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:

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


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



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


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



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


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 }


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


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

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.


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)


(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

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


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 ) = ∅.

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



→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))

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.

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


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 < +∞ ,


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 < +∞

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

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)

(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 )}.

|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


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
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

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


p

= 1,