Volume 2011, Article ID 496417, 16 pagesdoi:10.1155/2011/496417 Research Article Existence of Solutions for a Nonlinear Elliptic Equation with General Flux Term Hee Chul Pak Department o
Trang 1Volume 2011, Article ID 496417, 16 pages
doi:10.1155/2011/496417
Research Article
Existence of Solutions for a Nonlinear Elliptic
Equation with General Flux Term
Hee Chul Pak
Department of Applied Mathematics, Dankook University, Cheonan, Chungnam 330-714, Republic of Korea
Correspondence should be addressed to Hee Chul Pak,hpak@dankook.ac.kr
Received 25 September 2010; Revised 29 January 2011; Accepted 27 February 2011
Academic Editor: D R Sahu
Copyrightq 2011 Hee Chul Pak This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
We prove the existence of solutions for an elliptic partial differential equation having more
general flux term than either p-Laplacian or flux term of the Leray-Lions type conditions:
−n
j1∂/∂x j α|u x j |/u x j f Brouwer’s fixed point theorem is one of the fundamental tools
of the proof
1 Introduction
We are concerned with problems of partial differential equations such as a nonlinear elliptic equation
which contains the flux termJ The flux term J is a vector field that explains a movement of
some physical contents u such as temperature, chemical potential, electrostatic potential, or
fluid flows Physical observations tell us in general thatJ depends on u and approximately on
its gradient at each point x, that is, J Jx, ∇, ux For linear cases, one can simply represent
J as J c∇u on isotropic medium or J A∇u with a square matrix A on an isotropic
medium But for nonlinear cases, the situation can be much more complicated One of the common assumptions is thatJ |∇u| p−2∇u is to produce the p-Laplacian
Δp u : ∇ · |∇u| p−2∇u. 1.2
Trang 2Slightly more general conditions, for example, the Leray-Lions type conditions, might be placed onJ, but it is too good to be true that the flux term has those kinds of growth conditions
in reality
We prove the existence of solutions for the elliptic partial differential equation
−n
j1
∂
∂x j
⎛
⎜αu
x j
u x j
⎞
⎟
Galerkin’s approximation method and Brouwer’s fixed point theorem are employed for the proof
We introduce a new function space which is designed to handle solutions of nonlinear equations1.3 This space is arisen from a close look at the L p-normf L p X |fx| p dμ1/p
of the classical Lebesgue spaces L p X, 1 ≤ p < ∞ It can be rewritten as
f
L p : α−1
X
αf xdμ, with αx : x p 1.4
Even though the positive-real-variable function αx : x phas very beautiful and convenient algebraic and geometric properties, it also has some practical limitations to handle general nonlinear problems The new space is devised to overcome these limitations without hurting
the beauty of L p-norm too much Unfortunately the new space is only equipped with an
nonlinear problems such as1.3, the homogeneity property may not be an essential factor—
we try to explain that the new space accommodates the solutions of nonlinear problems without homogeneity
Although this space is similar to the Orlicz spaces, we present a di fferent approach of
discovering the new spaces which generalize the space L p
2 The Space LαX
We introduce some terminologies to define the Lebesgue-type function spaces L α X In the
following,X, M, μ always represents a given measurable space.
2.1 H ¨older’s Functions
A pre-H¨older’s function α :R → RR {x ∈ R : x ≥ 0} is an absolutely continuous bijective function satisfying α0 0 If there exists a pre-H¨older’s function β and λ satisfying
and c1x < λ x ≤ c2x x ∈ R for some constants 1 ≤ c1 < c2, then β is called the conjugate
pre-H¨older’s function of α linked by λ In the relation 2.1, the notations α−1, β−1are meant to
be the inverse functions of α, β, respectively Examples of pre-H ¨older’s pairs are αx, βx
x p , x q , p > 1, 1/p1/q 1 with λx x and αx, βx e x −x−1, 1x log1x−x.
Trang 3In fact, for any Orlicz N-function A together with complementary N-function A and any
positive constant c ≤ 1, cA, c A is a pre-H¨older’s pair with λx 1/cA−1x A−1x see
page 264 in1
Some basic identities for a pre-H ¨older’s pairα, β with respect to λ are in order: for
α : λ ◦ α, and β : λ ◦ β,
λ x
α−1x
or αx β
αx
x
x αx
β−1αx or
αx
x β−1◦ αx β−1◦ αx, 2.3
α−1x x
β−1x
α
α−1x
α−1x
β
y
x
β
y λ αx, for y : αx
α x αx
x αx
lim
x→ 0
λ αx
x→ ∞
λ αx
Let α be a given pre-H ¨older’s function For every link-function λ satisfying
lim
x→ 0
λ x
α−1x 0, xlim→ ∞
λ x
there exists a conjugate function β of α associated with λ.
In the following discussion, a functionΦ represents the two-variable function on R×
Rdefined by
Φx, y
: α−1xβ−1
y
provided that a pre-H ¨older’s pairα, β exists.
for a link-function λ is said to be a H ¨older’s function if for any positive constants a, b > 0, there exist constants θ1, θ2depending on a, b such that
Trang 4and that a comparable condition
Φx, y
≤ θ1
ab
λ ◦ αa x θ2
ab
holds for allx, y ∈ R× R
The following proposition and the proof may illustrate that the comparable condition
2.12 is not farfetched
Proposition 2.3 Let α be a convex pre-H¨older’s function with the convex conjugate function β.
Suppose that, for any a, b ≥ 0, there are constants p1, p2, q1, q2(depending on a, b) with 1/p11/p2≤
1≤ 1/q1 1/q2satisfying the slop conditions
p1λ ◦ αa
a ≤ α a ≤ q1α a
p2λ ◦ βb
b ≤ β b ≤ q2
β b
2.13
Then α is, in fact, a H¨older’s function (so is β).
z Φx
a, b
x − a Φ y
a, b
y − b Φa, b
β
−1
b
α
α−1a x − a
α−1a
β
β−1
b y − b α−1aβ−1
b
≡ Tx, y
.
2.14
Then, for α−1a : a and β−1b : b, Tx, y can be rewritten as
T
x, y
b
α a x
a
β b y ab −
bα a
aβ b
From the slop conditions2.13 together with the observation that
bα a
aβ b
1
q1ab 1
we have
T
x, y
≤ 1
p1
ab
λ ◦ αa x
1
p2 ab
Trang 5Since the restriction z Tx, a of the tangent plane z Tx, y is the tangent line to the
graphΦx, a aα−1x located inside x-z plane and α−1 is concave up onR, we observe
Φx, a ≤ Tx, a, which holds for all a Therefore, we conclude that
Φx, y
≤ θ1
ab
λ ◦ αa x θ2
ab
where we set θ1: 1/p1and θ2: 1/p2
Remark 2.4 We want to address the point that the convexity of pre-H ¨older’s functions is not
essential in the definition of H ¨older’s functions, which is different from the definition of the Orlicz spaces
The notationsα : λ ◦ α, β : λ ◦ β are used throughout the paper.
2.2 Basic Properties of the Space LαX
We now define the Lebesgue-Orlicz type function spaces L α X:
L α X :f : f is a measurable function on X satisfyingf
L α <∞, 2.19
where we set
f
L α : α−1
X
αf xdμ. 2.20
H ¨older-type inequality and Minkowski’s inequality on the new space L α X are
presented as follows
Remark 2.5 Let α be a H ¨older’s function, and let β be the corresponding H ¨older’s conjugate
function Then, for any f ∈ L α X and any g ∈ L β X, we have
X
f xgxdμ
≤f
L αg
and, for any f1, f2 ∈ L α X, we have
f1 f2
L α ≤f1
L αf2
The space L α X is a topological vector space with inhomogeneous norm · L α
Further-more, for k≥ 0,
k−1−1f
L ≤kf
L f
Trang 6the proof Also, the metric space Lα X is complete with respect to the metric
d
f, g :f − g
L α , for f, g ∈ L α X. 2.24
convexity assumption as pointed inRemark 2.4and by the choice of the conjugate function
In fact, for Orlicz space L A X, the complementary N-function A of A is designed to satisfy
the relation
A A−1
which implies, in turn,
c1x ≤ A−1x A−1x ≤ c2x 2.26
for some constants c1, c2 > 0see, e.g., page 265 in 1 Hence, the conjugate relation 2.1 is
devised so that the space L α X contains Orlicz spaces with the Δ2-condition
Whereas the Luxemburg norm for the Orlicz space L A X requires the convexity of the N-function A for the triangle inequality of the norm, the inhomogeneous norm for the space L α X does not ask the convexity of H¨older’s function, and it has indeed inherited the
beautiful and convenient properties from the classical Lebesgue’s norm1.4
Here we present some remarks on the dual space of L α X To each g ∈ L β X is
associated a bounded linear functionalFg on L α X by
Fg
f :
X
and the operatorinhomogeneous norm of Fgis at mostg L β:
Fg
L α : sup X fgdμ
f
L α
: f ∈ L α X, f / 0
≤g
L β 2.28
For 0 / g ∈ L β X, if we put fx : β|gx| sgngx/|gx|, then we have that f ∈ L α X
and
Fg
L α sup X fgdμ
f
L α
: f ∈ L α X, f / 0
≥ X fgdμ
f
L α
g
L β 2.29
This implies that the mapping g → Fg is isometric from L β X into the space of continuous linear functionals L α X Furthermore, it can be shown that the linear transformationF :
L β X → L α X is onto the following.
Trang 7Remark 2.7 dual space of L α X Let β be the conjugate H¨older’s function of a H¨older’s function α Then the dual space L α X is isometrically isomorphic to L β X.
2.3 Sobolev-Type Space W1
α
LetΩ be an open subset of Rn The Sobolev-type space W1
αΩ is employed by
W α1Ω :u ∈ L α Ω | ∂ x j u ∈ L α Ω, j 1, 2, , n 2.30 together with the norm
u W1: u L α α−1
⎛
⎝n
j1
α
∂ x j u
L α
⎞
where ∂ x j : ∂/∂xj Then it can be shown that the function space W1
αΩ is a separable
complete metric space and C∞Ω ∩ W1
α Ω is dense in W1
αΩ The proofs are very similar to the case of Orlicz spacessee page 274 in 1
The completion of the space C∞c Ω with respect to the norm · W1 is denoted by
α,0 Ω, where C∞
cΩ is the space of smooth functions with compact support
We are in the position of introducing the trace operator and Poincar´e’s inequality on
αΩ, which are important by themselves and also useful for the proof of the existence
theorem We say that a pre-H ¨older function β is to satisfy a slope condition if there exists some positive constant c > 1 for which
β x ≥ c βx
holds for almost every x > 0 The slope condition 2.32, in fact, corresponds to the Δ2 -condition for Orlicz spacespage 266 in 1
The boundary trace on C∞c Ω can be extended to the space W1
αΩ as follows For the caseΩ Rn
: {x, x n : x ∈ Rn−1, x n > 0 } and for a smooth function φ ∈ C∞
cRn
, we observe
αφ
x , 0 −∞
0
∂ x n αφ
x , x ndx n
≤
∞
0
αφ
x , x n∂ x
n φ
x , x ndx n
≤∂ x
n φ
x ,·
L α 0,∞ αφ
x ,·
L β 0,∞
2.33
Owing to the identity2.6, we have
α t s t α t
β s , s αt
Trang 8On the other hand, we can notice that the slope condition2.32 is equivalent to saying
β
αt
t
Reflecting this to the identity2.34, we have
αφ x ≤ c
c− 1
αφ x
φ x , x x , x n
Therefore, we have
β−1∞
0
βαφ
x , x ndx n≤ β−1∞
0
β
c
c− 1
αφ x , x n
φ x , x n
dx n
≤
c
c− 1
β−1∞
0
β
αφ x , x n
φ x , x n
dx n
c
c− 1
β−1∞
0
αφ
x , x ndx n.
2.37
Inserting this into the right side of2.33, we obtain
αφ
x , 0 ≤ C∂ x n φ
x ,·
L α 0,∞ β−1◦ αφx
,·
L α 0,∞
≤ C α∂
x n φ
x ,·
L α 0,∞
αφx
,·
L α 0,∞
,
2.38
for some positive constant C The comparable condition2.12 has been used in the second inequality Taking integration on both sides overRn−1, we obtain
αφ
L αRn−1
≤ C α∂
x n φ
L αRn
αφ
L αRn
This inequality says that the trace on C∞c Rn
can be uniquely extended to the space W1
αRn
as a metric space For the case Ω being a bounded open subset Ω can be more general,
e.g., it permits unbounded domains satisfying the uniform C m-regularity condition page
84 in1 , the partitions of unity can be employed to turn the case locally into that of Rn
with appropriate Jacobians Gluing a finite number of estimates2.39, we get the following propositionfor details, see page 164 in 1 or page 56 in 2
Proposition 2.8 Trace map on W1
α Let α, β be a H¨older pair obeying the slope condition 2.32,
and let Ω be a bounded open set with smooth boundary in R n Then the trace operator γ : W1
αΩ →
L α ∂Ω is continuous and uniquely determined by γu u| ∂Ωon those u ∈ C∞
c Ω.
We present Poincar´e’s inequality whose proof can be found in the appendix
Trang 9Proposition 2.9 Poincar´e’s inequality Let α, β be a H¨older pair with the slope condition 2.32,
and let Ω be an open set in R n which is bounded in some direction; that is, there is a vector v ∈ Rn
such that
sup{|x · v| : x ∈ Ω} < ∞ 2.40
Then there is a constant C > 0 such that, for any f ∈ W1
α Ω with fx 0 (in the sense of the trace
map) for x ∈ ∂Ω and x · v / 0,
f
L α ≤ Cv · ∇f
3 Nonlinear Elliptic Equations of General Flux Terms
In this sectionΩ is a fixed bounded open set in Rnwith smooth boundary We are concerned with an elliptic partial differential equation:
where the flux vector field is given by
Ju : α|∂ x1u|
∂ x1u ,
α|∂ x2u|
∂ x2u , ,
α|∂ x n u|
∂ x n u
We look for solutions of the elliptic equation 3.1 on an appropriate space In fact, the function space that can permit solutions of3.1 turns out to be the space W1
α,0 Ω : V
Now, we state the existence theorem of the nonlinear elliptic equation with general flux term3.1
Theorem 3.1 Let α, β be a H¨older pair satisfying the slope condition 2.32 Then, for any
functional f ∈ V , there exists a solution u ∈ V satisfying the elliptic partial differential equation
−n
j1
∂
∂x j
⎛
⎜αu
x j
u x j
⎞
⎟
We start to set up the functional equation associated with3.1 Let φ ∈ C∞
c Ω Then
we have
−
Ω∇ · J∇uφ dμ
Then, by Gauss-Green theorem, the left-hand side becomes
−
Ω∇ · J∇uφ dμ
ΩJ∇u · ∇φ dμ
Ω
n
j1
α∂
x j u
Trang 10We will consider the operatorA defined by
Auφ
:n
j1
Ω
αu
x j
u x j φ x j dμn
j1
Ωβ−1◦ αu
x j
φ x j dμ, 3.6
for u, v ∈ V : W1
α,0 Ω We investigate some properties of the operator A : V → V which will be used for the proof of the existence theorem
Lemma 3.2 One has an estimate: for u ∈ V ,
Auφ ≤ β−1◦ αu W1
·φ
In particular, the operator A : V → V is bounded; that is, for any bounded set S in V , the image
AS of S is bounded in V
Proof By H ¨older’s inequality and identity2.2, we have
Auφ ≤n
j1
Ω
α∂
x j u
∂ x j u ∂ x j φ dμ
≤n
j1
β−1
⎛
⎜
Ωβ
⎛
⎜α∂
x j u
∂ x j u
⎞
⎟
⎠dμ
⎞
⎟
⎠α−1
Ωα∂
x j φ
dμ
n
j1
β−1
Ωα∂
x j u
∂ x j φ
L α
n
j1
β−1◦ α
∂ x j u
L α
∂ x j φ
L α
.
3.8
H ¨older’s inequality with respect to the counting measure reads as for a j , b j > 0
n
j1
a j b j ≤ α−1
⎛
⎝n
j1
αa j⎞⎠β−1
⎛
⎝n
j1
βb j
⎞
Apply this inequality to3.8, and we get
Auφ ≤ β−1
⎛
⎝n
j1
α
∂ x j u
L α
⎞
⎠α−1
⎛
⎝n
j1
α
∂ x j φ
L α
⎞
⎠
≤ β−1◦ αu W1
·φ
W1.
3.10
Since α and β−1are continuous onR, the operatorA is bounded