g| ∀s∈(0,|D|) :
Z s
0
g∗(t)dt≤
Z s
0
g∗0(t)dt, and
Z |D|
0
g∗(t)dt=
Z |D|
0
g∗0(t)dt
)
attributed to Ryff[13], giveng : D→ R, there existsφ ∈ M(D,[0,|D|]) such that g=g∗◦φalmost everywhere inD.
We now introduce the function that is the main drive behind writing this note. To this end, we assumeu∈Wm,p(D)∩C(D) is a strong solution of (1). We are interested in the functionξ: [0,|D|]→Rdefined by:
ξ(s)=Z
{u≥u∗(s)}
f(x)dx. (2)
Thanks to property(P)on page 1, and of course the fact that f is positive, the level sets{u =γ}must have zero measure, henceξis well-defined. This function is fre-quently referred to in partial differential equations, particularly when one is interested in comparing the solution of a boundary value problem to that of a symmetrized problem, the latter being readily solved. There are many references in this regard, e. g. [2, 6, 14], to mention a few. In this note we prove thatξis absolutely con-tinuous, hence it can be represented by a definite integral of the formR0sF(τ)dτ. Then, we will prove that the integrandF composed with any measure-preserving mapφ ∈ M(D,[0,|D|]) belongs toR(f). Using these two results we point out a couple of applications.
Throughout this paper we use some standard notations. For example,Wm,p(D) andWm(D) denote the usual Sobolev spaces. The spaceLp(D) comprises functions whose p-th powers are integrable, and the norm in this space is defined bykfkp =
R
D|f| pdx1/p
. Moreover,C(D) andC(D) denote the spaces of continuous functions overD and its closureD, respectively, and the corresponding norm is denoted by
k · k∞. The arrow “→” indicates strong convergence, whilst “*” indicates weak
convergence in spaces under discussion.
2
Main results
Our first main result is the following:
Theorem 2. The functionξ, as defined in (2), is absolutely continuous on[0,|D|].
Proof. Let > 0, and consider a finite sequence{(αi, βi) | 1 ≤ i ≤ N}of
non-overlapping subintervals of [0,|D|] such thatPN
i=1(βi−αi)< δ, whereδis a positive
number to be determined later. By settingt(αi) =u∗(αi) andt(βi) =u∗(βi) we will
have:
N
X
i=1
|ξ(βi)−ξ(αi)|= N
X
i=1
Z
{t(βi)<u<t(αi)}
f(x)dx
=Z E
whereE =∪N i=1{x: u
∗(β
i)<u(x)<u∗(αi)}. By applying the H¨older inequality we
obtain:
Z
E
f(x)dx≤ |E|1q kfkp, (4)
where1p+1q =1.Note that|E|=PN
i=1(βi−αi).This, along with (3) and (4), will give
the desired result, provided thatδ <
kfkp
q
.
Corollary 3. The functionξ, as defined in (2), satisfies:
ξ(s)=Z s 0
F(τ)dτ, (5)
for some integrable function F.
Proof. By Theorem 2,ξis absolutely continuous. Hence we can apply Corollary 14
in [12], together with the fact thatξ(0)=0, to deduce
ξ(s)=Z s 0
ξ0(τ)dτ,
almost everywhere in [0,|D|]. So by settingF(s)=ξ0(s), we get the desired result.
We now state our second main result:
Theorem 4. Let F be the function in Corollary 3 andφ ∈ M(D,[0,|D|]). Then
F◦φ∈ R(f).
Proof. Note thatλF◦φ(α)=λF(α), for everyα ∈R. Thus, (F◦φ)∗(s)= F∗(s), for
almost everys∈[0,|D|]. Hence, in view of item (3) of Lemma 1, it suffices to prove:
(i) R0|D|F∗(s)ds=R|D| 0 f
∗(s)ds.
(ii) R0sF∗(t)dt≤Rs 0 f
∗(t)dt, ∀s∈(0,|D|).
Proving (i) is straightforward as
Z |D|
0
F∗(t)dt=
Z |D|
0
F(t)dt=ξ(|D|)
=Z
{u≥t(|D|)} f dx=
Z
{u≥0} f dx=
Z
D
f dx=
Z |D|
0
f∗(t)dt,
where we have used Corollary 3.
Step1. LetU be an open subset of (0,|D|). Then, we can writeU =∪∞
i=1(Ai,Bi),
where (Ai,Bi) are mutually disjoint. Hence,
Z
U
F(τ)dτ =
∞
X
i=1
Z Bi
Ai
F(τ)dτ=
∞
X
i=1
Z Bi
0
F(τ)dτ−
Z Ai
0 F(τ)dτ ! = ∞ X
i=1
Z
{u≥t(Bi)}
f dx−
Z
{u≥t(Ai)}
f dx
!
=
∞
X
i=1
Z
{t(Bi)≤u<t(Ai)}
f dx
= Z
S{t(Bi)≤u<t(Ai)} f dx≤
Z |S{t(B
i)≤u<t(Ai)| 0
f∗(s)ds
= Z
P(B
i−Ai)
0
f∗(s)ds=
Z | U |
0
f∗(s)ds.
Step2. LetVbe a measurable subset of (0,|D|), and >0. By Theorem 3.6 in [15], there exists an open setGcontainingVsuch that|G\ V|< . Whence
Z
V
F(t)dt≤
Z
G
F(t)dt≤
Z |G|
0
f∗(s)ds
=Z |V| 0
f∗(s)ds+
Z |G|
|V|
f∗(s)ds
≤
Z |V|
0
f∗(s)ds+kfkp(|G| − |V|)1/q, (6)
where we have used Step 1, and H¨older’s inequality. Since|G| − |V| =|G\ V|< , from (6) we infer
Z
V
F(t)dt≤
Z | V |
0
f∗(s)ds+1/qkfkp. (7)
Sinceis arbitrary, (7) implies
Z
V
F(t)dt≤
Z | V |
0
f∗(s)ds.
Step3. We recall the following maximization from [1]:
sup {ω⊆[ 0,|D|] :|ω|=γ}
Z
ωF(t)dt=
Z |ω|
0
F∗(s)ds.
Now, fixs∈(0,|D|), and apply Step 3 to obtain
sup {ω⊆[ 0,|D|] :|ω|=s}
Z
ω
F(t)dt=
Z s
0
On the other hand, from Step 2, we have:
Z
ωF(t)dt
≤
Z |ω|
0
f∗(s)ds. (9)
From (8) and (9) we deduce
Z s
0
F∗(t)dt≤
Z s
0
f∗(t)dt,
as desired.
Corollary 5. Suppose the hypotheses of Theorem 4 hold. Then, there exists a
se-quence of functions{Fn}such that F∗n(s)=f∗(s)and Fn*F in Lp(0,|D|).
Proof. By Ryff’s result, f = f∗◦φ, for someφ∈ M(D,[0,|D|]). From Theorem 4,
we inferF◦φ∈ R(f). So, there exists a sequence{fn} ⊆ R(f) such that fn *F◦φ
inLp(D). Therefore, f
n◦φ−1 * F inLp(0,|D|). Clearly, λfn◦φ−1(α) = λf(α), so
(fn◦φ−1)∗(s)= f∗(s). This completes the proof.
3
Applications
In this section we will present a couple of applications of the results of the previous section. Throughout we will assume the extra condition f ∈C(D). Let us consider the following Hamilton-Jacobi system:
| ∇u|= f(x), inD
u=0 on∂D. (10)
Lemma 6. The system (10) has a strong positive solution u∈W1,∞(D).
Proof. From [10] we know that the system (10) has a strong solutionu ∈W1,∞(D).
Replacinguby|u| ∈W1,∞(D) if necessary, taking into account that| ∇(|u|)|=| ∇u|, we can assumeuis non-negative. On the other hand, since f is positive, we can apply Lemma 7.7 in [7], to ensure that the level sets{u=γ}have zero measure. Thus,uis
essentially positive, as desired.
Remark1. Forf anduas in Lemma 6, the function:
ξ(s)=Z
{u≥t}
f(x)dx, (wheres=λu(t)),
Our first application is as follows:
Theorem 7. Let u ∈ W1,∞(D)be a strong positive solution of the Hamilton-Jacobi
system (10) and let v be the unique solution of the following system:
| ∇Z|=F(ωn|x|n), in B
Z =0, on∂B, (11)
in which:
• B is the ball centred at the origin with radius(|D|/ωn)1/n, andωnindicates the
volume of the unit n-dimensional ball.
• The function F is as in Corollary 3, which is well defined by Remark 1.
Also, let u](x) ≡ u∗(ωn|x|n), which in the literature is referred to as the Schwarz
symmetrization of u. Then, u](x)≤v(x), for x∈B.
Proof. The proof is a consequence of Corollary 3, along the same lines as in the
proof of Lemma 2.2 in [6].
Example 1. Choosing f(x)=1 in Theorem 7 yieldsF(t)=1. Thus, the conclusion
of Theorem 7 states:
u](x)≤v(x)=R− |x|, x∈B,
whereR=(|D|/ωn)1/n. This estimate can be obtained directly as follows:
λu(t)=
Z
{u≥t} dx=
Z
{u≥t}
| ∇u|dx
=Z kuk∞ t
Z
{u=τ} dHn−1
!
dτ=Z kuk∞ t
P({u≥τ})dτ,
(12)
where we have used the co-area formula (e. g. see [11]). Here,P(E) stands for the perimeter ofEin the sense of De Giorgi. By differentiating (12), and applying the classical Isoperimetric Inequality (e. g. see [8]), we derive:
λ0
u(t)=−P({u≥t})≤ −nω
1
n nλ
1−1
n u (t).
Thus, we obtain:
1≤ − λ
0
u(t)
nω1n nλ
1−1
n u (t)
Integrating (13) from 0 totleads to:
t≤ − 1
nω1/n n
Z t
0
λ0
u(τ)
λ1−1
n u (τ)
dτ=− 1
nω1/n n
Z λu(t) |D|
ds s1−1
n
= 1
ω1/n n
(|D|1/n−λu1/n(t))=R− λu(t)
ωn
!1/n
.
(14)
By lettingt =u∗(ωn|x|n) in (14), and recallingλu(u∗(ωn|x|n))=ωn|x|n, we obtain
u](x)≤R− |x|forx∈B, as expected.
The second application is stated in the following Theorem:
Theorem 8. Let u be as in Theorem 7. Then
kuk∞≤C|D|1/nkfk∞.
Proof. The proof is a consequence of Corollary 5, along the same lines as in the
proof of Corollary 2.1 in [6].
Acknowledgment. The authors wish to thank the referees for their constructive
cri-tique of the first draft.
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