Điều kiện tối ưu không cách biệt và tính ổn định nghiệm của các bài toán điều khiển tối ưu được cho bởi các phương trình elliptic nửa tuyến tính

MINISTRY OF EDUCATION AND TRAINING
HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY

NGUYEN HAI SON

NO-GAP OPTIMALITY CONDITIONS
AND SOLUTION STABILITY
FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY
SEMILINEAR ELLIPTIC EQUATIONS

DOCTORAL DISSERTATION OF MATHEMATICS

Hanoi - 2019


MINISTRY OF EDUCATION AND TRAINING
HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY

NGUYEN HAI SON

NO-GAP OPTIMALITY CONDITIONS
AND SOLUTION STABILITY
FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY
SEMILINEAR ELLIPTIC EQUATIONS

Major: MATHEMATICS
Code: 9460101

DOCTORAL DISSERTATION OF MATHEMATICS

SUPERVISORS:

1. Dr. Nguyen Thi Toan
2. Dr. Bui Trong Kien

Hanoi - 2019


COMMITTAL IN THE DISSERTATION
I assure that my scientific results are new and righteous. Before I published these
results, there had been no such results in any scientific document. I have responsibilities for my research results in the dissertation.
Hanoi, April 3rd , 2019
On behalf of Supervisors

Author

Dr. Nguyen Thi Toan

Nguyen Hai Son

i


ACKNOWLEDGEMENTS
This dissertation has been carried out at the Department of Fundamental Mathematics, School of Applied Mathematics and Informatics, Hanoi University of Science
and Technology. It has been completed under the supervision of Dr. Nguyen Thi Toan
and Dr. Bui Trong Kien.
First of all, I would like to express my deep gratitude to Dr. Nguyen Thi Toan and
Dr. Bui Trong Kien for their careful, patient and effective supervision. I am very lucky
to have a chance to work with them, who are excellent researchers.
I would like to thank Prof. Jen-Chih Yao for his support during the time I visited and
studied at Department of Applied Mathematics, Sun Yat-Sen University, Kaohsiung,

Taiwan (from April, 2015 to June, 2015 and from July, 2016 to September, 2016). I
would like to express my gratitude to Prof. Nguyen Dong Yen for his encouragement
and many valuable comments.
I would also like to especially thank my friend, Dr. Vu Huu Nhu for kind help and
encouragement.
I would like to thank the Steering Committee of Hanoi University of Science and
Technology (HUST), and School of Applied Mathematics and Informatics (SAMI) for
their constant support and help.
I would like to thank all the members of SAMI for their encouragement and help.
I am so much indebted to my parents and my brother for their support. I thank my
wife for her love and encouragement. This dissertation is a meaningful gift for them.
Hanoi, April 3rd , 2019
Nguyen Hai Son

ii


CONTENTS
. . . . . . . . . . . . . . . . . . . . . .

i

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ii

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iii

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

COMMITTAL IN THE DISSERTATION
ACKNOWLEDGEMENTS
CONTENTS

TABLE OF NOTATIONS
INTRODUCTION

Chapter 0.
0.1

0.2

0.3

PRELIMINARIES AND AUXILIARY RESULTS

8

Variational analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

0.1.1

Set-valued maps . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

0.1.2

Tangent and normal cones . . . . . . . . . . . . . . . . . . . . .

9

Sobolev spaces and elliptic equations . . . . . . . . . . . . . . . . . . .

13

0.2.1

Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

0.2.2

Semilinear elliptic equations . . . . . . . . . . . . . . . . . . . .

20

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

Chapter 1.

NO-GAP OPTIMALITY CONDITIONS FOR DISTRIBUTED CONTROL

PROBLEMS

25

1.1

Second-order necessary optimality conditions . . . . . . . . . . . . . . .

26

1.1.1

An abstract optimization problem . . . . . . . . . . . . . . . . .

26

1.1.2

Second-order necessary optimality conditions for optimal control
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

1.2

Second-order sufficient optimality conditions . . . . . . . . . . . . . . .

40

1.3

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

Chapter 2.

NO-GAP OPTIMALITY CONDITIONS FOR BOUNDARY CONTROL

PROBLEMS

58

2.1

Abstract optimal control problems . . . . . . . . . . . . . . . . . . . . .

59

2.2

Second-order necessary optimality conditions . . . . . . . . . . . . . . .

66

2.3

Second-order sufficient optimality conditions . . . . . . . . . . . . . . .

75

2.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

Chapter 3.

UPPER SEMICONTINUITY AND CONTINUITY OF THE SOLUTION

MAP TO A PARAMETRIC BOUNDARY CONTROL PROBLEM

91

3.1

Assumptions and main result . . . . . . . . . . . . . . . . . . . . . . .

92

3.2

Some auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

iii


3.2.1

Some properties of the admissible set . . . . . . . . . . . . . . .

94

3.2.2

First-order necessary optimality conditions

98

. . . . . . . . . . .

3.3

Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.4

Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

GENERAL CONCLUSIONS
LIST OF PUBLICATIONS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

iv


TABLE OF NOTATIONS

N := {1, 2, . . .}
R

set of positive natural numbers

|x|

absolute value of x ∈ R

Rn

n-dimensional Euclidean vector space

∅

empty set

x∈A

x is in A

x∈
/A

x is not in A

A ⊂ B(B ⊃ A)

A is a subset of B

A

A is not a subset of B

B

set of real numbers

A∩B

intersection of the sets A and B

A∪B

union of the sets A and B

A\B

set difference of A and B

A×B

Descartes product of the sets A and B

[x1 , x2 ]

the closed line segment between x1 and x2

x
x

norm of a vector x
norm of vector x in the space X

X

X∗

topological dual of a normed space X

X ∗∗

topological bi-dual of a normed space X

x∗ , x

canonical pairing

x, y

canonical inner product

B(x, δ)

open ball with centered at x and radius δ

B(x, δ)

closed ball with centered at x and radius δ

BX

open unit ball in a normed space X

BX

closed unit ball in a normed space X

dist(x; Ω)

distance from x to Ω

{xk }

sequence of vectors xk

xk → x

xk converges strongly to x (in norm topology)

xk

xk converges weakly to x

x

∀x

for all x

∃x

there exists x

A := B

A is defined by B

f :X→Y

function from X to Y

f (x), ∇f (x)

Fr´echet derivative of f at x

f (x), ∇2 f (x)

Fr´echet second-order derivative of f at x

1


Lx , ∇x L

Fr´echet derivative of L in x

Lxy , ∇2xy L

Fr´echet second-order derivative of L in xand y

ϕ : X → IR

extended-real-valued function

domϕ

effective domain of ϕ

epiϕ

epigraph of ϕ

suppϕ

the support of ϕ

F :X⇒Y

multifunction from X to Y

domF

domain of F

rgeF

range of F

gphF

graph of F

kerF

kernel of F

T (K, x)

Bouligand tangent cone of the set K at x

T (K, x)

adjoint tangent cone of the set K at x

T 2 (K, x, d)

second-order Bouligand tangent set of the set
K at x in direction d

T2

(K, x, d)

second-order adjoint tangent set of the set K
at x in direction d

N (K, x)

normal cone of the set K at x

∂Ω
¯
Ω

boundary of the domain Ω

Ω ⊂⊂ Ω

Ω ⊂ Ω and Ω is compact.

Lp (Ω)

the space of Lebesgue measurable functions f

closure of the set Ω

and

Ω

|f (x)|p dx < +∞

L∞ (Ω)
¯
C(Ω)

the space of bounded functions almost every Ω
¯
the space of continuous functions on Ω

¯
M(Ω)

the space of finite regular Borel measures


m,p
m,p


W (Ω), W0 (Ω),
W s,r (Γ),

Sobolev spaces





H m (Ω), H0m (Ω)

W −m,p (Ω)(p−1 + p −1 = 1)

the dual space of W0m,p (Ω)

X →Y

X is continuous embedded in Y

X →→ Y

X is compact embedded in Y

i.e.

id est (that is)

a.e.

almost every

s.t.

subject to

p. 5

page 5

w.r.t

with respect to

✷

The proof is complete

2


INTRODUCTION

1. Motivation
Optimal control theory has many applications in economics, mechanics and other
fields of science. It has been systematically studied and strongly developed since the
late 1950s, when two basic principles were made. One was the Pontryagin Maximum
Principle which provides necessary conditions to find optimal control functions. The
other was the Bellman Dynamic Programming Principle, a procedure that reduces
the search for optimal control functions to finding the solutions of partial differential
equations (the Hamilton-Jacobi equations). Up to now, optimal control theory has
developed in many various research directions such as non-smooth optimal control,
discrete optimal control, optimal control governed by ordinary differential equations
(ODEs), optimal control governed by partial differential equations (PDEs),...(see [1, 2,

3]).
In the last decades, qualitative studies for optimal control problems governed by
ODEs and PDEs have obtained many important results. One of them is to give optimality conditions for optimal control problems. For instance, J. F. Bonnans et al.
[4, 5, 6], studied optimality conditions for optimal control problems governed by ODEs,
while J. F. Bonnans [7], E. Casas et al. [8, 9, 10, 11, 12, 13, 14, 15, 16, 17], C. Meyer
and F. Tr¨oltzsch [18], B. T. Kien et al. [19, 20, 21, 22], A. R¨osch and F. Tr¨oltzsch
[23, 24]... derived optimality conditions for optimal control problems governed by elliptic equations.
It is known that if u¯ is a local minimum of F , where F : U → R is a differentiable
functional and U is a Banach space, then F (¯
u) = 0. This a first-order necessary
optimality condition. However, it is not a sufficient condition in case of F is not
convex. Therefore, we have to invoke other sufficient conditions and should study the
second derivative (see [17]).
Better understanding of second-order optimality conditions for optimal control problems governed by semilinear elliptic equations is an ongoing topic of research for several
researchers. This topic is great value in theory and in applications. Second-order sufficient optimality conditions play an important role in the numerical analysis of nonlinear
optimal control problems, and in analyzing the sequential quadratic programming algorithms (see [13, 16, 17]) and in studying the stability of optimal control (see [25, 26]).
Second-order necessary optimality conditions not only provide criterion of finding out
stationary points but also help us in constructing sufficient optimality conditions. Let
us briefly review some results on this topic.

3


For distributed control problems, i.e., the control only acts in the domain Ω in Rn ,
E. Casas, T. Bayen et al. [11, 13, 16, 27] derived second-order necessary and sufficient
optimality conditions for problem with pure control constraint, i.e.,
a(x) ≤ u(x) ≤ b(x) a.e. x ∈ Ω,

(1)

and the appearance of state constraints. More precisely, in [11] the authors gave
second-order necessary and sufficient conditions for Neumann problems with constraint
(1) and finitely many equalities and inequalities constraints of state variable y while
the second-order sufficient optimality conditions are established for Dirichlet problems
with constraint (1) and a pure state constraint in [13]. T. Bayen et al. [27] derived
second-order necessary and sufficient optimality conditions for Dirichlet problems in the
sense of strong solution. In particular, E. Casas [16] established second-order sufficient
optimality conditions for Dirichlet control problems and Neumann control problems
with only constraint (1) when the objective function does not contain control variable u.
In [18], C. Meyer and F. Tr¨oltzsch derived second-order sufficient optimality conditions
for Robin control problems with mixed constraint of the form a(x) ≤ λy(x) + u(x) ≤
b(x) a.e. x ∈ Ω and finitely many equalities and inequalities constraints.
For boundary control problems, i.e., the control u only acts on the boundary Γ, E.
Casas and F. Tr¨oltzsch [10, 12] derived second-order necessary optimality conditions
while the second-order sufficient optimality conditions were established by E. Casas et
al. in [12, 13, 17] with pure pointwise constraints, i.e.,
a(x) ≤ u(x) ≤ b(x)

a.e. x ∈ Γ.

A. R¨osch and F. Tr¨oltzsch [23] gave the second-order sufficient optimality conditions
for the problem with the mixed pointwise constraints which has unilateral linear form
c(x) ≤ u(x) + γ(x)y(x) for a.e. x ∈ Γ.
We emphasize that in above papers, a, b ∈ L∞ (Ω) or a, b ∈ L∞ (Γ). Therefore,
the control u belongs to L∞ (Ω) or L∞ (Γ). This implies that corresponding Lagrange
multipliers are measures rather than functions (see [19]). In order to avoid this disadvantage, B. T. Kien et al. [19, 20, 21] recently established second-order necessary
optimality conditions for distributed control of Dirichlet problems with mixed statecontrol constraints of the form
a(x) ≤ g(x, y(x)) + u(x) ≤ b(x) a.e x ∈ Ω
with a, b ∈ Lp (Ω), 1 < p < ∞ and pure state constraints. This motivates us to develop
and study the following problems.

(OP 1) : Establish second-order necessary optimality conditions for Robin boundary
control problems with mixed state-control constraints of the form
a(x) ≤ g(x, y(x)) + u(x) ≤ b(x) a.e. x ∈ Γ,
4


where a, b ∈ Lp (Γ), 1 < p < ∞.
(OP 2) : Give second-order sufficient optimality conditions for optimal control problems with mixed state-control constraints when the objective function does not depend
on control variables.
Solving problems (OP 1) and (OP 2) is the first goal of the dissertation.
After second-order necessary and sufficient optimality conditions are established,
they should be compared to each other. According to J. F. Bonnans [4], if the change
between necessary and sufficient second-order optimality conditions is only between
strict and non-strict inequalities, then we say that the no-gap optimality conditions are
obtained. Deriving second-order optimality conditions without a gap between secondorder necessary optimality conditions and sufficient optimality conditions, is a difficult
problem which requires to find a common critical cone under which both second-order
necessary optimality conditions and sufficient optimality conditions are satisfied. In [7],
J. F. Bonnans derived second-order necessary and sufficient optimality conditions with
no-gap for an optimal control problem with pure control constraint and the objective
function is quadratic in both state variable y and control variable u. The result in
[7] was established by basing on polyhedric property of admissible sets and the theory
of Legendre forms. Recently, the result has been extended by [27] and [28]. However,
there is an open problem in this area. Namely, we need to study the following problem:
(OP 3) : Find a theory of no-gap second-order optimality conditions for optimal control problems governed by semilinear elliptic equations with mixed pointwise constraints.
Solving problem (OP 3) is the second goal of this dissertation.
Solution stability of optimal control problem is also an important topic in optimization and numerical method of finding solutions (see [25, 29, 30, 31, 32, 33, 34, 35, 36,
37, 38, 39, 40, 41]). An optimal control problem is called stable if the error of the
output data is small in some sense for a small change in the input data. The study of
solution stability is to investigate continuity properties of solution maps in parameters
such as lower semicontinuity, upper semicontinuity, H¨older continuity and Lipschitz

continuity.
Let us consider the following parametric optimal problem:
P (µ, λ)


F (y, u, µ) → inf,

(2)

(y, u) ∈ Φ(λ),
where y ∈ Y, u ∈ U are state and control variables, respectively; µ ∈ Π, λ ∈ Λ are
parameters, F : Y × U × Π → R is an objective function on Banach space Y × U × Π
and Φ(λ) is an admissible set of the problem.
It is well-known that if the objective function F (·, ·, µ) is strongly convex, and the
admissible set Φ(λ) is convex, then the solution map of problem (2) is single-valued (see
[29], [30], [31]). Moreover, A. Dontchev [30] showed that under some certain conditions,
5


the solution map is Lipschitz continuous w.r.t. parameters. By using implicit function
theorems, K. Malanowski [35]-[40] proved that the solution map of problem (2) is also a
Lipschitz continuous function in parameters if weak second-order optimality conditions
and standard constraint qualifications are satisfied at the reference point. Notice that
the obtained results in [37]-[40] are for problems with pure state constraints, while the
one in [35] is for problems with pure control constraints.
When the conditions mentioned above are invalid, the solution map may not be
singleton (see [32, 33]). In this situation, we have to use tools of set-valued analysis and
variational analysis to deal with the problem. In 2012, B. T. Kien et al. [32] and [33]
obtained the lower semicontinuity of the solution map to a parametric optimal control
problem for the case where the objective function is convex in both variables and the

admissible sets are also convex. Recently, the upper semicontinuity of the solution map
has been given by B. T. Kien et al. [34] and V. H. Nhu [42] for problems, where the
objective functions may not be convex in the both variables and the admissible sets
are not convex. Notice that in [34] the authors considered the problem governed by
ordinary differential equations meanwhile in [42] the author investigated the problem
governed by semilinear elliptic equation with distributed control. From the above, one
may ask to study the following problem:
(OP 4) : Establish sufficient conditions under which the solution map of parametric
boundary control problem is upper semicontinuous and continuous.
Giving a solution for (OP 4) is the third goal of this dissertation.
2. Objective
The objective of this dissertation is to study no-gap second-order optimality conditions and stability of solution to optimal control problems governed by semilinear
elliptic equations with mixed pointwise constraints. Namely, the main content of the
dissertation is to concentrate on
(i) establishing second-order necessary optimality conditions for boundary control
problems with the control variables belong to Lp (Γ), 1 < p < ∞;
(ii) deriving second-order sufficient optimality conditions for distributed control problems and boundary control problems when objective functions are quadratic forms
in the control variables, and showing that no-gap optimality condition holds in
this case;
(iii) deriving second-order sufficient optimality conditions for distributed control problems and boundary control problems when objective functions are independent of
the control variables, and showing that in general theory of no-gap conditions does
not hold;
(iv) giving sufficient conditions for a parametric boundary control problem under which
6


the solution map is upper semicontinuous and continuous in parameters.
3. The structure and results of the dissertation
The dissertation has four chapters and a list of references.
Chapter 0 collects several basic concepts and facts on variational analysis, Sobolev

spaces and partial differential equations.
Chapter 1 presents results on the no-gap second-order optimality conditions for
distributed control problems.
Chapter 2 provides results on the no-gap second-order optimality conditions for
boundary control problems.
The obtained results in Chapters 1 and 2 are answers for problems (OP 1), (OP 2)
and (OP 3), respectively.
Chapter 3 presents results on the upper semicontinuity and continuity of the solution
map to a parametric boundary control problem, which is a positive answer for problem
(OP 4).
Chapter 1 and Chapter 2 are based on the contents of papers [1] and [2] in the
List of publications which were published in the journals Set-Valued and Variational
Analysis and SIAM Journal on Optimization, respectively. The results of Chapter 3
were content of article [3] in the List of publications which is published in Optimization.
These results have been presented at:
• The Conference on Applied Mathematics and Informatics at Hanoi University of
Science and Technology in November 2016.
• The 15th Conference on Optimization and Scientific Computation, Ba Vi in April
2017.
• The 7th International Conference on High Performance Scientific Computing in
March 2018 at Vietnam Institute for Advanced Study in Mathematics (VIASM).
• The 9th Vietnam Mathematical Congress, Nha Trang in August 2018.
• Seminar ”Optimization and Control” at the Institute of Mathematics, Vietnam
Academy of Science and Technology.

7


Chapter 0
PRELIMINARIES AND AUXILIARY RESULTS

In this chapter, we review some background on Variational Analysis, Sobolev spaces,
and facts of partial differential equations relating to solutions of linear elliptic equations
and semilinear elliptic equations. For more details, we refer the reader to [1], [2], [3],
[27], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], and [56] .

0.1

Variational analysis

0.1.1

Set-valued maps

Let X and Y be nonempty sets. A set-valued map/multifunction F from X to Y ,
denoted by F : X ⇒ Y , which assigns for each x ∈ X a subset F (x) ⊂ Y . F (x) is
called the image or the value of F at x.
Let F : X ⇒ Y be a set-valued map between topological spaces X and Y . We call
the sets
gph(F ) := (x, y) ∈ X × Y | y ∈ F (x)
dom(F ) := x ∈ X | F (x) = ∅ ,
rge(F ) := y ∈ Y | y ∈ F (x) for some x ∈ X :=

F (x)
x∈X

the graph, the domain and the range of F , respectively.
The inverse F −1 : Y ⇒ X of F is the set-valued map, defined by
F −1 (y) := {x ∈ X | y ∈ F (x)} for all y ∈ Y.
The set-valued map F is called proper if dom(F ) = ∅.

Definition 0.1.1. ([46, p. 34]) Let F : X ⇒ Y be a set-valued map between topological spaces X and Y .
(i) If gph(F ) is a closed subset of the topological space X × Y then F is called closed
map (or graph-closed map).
(ii) If X, Y are linear topological spaces and gph(F ) is a convex subset of the topological space X × Y then F is called convex set-valued map.
(iii) If F (x) is a closed subset of Y for all x ∈ X then F is called closed-valued map.
(iv) If F (x) is a compact subset of Y for all x ∈ X then F is called compact-valued
map.
8


The concepts of semicontinuous set-valued maps had been introduced in 1932 by G.
Bouligand and K. Kuratowski (see [44]).
Definition 0.1.2. ([45, Definition 1, p. 108] and [44, Definition 1.4.1, p.38]) Let
F : X ⇒ Y be a set-valued map between topological spaces and x0 ∈ dom(F ).
(i) F is said to be upper semicontinuous at x0 if for any open set W in Y satisfying
F (x0 ) ⊂ W , there exists a neighborhood V of x0 such that
F (x) ⊂ W

for all x ∈ V.

(ii) F is said to be lower semicontinuous at x0 if for any open set W in Y satisfying
F (x0 ) ∩ W = ∅, there exists a neighborhood V of x0 such that
F (x) ∩ W = ∅ for all x ∈ V ∩ dom(F ).
(iii) F is continuous at x0 if it is both lower semicontinuous and upper semicontinuous
at x0 .
The map F is called upper semicontinuous (resp., lower semicontinuous, continuous)
if it is upper semicontinuous (resp., lower semicontinuous, continuous) at every point
x ∈ dom(F ).
Notice that in case of single-valued map F : X → Y , the above concepts are
coincident.

When X, Y are metric spaces, set-valued map F : X ⇒ Y is lower semicontinuous
at x ∈ dom(F ) if and only if for all y ∈ F (x) and sequence {xn } ∈ dom(F ), xn → x,
there exists a sequence {yn } ⊂ Y , yn ∈ F (xn ) such that yn → y.
0.1.2

Tangent and normal cones

Let X be a normed space with the norm · . For each x0 ∈ X and δ > 0, we denote
by B(x0 , δ) the open ball {x ∈ X | x − x0 < δ}, and by B(x0 , δ) the corresponding
closed ball. We will write BX and B X for B(0X , 1) and B(0X , 1), respectively. Let D
be a nonempty subset of X. The distance from x ∈ X to D is defined by
dist(x; D) = inf x − u .
u∈D

Definition 0.1.3. ([44, Definition 4.1.1, p. 121]) Let D ⊂ X be a subset of a normed
space X and a point x ∈ D. The set
T (D, x) :=

v ∈ X | lim inf
t→0+

dist(x + tv, D)
=0 .
t

is called Bouligand (contingent) cones of D at x.

9

From Definition 0.1.3, it follows that T (D, x) is a closed cone and T (D, x) ⊂
cone(D − x), where cone(A) := {λa | λ ≥ 0, a ∈ A} is the cone generated by the set A.
Moreover, the following property characterizes the Bouligand cone:
T (D, x) = {v ∈ X | ∃tn → 0+ , ∃vn → v s.t. x + tn vn ∈ D for all n ∈ N}.
Definition 0.1.4. ([44, Definition 4.1.5, p. 126]) Let D ⊂ X be a subset of normed
space X and x ∈ D. The adjoint tangent cone or the intermediate cone T (K, x) of D
at x is defined by
T (D, x) :=

v ∈ X | lim

t→0+

dist(x + tv, D)
=0 .
t

The Clarke tangent cone TC (D, x) of D at x is defined by

TC (D, x) :=





v ∈ X | lim+

t→0
D
x−

→x








d(x + tv, D)
=0 ,
t



D

where x −
→ x means that x ∈ D and x → x.
From Definition 0.1.4, we have the following characters of the adjoint cones and the
Clarke tangent cones (see [44, p. 128]):
T (D, x) = {v ∈ X | ∀tn → 0+ , ∃vn → v s.t. x + tn vn ∈ D ∀n ∈ N},
and
D

TC (D, x) = {v ∈ X | ∀tn → 0+ , ∀xn −
→ x, ∃vn → v s.t. xn + tn vn ∈ D ∀n ∈ N}.
It is clear that
TC (D, x) ⊂ T (D, x) ⊂ T (D, x) ⊂ cone(D − x).
Example 0.1.5. ( TC (D, x) = T (D, x) = T (D, x) = cone(D − x))

Putting D = {(x1 , x2 ) | x2 = x21 } ⊂ R2 and taking x = (0, 0), we have
TC (D, x) = {(0, 0)},
T (D, x) = T (D, x) = {(x1 , 0) | x1 ∈ R},
cone(D − x) = {(x1 , x2 ) | x2 ≥ 0}.
Example 0.1.6. ( TC (D, x) = T (D, x) = T (D, x) = cone(D − x))
¯ we have
Putting D = { 1 | n = 1, 2, ...} ⊂ R and taking x = 0 ∈ D,
n

TC (D, x) = T (D, x) = {0},
T (D, x) = cone(D − x) = R+

10


Example 0.1.7. ( TC (D, x) = T (D, x) = T (D, x) = cone(D − x))
Putting D = {(x1 , x2 ) | x2 = 0} ∪ {(x1 , x2 ) | x1 = 0} ⊂ R2 and taking x = (0, 0), we
have
TC (D, x) = {(0, 0)},
T (D, x) = T (D, x) = cone(D − x) = D.
Let x1 , x2 ∈ X. The segment [x1 , x2 ] connect x1 and x2 is defined by
[x1 , x2 ] := {x ∈ X | x = αx1 + (1 − α)x2 , 0 ≤ α ≤ 1}.
A subset K of X is said to be convex if [x1 , x2 ] ⊂ K for all x1 , x2 ∈ K. (see [1,
section 0.3.1, p. 45]). According to [44, Chapter 4], the Clarke cone TC (D, x) is a
convex and closed cone while the adjoint cone T (D, x) is a closed cone. Moreover, if
D is convex then
TC (D, x) = T (D, x) = T (D, x) = cone(D − x).
The above tangent cones has important roles in the study of first-order optimality
conditions for optimal control problems with constraints. However, in order to obtain
second-order optimality conditions for optimal control problems, we need to use secondorder tangent sets.

Definition 0.1.8. ([44, Definition 1.1.1, p.

17]) Let X be a normed space and

(Dt )t∈T ⊂ X be a sequence of sets depend on parameters t ∈ T, where T is a metric
space. Suppose that t0 ∈ T. The set
Limsup Dt := {x ∈ X | lim inf dist(x, Dt ) = 0}
t→t0

t→t0

is called Painlev´e-Kuratowski upper limit of (Dt ) as t → t0 .
The set
Liminf Dt := {x ∈ X | lim dist(x, Dt ) = 0}
t→t0

t→t0

is called Painlev´e-Kuratowski lower limit of (Dt ) as t → t0 .
Definition 0.1.9. ( [44, Definition 4.7.1 and 4.7.2, p. 171]). Let D be a subset in the
¯ v ∈ X.
normed space X and x ∈ D,
The set
T 2 (D, x, v) := Limsup
t→0+

D − x − tv
t2

is called Bouligand second-order tangent set of D at x in direction v.

The set

D − x − tv
t2
t→0+
is called second-order adjoint tangent set D at x in direction v.
T 2 (D, x, v) := Liminf

11


Obviously, T 2 (D, x, v) and T 2 (D, x, v) are closed and
T 2 (D, x, 0) = T (D, x),

T 2 (D, x, 0) = T (D, x).

Moreover, we have
T 2 (D, x, v) = {w|∃tn → 0+ , ∃wn → w, x + tn v + t2n wn ∈ D},
T 2 (D, x, v) = {w|∀tn → 0+ , ∃wn → w, x + tn v + t2n wn ∈ D}.
Notice that T 2 (D, x, v) and T 2 (D, x, v) are nonempty only if v ∈ T (D, x) and v ∈
T (D, x), respectively. Moreover, when D is convex, T 2 (D, x, v) is convex but T (D, x, v)
may not be convex (see [47, Subsection 3.2.1]).
The following example shows that in general (T 2 (D, x, v) is different from T 2 (D, x, v))
(see [47, Example 3.31]).
Example 0.1.10. (T 2 (D, x, v) = T 2 (D, x, v))
Let us first construct a convex piecewise linear function x2 = ϕ(x1 ), x1 ∈ R, oscillating between two parabolas x2 = x21 and x2 = 2x21 in the following way: ϕ(x1 ) =
ϕ(−x1 ), ϕ(0) = 0 and the function ϕ(x1 ) is linear on every interval [x1,k+1 , x1,k ],
ϕ(x1,k ) = x21,k and its graph on [x1,k+1 , x1,k ] is tangent to the curve x2 = 2x21 for some
monotonically decreasing to zero sequence {x1,k }. It is evident how such a function
can be constructed. Indeed, for a given point x1,k > 0 consider the straight line passing

through the point (x1,k , x21,k ) and is tangent to the curve x2 = 2x21 . It intersects the
curve x2 = x21 at a point x1,k+1 . We can iterate this process and obtain a sequence
{x1,k }. It is easily seen that x1,k > x1,k+1 > 0 and x1,k → 0 as k → ∞.
Taking K = {(x1 , x2 ) ∈ R2 | x2 ≥ ϕ(x1 )} and x = (0, 0), v = (1, 0), we have
T 2 (D, x, v) = {(x1 , x2 ) | x2 ≥ 2} and T 2 (D, x, v) = {(x1 , x2 ) | x2 ≥ 4}.
The following result allows us to compute tangent cones of a convex and closed
subset K in Lp (Ω) with 1 ≤ p < +∞ (see Definition Lp (Ω) in next section).
Theorem 0.1.11. ([44, Theorem 8.5.1, p. 324]). Let K be a subset of Lp (Ω) such that
M (x) := {u(x) | u ∈ K} is measurable and closed in R for a.e. x ∈ Ω. Then for all
u0 ∈ K, one has
v ∈ Lp (Ω) | v(x) ∈ T (M (x), u0 (x)) a.e. x ∈ Ω
⊂ T (K, u0 ) ⊂ T (K, u0 )
⊂ {v ∈ Lp (Ω) | v(x) ∈ T (M (x), u0 (x)) a.e. x ∈ Ω} .
Corollary 0.1.12. ([27, Lemma 4.11] Let 1 ≤ p < +∞, and K := {u ∈ Lp (Ω) |
a(x) ≤ u(x) ≤ b(x) a.e. x ∈ Ω}, with a, b ∈ Lp (Ω) and u0 ∈ K. Then
T (K, u0 ) = T (K, u0 )
= v ∈ Lp (Ω) | v(x) ∈ T ([a(x), b(x)], u0 (x)) a.e. x ∈ Ω .
12


In the sequel, we shall use concept normal cone which is dual concept of Clarke
tangent cones. We denote by X ∗ the dual space of the normed space X, i.e., the space
of all continuous linear functionals on X; the (dual) norm on X ∗ is defined by
f

X∗

= sup{f (x) | x ∈ X, x ≤ 1}.

Then X ∗ is a Banach space, i.e., X ∗ is complete even if X is not (see [48, p.3]). Let

us denote by ·, · the canonical pairing between X ∗ and X.
Definition 0.1.13. ([44, Definition 4.4.2, p. 157]). Let X be a Banach space, a subset
D ⊂ X and a point x ∈ D. We shall say that the polar cone
N (D, x) := TC (D, x)− = {p ∈ X ∗ | p, v ≤ 0 ∀v ∈ TC (D, x)}
is (Clarke) normal cone of D at x.
When D is convex, N (D, x) coincides with the normal cone of D at x in Convex
Analysis, i.e.,
N (D, x) = {x∗ ∈ X ∗ | x∗ , d − x ≤ 0 ∀d ∈ D}.

0.2
0.2.1

Sobolev spaces and elliptic equations
Sobolev spaces

First, we recall some relative concepts and properties which are introduced in many
books on Sobolev spaces, elliptic equations and partial differential equations.
For any multiindex α := (α1 , α2 , ..., αN ) ∈ NN , let us denote by xα := xα1 1 xα2 2 · · · xαNN ,
with x = (x1 , x2 , ..., xN ) ∈ RN , a monomial of order |α| =

N
i=1 αi ,

and denote by

αN
Dα := D1α1 D2α2 · · · DN

a differential operator of order |α|, where Dj =
convention that

D(0,...,0) u

∂
∂xj

for 1 ≤ j ≤ N . We adopt the

= u for all function u defined on RN .

Let Ω be an open subset in RN . For each function u : Ω → R, we call suppu :=
{x ∈ Ω : u(x) = 0} the support of u.
For each non-negative integer number m, we have the following classical function
spaces:
C m (Ω) := {u : Ω → R | Dα u is continuous on Ω, ∀|α| ≤ m},
m
C ∞ (Ω) := ∩∞
m=0 C (Ω),

C0 (Ω) := {u ∈ C 0 (Ω) | suppu is a compact subset in Ω},
C0∞ (Ω) := {u ∈ C ∞ (Ω) | suppu is a compact subset in Ω}.
Notice that C 0 (Ω) ≡ C(Ω).
13


Definition 0.2.1. ([43, Chapter 2] and [49, Definition 2.1, p. 14]) Let Ω be an open
set in RN , N ≥ 1, and p ≥ 1.
Lp (Ω) :=

|u(x)|p dx < +∞ ,

u : Ω → R | u is Lebesgue measurable and
Ω

∞

L (Ω) := u : Ω → R | u is Lebesgue measurable and
∃C > 0 such that |u(x)| ≤ C a.e. x ∈ Ω ,
with respectively norms
1/p

u

Lp (Ω)

|u(x)|p dx

:=

,

Ω

u

L∞ (Ω)

:= inf {C | |u(x)| ≤ C a.e. x ∈ Ω} .

For p ∈ [1, +∞], let us denote by p the adjoint number of p, i.e.,

p :=


p


 p−1

if p ∈ (1, ∞),





if p = +∞.

+∞ if p = 1,
1

The spaces Lp (Ω), 1 ≤ p ≤ ∞, are Banach spaces. Moreover, Lp (Ω) with 1 < p < +∞
are reflexive and separable, while L1 (Ω) is separable. Besides, L2 (Ω) is a Hilbert space
with the scalar product
u(x)v(x)dx ∀u, v ∈ L2 (Ω).

(u, v)L2 (Ω) :=
Ω

It is noted that C0 (Ω) is dense in Lp (Ω) for 1 ≤ p < +∞. The topological dual spaces of
Lp −spaces for (1 ≤ p < +∞) are Lp −space too, namely, Lp (Ω)∗ = Lp (Ω), 1 < p < +∞

and L1 (Ω)∗ = L∞ (Ω) (see [43, Chapter 2] and [48, Section 4.3]).
In the sequel, we will write Ω ⊂⊂ Ω if Ω is included in Ω and compact. We denoted
by L1loc (Ω) the space of local integrable functions on Ω, i.e.,
L1loc (Ω) := u : Ω → R | u is Lebesgue measurable and

|u(x)|dx < +∞
Ω

for all measurable subset Ω ⊂⊂ Ω .
Then, for any open set Ω in RN and for all p ∈ [1, +∞], we have Lp (Ω) ⊂ L1loc (Ω) (see
[43, Chapter 2, p. 26]).
Recall that C0∞ (Ω) the space of functions infinitely differentiable in Ω with compact
support in Ω. We introduce a notion of convergence in the space C0∞ (Ω) which can be
defined by a topology on C0∞ (Ω). Then C0∞ (Ω) is denoted by D(Ω).
Definition 0.2.2. ([43, Chapter 1, p. 19] and [49, Definition 2.3, p. 18]) Let (ϕi ), i ∈ N
be a sequence of functions in D(Ω). We say that (ϕi ) converges to ϕ in D(Ω) when
14


i → +∞, if there exists a compact set K ⊂⊂ Ω satisfying suppϕ ⊂ K, suppϕi ⊂ K
for all i ∈ N and
Dα ϕi → Dα ϕ uniformly in K

∀α ∈ NN ,

i.e.,
lim sup |Dα ϕi (x) − Dα ϕ(x)| = 0 ∀α ∈ NN .

i→+∞ x∈K

Functions ϕ ∈ D(Ω) is called test functions.
Definition 0.2.3. ([43, Chapter 1, p. 19] and [49, Definition 2.4, p. 19]). A distribution T on Ω is a continuous linear form on D(Ω), i.e., T : D(Ω) → R is a linear map
such that
lim T (ϕi ) = T (ϕ)

i→+∞

for every sequence ϕi → ϕ in D(Ω) when i → +∞. T (ϕ) will be denoted by T, ϕ and
the space of distributions on Ω by D (Ω).
For example, for each T ∈ L1loc (Ω), the equality
T (x)ϕ(x)dx ∀ϕ ∈ D(Ω)

T, ϕ :=
Ω

defines a distribution on Ω. Thus, we have L1loc (Ω) ⊂ D (Ω) (see [49, Example, p. 22]).
Definition 0.2.4. ([43, Chapter 1, p. 20] and [49, Definition 2.5, p. 20]). For α =
(α1 , α2 , ..., αN ) ∈ NN and T ∈ D (Ω), the map
ϕ → (−1)|α| T, Dα ϕ
defines a distribution on Ω which we denoted by Dα T. Distribution Dα T called the
derivative in the distributional sense of T. Moreover, we have
Dα T, ϕ = (−1)|α| T, Dα ϕ

∀ϕ ∈ D(Ω).

It can show that if T is a k-time differentiable function on Ω then the classical
derivative Dα T of T coincides with the derivative in the distributional sense of T for
any multiindex α ∈ NN with |α| ≤ k. Therefore, notion of the derivative in the
distributional sense is an extension of notion of the derivative in the usual sense.
Definition 0.2.5. ([49, Definition 2.6, p. 21]) Let (Ti ) be a sequence of distributions

in D (Ω). We say that
lim Ti = T

i→+∞

in D (Ω),

iff
lim

i→+∞

Ti , ϕ = T, ϕ

∀ϕ ∈ D(Ω).

The following proposition shows continuity of derivative operator in the distributional sense.
15


Proposition 0.2.6. ([43, Chapter 1, p. 20] and [49, Proposition 2.5, p. 22]). The
operator Dα with α ∈ NN is continuous on D (Ω), i.e., if Ti → T in D (Ω) then
Dα Ti → Dα T

in

D (Ω).

We now give definition of weak partial derivative for locally integrable functions.
Definition 0.2.7. ([43, Chapter 1, p. 21] and [50, Chapter 5]) Let u, v ∈ L1loc (Ω) and

α be a multiindex. We say that v is α-order weak partial derivative (or α-order general
partial derivative) of u, written by v = Dα u, if
u(x)Dα ϕ(x)dx = (−1)|α|
Ω

v(x)ϕ(x)dx ∀ϕ ∈ D(Ω).
Ω

From Definition 0.2.4 and 0.2.7, it is easily seen that if v = Dα u is α-order weak
partial derivative then v is α-order partial derivative in the distributional sense of u.
Next, we give definition of Sobolev spaces.
Definition 0.2.8. ([43, Chapter 3, p. 44] and [50, Chapter 5]) Let m ∈ N, p ∈ [1, +∞].
We consider the space
W m,p (Ω) := {u ∈ Lp (Ω) | Dα u ∈ Lp (Ω) with 0 ≤ |α| ≤ m
and Dα u is α-order weak partial derivative of u}
with corresponding norm

u

W m,p (Ω)

:=






p
Lp (Ω)

Dα u
0≤|α|≤m



 max

Dα u

0≤|α|≤m

L∞ (Ω)

1/p

if 1 ≤ p < +∞,
if p = +∞

and a subspace of W m,p (Ω),
W0m,p (Ω) := Closure of C0∞ (Ω) in W m,p (Ω).
We call W m,p (Ω) and W0m,p (Ω) Sobolev spaces.
Remark 0.2.9. (i) In case of p = 2, we write H m (Ω) := W m,2 (Ω) and H0m (Ω) :=
W0m,2 (Ω).
(ii) In case of m = 0, we have W 0,p (Ω) = Lp (Ω). Moreover, if Ω is bounded and
p ∈ [1, +∞) then we have W00,p (Ω) = Lp (Ω) (see [43, Chapter 3, p. 44]).
(iii) Sobolev spaces W m,p (Ω) and W0m,p (Ω) are Banach spaces, W0m,p (Ω) is a closed
subspace of W m,p (Ω). Moreover, H m (Ω), H0m (Ω) are Hilbert spaces with scalar product
(Dα u, Dα v)L2 (Ω) .

(u, v)H m (Ω) =

0≤|α|≤m

(iv) Sobolev spaces W m,p (Ω) and W0m,p (Ω) are reflexive and uniformly convex (and so,
strictly convex) if p ∈ (1, +∞); is separable if p ∈ [1, +∞) (see Theorems 1.21 and 3.5
in [43]).
16


The following is definition on the regularity of boundary Γ of domain Ω.
Definition 0.2.10. ([52, Definition 1.2.1.1, p. 5] and [3, Subsection 2.2.2, p.26]) Let
Ω be an open set in RN . Boundary Γ of Ω is called continuous (respectively Lipschitz,
continuously differential, of class C k,l , m times continuously differential) if for each
x ∈ Γ, there exist a neighborhood V ⊂ RN of x and a new orthogonal coordinate
{y1 , y2 , ..., yN } such that
(i) V is a hypercube in the new coordinate {y1 , y2 , ..., yN } :
V = {(y1 , y2 , ..., yN ) | −ai < yi < ai , 1 ≤ i ≤ N };
ii) there exists a continuous (respectively Lipschitz, continuously differential, of class
C k,l , m times continuously differential) function ϕ, defined in
V := {(y1 , y2 , ..., yN −1 ) | −ai < yi < ai , 1 ≤ i ≤ N − 1}
and such that
aN
for every y := (y1 , y2 , ..., yN −1 ) ∈ V ,
2
Ω ∩ V = {y = (y , yN ) ∈ V | yN < ϕ(y )},

|ϕ(y )| ≤

Γ ∩ V = {y = (y , yN ) ∈ V | yN = ϕ(y )}.

In other words, in a neighborhood of x ∈ Γ, Ω is below the graph of ϕ and the
boundary Γ is the graph of ϕ.
Using new locally coordinate systems, we can derive Lebesgue measure on Γ in
natural way (see [3, Subsection 2.2.2]). Assume that the set E ⊂ Γ can be completely
represented by a locally coordinate system S = {y1 , y2 , · · · , yN }, i.e., for every point
P ∈ E, there is y ∈ V such that P = (y, ϕ(y)). Let D := ϕ−1 (E) ⊂ V . Then we say
that E is measurable if D is measurable with respect to (N − 1)−dimensional Lebesgue
measure. The measure of E is defined by
1 + |∇ϕ(y1 , y2 , · · · , yN −1 )|2 dy1 dy2 ...dyN −1 .

|E| :=
D

We denoted by dσ is measure on Γ.
Let us introduce Sobolev spaces on the set Γ, (see [2, Chapter 2, p. 75] and [52,
Definition 1.3.2.1 and Definition 1.3.3.2]). For s ∈ (0, 1), p ≥ 1 and u ∈ C ∞ (Γ), we
consider the norm
u

W s,p (Γ)

|u(x)|p dσ(x) +

:=
Γ

Γ×Γ

|u(x) − u(x )|
dσ(x)dσ(x )

|x − x |N −1+sp

1/p

,

(1)

where dσ is measure on Γ. Let us denoted by W s,p (Γ) the closed space generated by
C ∞ (Γ) under norm (1). Thus, W s,p (Γ) is a Banach space.
We denote by W −m,p (Ω) ans W −r,s (Γ) the dual spaces of the spaces W m,p (Ω) and
W r,s (Γ), respectively, where

1
p

+

1
p

=

1
s

+

1
s

17

= 1.


Definition 0.2.11. ([43, Chapter 1, p. 9]) Let X, Y be the normed spaces. We say
that X is imbedded in Y and write X → Y, if there a linear continuous injection
j : X → Y.
Moreover, if j is compact then we say that X compactly imbedded in Y and write
X → → Y.
We are ready to present some imbedding results for Sobolev spaces.
Theorem 0.2.12. (Sobolev and Rellich embedding theorem, [43, Theorem 5.4,
p. 97 and Theorem 6.2, p. 144], [48, Theorem 9.16, p. 285] and [52, Chapter 1, p. 27])
Let Ω ⊂ RN be a bounded Lipschitz domain, 1 ≤ p ≤ +∞ and 1 ≤ p ≤ +∞.
• If 1 ≤ p < N then
W 1,p (Ω) → Lq (Ω) ∀1 ≤ q ≤
and this embedding is compact for 1 ≤ q <

Np
N −p

Np
N −p .

• If p = N then W 1,p (Ω) → → Lq (Ω) ∀q ∈ [1, +∞).
• If p > N then W 1,p (Ω) → → C(Ω).
∗

Remark 0.2.13. (i) The injection W 1,p (Ω) → Lp (Ω), p∗ =

Np
N −p ,

is never compact

even if Ω is bounded and smooth (see [48]).
(ii) If W 1,p (Ω) is replaced by W01,p (Ω) then Theorem 0.2.12 is valid even if Γ is not
Lipschitz.
This results can be extended for the spaces W m,p (Ω) with m is a non negative
integer.
Theorem 0.2.14. (see [3, Theorem 7.1, p. 355]) Let Ω ⊂ RN be a bounded Lipschitz
domain, 1 ≤ p ≤ +∞, and m be a one negative integer.
• If mp < N then
W m,p (Ω) → Lq (Ω) ∀1 ≤ q ≤
and this embedding is compact for 1 ≤ q <

Np
N − mp

Np
N −mp .

• If mp = N then W m,p (Ω) → → Lq (Ω) ∀q ∈ [1, ∞).
• If mp > N then W m,p (Ω) → → C(Ω).
In particular, if N = 2 then H 1 (Ω) → Lq (Ω) for all 1 ≤ q < +∞, and if N = 3
then H 1 (Ω) → L6 (Ω).
An important issue when studying the boundary value problems defined on Ω for an
operator equation, is to determine spaces containing trace functions u |Γ , restriction
¯ then it is easily seen that

of u ∈ W 1,p (Ω) on Γ. For instance, if W 1,p (Ω) → C(Ω)
u |Γ ∈ C(Γ) for all u ∈ W 1,p (Ω).
18


The following theorem was proved by Gagliardo in 1975 and it is called trace theorem
.
Theorem 0.2.15. ([52, Theorem 1.5.1.3, p. 38]) Suppose that Ω is a bounded open
subset of RN with Lipschitz boundary Γ. Then there is a unique linear bounded map
T : W 1,p (Ω) → W 1−1/p,p (Γ) such that
T u(x) = u |Γ (x),

¯
∀x ∈ Γ, u(·) ∈ C 0,1 (Ω).

Moreover, the map T is surjective and has a right inverse which does not depend on p,
i.e., there exists a unique bounded linear map η : W 1−1/p,p (Γ) → W 1,p (Ω) such that
∀ξ(·) ∈ W 1−1/p,p (Γ).

T (η(ξ))(·) = ξ(·),

We shall call T the trace operator and T u the trace of u on Γ.
The below results represent the relation between Sobolev spaces W01,p (Ω) and the
set Ker(T ) of trace operator T .
Theorem 0.2.16. ([52, Theorem 1.5.1.5, p. 38] and [50, Chapter 5]) Suppose that Ω
is a bounded open subset of RN with Lipschitz boundary Γ . Then
u ∈ W01,p (Ω) if and only if T (u) = 0 on Γ.
Theorem 0.2.17. ([48, Theorem 9.17, p. 288]) Let Ω be of class C 1 and u ∈ W 1,p (Ω)∩
¯ Then u = 0 on Γ if and only if u ∈ W 1,p (Ω).
C(Ω).

0

The smoothness of boundary Γ plays an important role in the following results.
Theorem 0.2.18. ([3, Theorem 7.2, p.355 ]) Let m ∈ N with m > 0, and let Γ be of
class C m−1,1 . Then for all mp < N the trace operator T is continuous from W m,p (Ω)
into Lr (Γ), provided by 1 ≤ r ≤

(N −1)p
N −mp .

If mp = N then T is continuous for all

1 ≤ r < ∞.
Theorem 0.2.19. ([3, Theorem 7.3, p.356 ]) Suppose that Ω is a domain of class C m
with integers m ≥ 1 and let 1 < p < ∞. Then the trace operator T is continuous from
1

W m,p (Ω) onto W m− p ,p (Γ).
In particular, for m = 1, p = 2 it follows that H 1 (Ω) → L2 (Γ) and T : H 1 (Ω) →
1
2

H (Γ) is surjective.
We have the following Green’ formula on the relationship between integrals on domain Ω and integrals on its boundary Γ.
Theorem 0.2.20. ([52, Theorem 1.5.3.1]) Let Ω be bounded open subset in RN with
Lipschitz boundary Γ. Then, for every u ∈ W 1,p (Ω) and v ∈ W 1,p (Ω) with
we have

where

νi

∂u
vdx +
∂xi

denotes the

Ω
nth

u
Ω

∂v
dx =
∂xi

1
p

+

T (u)T (v)ν i dσ,
Γ

complement of the unit outward normal vector ν to Γ.
19

1

p

= 1,