Research Article
Shengda Zeng, Yunru Bai, Patrick Winkert*, and Jen-Chih Yao
Identification of discontinuous parameters
in double phase obstacle problems
https://doi.org/10.1515/anona-2022-0223
received November 4, 2021; accepted December 30, 2021
Abstract: In this article, we investigate the inverse problem of identification of a discontinuous parameter
and a discontinuous boundary datum to an elliptic inclusion problem involving a double phase differential
operator, a multivalued convection term (a multivalued reaction term depending on the gradient), a multi-
valued boundary condition and an obstacle constraint. First, we apply a surjectivity theorem for multi-
valued mappings, which is formulated by the sum of a maximal monotone multivalued operator and a
multivalued pseudomonotone mapping to examine the existence of a nontrivial solution to the double
phase obstacle problem, which exactly relies on the first eigenvalue of the Steklov eigenvalue problem for
the
p
-Laplacian. Then, a nonlinear inverse problem driven by the double phase obstacle equation is
considered. Finally, by introducing the parameter-to-solution-map, we establish a continuous result of
Kuratowski type and prove the solvability of the inverse problem.
Keywords: discontinuous parameter, double phase operator, elliptic obstacle problem, inverse problem,
mixed boundary condition, multivalued convection, Steklov eigenvalue problem
MSC 2020: 35J20, 35J25, 35J60, 35R30, 49N45, 65J20
Special Issue –Nonlinear analysis: perspectives and synergies
1 Introduction
The aim of this article is to study an inverse problem to an elliptic differential inclusion problem involving a
double phase differential operator, a multivalued convection term (dependence on the gradient of the
solution), a multivalued boundary condition and an obstacle constraint. To this end, let
Ω
be a bounded
domain in
N
(
≥
N2
)with Lipschitz boundary ≔∂
Γ
Ωsuch that
Γ
is divided into three mutually disjoint
parts
Γ
1
,
Γ
2
and
Γ
3
with
Γ
1
having positive Lebesgue measure. We study the problem
(()∣ ∣ ()∣ ∣ ) ( ) ()∣∣ ( )
()
()
() ()
−∇∇+∇∇++ ∈∇
=
∂∂=
∂∂∈≤
−− −
ax u u μx u u gxu μx u u fxu u
u
u
νhx
u
νUxu
ux x
div , , , in Ω,
0onΓ,
on Γ ,
,onΓ,
ΦinΩ,
pq q
a
a
22 2
1
2
3
(1.1)
Shengda Zeng: Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing,
Yunru Bai: School of Science, Guangxi University of Science and Technology, Liuzhou, Guangxi 545006, China,
e-mail: [email protected]
* Corresponding author: Patrick Winkert, Technische Universität Berlin, Institut für Mathematik, Strasse des 17. Juni 136, 10623
Jen-Chih Yao: Center for General Education, China Medical University, Taichung, Taiwan, e-mail: [email protected]
Advances in Nonlinear Analysis 2023; 12: 1–22
Open Access. © 2023 Shengda Zeng et al., published by De Gruyter. This work is licensed under the Creative Commons
Attribution 4.0 International License.
where
<<pN
1
,<
pq
,
[
)
→∞
μ
:Ω 0,
is a bounded function,
(()∣ ∣ ()∣ ∣ )
∂∂≔∇∇+∇∇⋅
−−
u
νax u u μx u u ν
,
a
pq22
with
ν
being the outward unit normal vector on
Γ
,
××→
f
:Ω 2
N
and
×→
U
:Γ 2
3
are two given
multivalued functions,
→Φ:Ω
is an obstacle function and (
)
→+∞
a
:Ω 0, ,
→h:Γ
2
are two pos-
sibly discontinuous parameters.
The main contribution of this article is twofold. The first intention of the article is to establish the
nonemptiness, boundedness and closedness of the solution set to problem (1.1)(in the weak sense),in
which our main methods are based on a surjectivity theorem for multivalued mappings, which is formu-
lated by the sum of a maximal monotone multivalued operator and a multivalued pseudomonotone map-
ping, the theory of nonsmooth analysis and the properties of the Steklov eigenvalue problem for the
p
-Laplacian. The second contribution of the article is to develop a general framework for studying the
inverse problem under consideration and to establish the solvability for such inverse problems. To the best
of our knowledge, this is the first work studying the identification of discontinuous parameters for such
general nonlinear elliptic equations. The problem under consideration combines several interesting phe-
nomena such as double phase operators, multivalued right-hand sides, mixed boundary conditions and
obstacle constraints.
First we point out that, motivated by several applications, the inverse problem of parameter identifica-
tion in partial differential equations is an important field in mathematics and even though such problems in
form of equations and inequalities have been studied a lot, there are still several open problems to be
solved. Our work is motivated by the article of Migórski et al. [39], who studied the inverse problem of mixed
quasi-variational inequalities of the form
( ) () () ()
⟨
−⟩+ − ≥⟨ −⟩ ∈Tau v u φv φu mv u v Ku,, , for all
,
where →
K
C:2
Cis a set-valued map, ×→
∗
TB V V:is a nonlinear map, {}→∪+∞φV:is a func-
tional and ∈∗
mV
, while
V
is a real reflexive Banach space,
B
is another Banach space and
C
is a nonempty,
closed, convex subset of
V
. Their abstract result applies to
p
-Laplacian inequalities, see also [38]for
hemivariational inequalities. We also mention the works of Clason et al. [10]for noncoercive variational
problems, Gwinner [27]for variational inequalities of second kind, Gwinner et al. [28]for an optimization
setting and Migórski and Ochal [40]for nonlinear parabolic problems, see also references therein. In addition,
we refer to the recent work of Papageorgiou and Vetro [45]about existence and relaxation theorems for
different types of problems which can be applied to variational inequalities and control systems.
A second interesting phenomenon is the occurrence of the weighted double phase operator, namely,
(()∣ ∣ ()∣ ∣ ) ( )∇∇+∇∇ ∈
−−
ax u u μx u u u Wdiv for Ω
.
pq22 1,
(1.2)
For
≡
a1
, this operator corresponds to the energy functional given by
(∣ ∣ ()∣ ∣)
∫
∇+∇uμxuxd
.
pq
Ω
(1.3)
Functionals of the form (1.3)have been initially introduced by Zhikov [56]in 1986 in order to describe
models for strongly anisotropic materials and it also turned out its relevance in the study of duality theory
as well as in the context of the Lavrentiev phenomenon, see Zhikov [57]. Observe that the energy density in
(1.3)changes its ellipticity and growth properties according to the point in the domain. In general, double
phase differential operators and corresponding energy functionals interpret various comprehensive natural
phenomena and model several problems in Mechanics, Physics and Engineering Sciences. For example, in
the elasticity theory, the modulating coefficient
(
)
⋅
μ
dictates the geometry of composites made of two
different materials with distinct power hardening exponents
p
and
q
, see Zhikov [58]. Functionals given
in (1.3)have been intensively studied in the last few years concerning regularity of local minimizers. We
mention the famous works of Baroni et al. [2,3], Byun and Oh [7], Colombo and Mingione [12,13], Marcellini
[36,37]and Ragusa and Tachikawa [49].
2Shengda Zeng et al.
A third interesting phenomenon is not only the multivalued right-hand side, which is motivated by
several physical applications (see, e.g., Panagiotopoulos [42,43], Carl and Le [8]and references therein)but
also its dependence on the gradient of the solutions often called convection term. The main difficulty in the
study of gradient dependent right-hand sides is their nonvariational character, that is, the standard varia-
tional tools to the corresponding energy functionals are not applicable. In the past few years, several
interesting works have been published with convection terms, we refer to the papers of El Manouni et al.
[16], Faraci et al. [17], Faraci and Puglisi [18], Figueiredo and Madeira [20], Gasiński and Papageorgiou [23],
Liu et al. [32], Liu and Papageorgiou [33], Marano and Winkert [35], Papageorgiou et al. [44]and Zeng and
Papageorgiou [55].
Finally, we mention some existence results on the recent topic of double phase operators published
within the last few years. We refer to Bahrouni et al. [1], Benslimane et al. [4], Biagi et al. [5], Colasuonno
and Squassina [11], Fiscella [21], Farkas and Winkert [19], Gasiński and Papageorgiou [22], Gasiński and
Winkert [24–26], Liu and Dai [31], Liu and Winkert [34], Papageorgiou et al. [46], Perera and Squassina
[48], Stegliński [51]and Zeng et al. [52–54].
This article is organized as follows. Section 2 recalls preliminary material including Musielak-Orlicz
Lebesgue and Musielak-Orlicz Sobolev spaces, the
p
-Laplacian eigenvalue problem with Steklov boundary
condition, pseudomonotone operators and a surjectivity theorem for multivalued mappings. Under very
general assumptions on the data, Section 3 proves the nonemptiness and compactness of the solution set to
problem (1.1). In Section 4, we present a new existence result to the inverse problem of (1.1).
2 Preliminaries
This section is devoted to recall some basic definitions and preliminaries, which will be used in the next
sections to derive the main results of the article. To this end, let
⊂Ω
N
be a bounded domain with Lipschitz
boundary ≔∂
Γ
Ωsuch that
Γ
is decomposed into three mutually disjoint parts
Γ
1
,
Γ
2
and
Γ
3
with
Γ
1
having
positive Lebesgue measure. In what follows, we denote by ()
M
Ωthe space of all measurable functions
→
u
:Ω . As usual, we identify two functions which differ on a Lebesgue-null set. Let
[
)
∈∞r1,
and
D
be
a nonempty subset of
Ω
. We denote the usual Lebesgue spaces by () (
)
≔
L
DLD;
rr
and
(
)L
D;
rN
equipped
with the standard
r
-norm
‖
⋅‖
r
D
,
and (
)L
Γ
rstands for the boundary Lebesgue spaces with norm
‖
⋅‖r,Γ
.
Let
() { () () }≔∈ ≥ ∈
+
L
DuLDux x: 0 for a.a. Ω
rr
.By
(
)
WΩ
r1,
we define the corresponding Sobolev
space endowed with the norm
‖
⋅‖
r1, ,
Ω
given by ()
‖
‖≔‖‖+‖∇‖ ∈uuu uWfor all Ω
.
rr r r
1, ,Ω ,Ω ,Ω 1,
For any fixed >
s1
, the conjugate of
s
is defined by
′>
s1
such that +=
′
1
ss
11 . Moreover, we use the symbols
∗
s
and ∗
s
to represent the critical exponents to
s
in the domain and on the boundary, respectively, given by
⎧⎨⎩⎧⎨⎩()
=−<
+∞ ≥ =−−<
+∞ ≥
∗∗
s
Ns
Ns sN
sN sNs
Ns sN
sN
if ,
if , and 1if ,
if .
(2.1)
Let us comment on the
r
-Laplacian eigenvalue problem with Steklov boundary condition given by
∣∣
∣∣ ∣∣
−=−
⋅=
−
−−
uuu
uuνλuu
ΔinΩ,
on Γ,
rr
rr
2
22 (2.2)
for <<∞r
1
. From Lê [30]we know that (2.2)has a smallest eigenvalue
>λ
0
r
S
1,
, which is isolated and
simple. Besides, we know that
>λ
0
r
S
1,
can be characterized by
(){}
=‖∇‖ +‖‖
‖‖
∈⧹
λuu
u
inf
.
r
S
uW
r
rr
r
r
r
1, Ω0
,Ω ,Ω
,Γ
r1, (2.3)
Identification of discontinuous parameters in double phase obstacle problems 3
The following assumptions are supposed in the entire article:
() ()<< << ≤⋅∈
∗∞
pN pqp μ L
1
,and0Ω
.
(2.4)
Now we define the nonlinear function
[)[)×∞→∞: Ω 0, 0,
given by
() () () [ )=+ ∈×∞xt t μxt xt,for all,Ω0,
.
pq
Then, the Musielak-Orlicz Lebesgue space
(
)L
Ω
driven by the function
is given by
(){ () () }=∈ <+∞
L
uM ρuΩΩ:
equipped with the Luxemburg norm
⎧⎨⎩⎛⎝⎞⎠⎫⎬⎭
‖
‖= > ≤uτρ
u
τ
inf 0 : 1
.
Here, the modular function is given by
() ( ∣∣) (∣∣ ()∣∣)
∫∫
≔=+
ρ
uxuxuμxux,d d
.
pq
ΩΩ
We know that
(
)L
Ω
is uniformly convex and so a reflexive Banach space. Moreover, we introduce the
seminormed space
(
)L
Ω
μ
q
()⎧⎨⎩() ()∣∣ ⎫⎬⎭
∫
=∈ <+∞
L
uM μxu xΩΩ:d
μ
qq
Ω
endowed with the seminorm
⎛⎝⎜⎜()∣∣ ⎞⎠⎟⎟
∫
‖
‖=uμxuxd
.
qμ q
,
Ω
q
1
The Musielak-Orlicz Sobolev space ()WΩ
1, is given by
(){ ()∣∣ ()
}
=∈ ∇∈WuLuLΩΩ:Ω
1,
equipped with the norm
‖
‖=‖∇‖+‖‖uuu,
1,
where
∣∣
‖
∇‖=‖∇‖uu
. As before, it is known that ()WΩ
1, is a reflexive Banach space.
Next, we introduce a closed subspace
V
of ()WΩ
1, given by
{()
}
≔∈ =VuW uΩ: 0 onΓ
1, 1
endowed with the norm
‖
‖=‖‖uu
V1, for all
∈
uV
. Of course,
V
is also a reflexive Banach space. In the
following, we denote the norm of the dual space ∗
Vof
V
by
‖
⋅‖∗
V
.
Let us recall some embedding results for the spaces
(
)L
Ω
and ()WΩ
1, , see Gasiński and Winkert [26]
or Crespo-Blanco et al. [14].
Proposition 2.1. Let (2.4)be satisfied and denoted by ∗
p
,
∗
p
the critical exponents to
p
as given in (2.1)
for =
sp
.
(i)
() ()↪
L
LΩΩ
rand
() ()↪WWΩΩ
r1, 1,
are continuous for all
[]∈rp1,
;
(ii)() (
)
↪WLΩΩ
r1, is continuous for all []∈∗
rp1, and compact for all
[
)
∈
∗
rp1,
;
(iii)() (
)
↪WLΩΓ
r1, is continuous for all []∈∗
rp1, and compact for all
[
)
∈
∗
rp1,
;
(iv)
() ()↪
L
LΩΩ
μ
qis continuous;
(v)
() ()↪
L
LΓΩ
qis continuous.
4Shengda Zeng et al.
We point out that if we replace the space ()WΩ
1, by
V
in Proposition 2.1, then the embeddings (ii)and
(iii)remain valid.
The following proposition is due to Liu and Dai [31, Proposition 2.1].
Proposition 2.2. Let (2.4)be satisfied and let ()∈
y
LΩ.Then the following hold:
(i)if
≠
y
0
,then
‖
‖=y
λ
if and only if
()
=
ρ1
y
λ;
(ii)
‖
‖<y
1
(resp.
>1
and
=1
)if and only if
()<
ρ
y
1
(resp.
>1
and
=1
);
(iii)if
‖
‖<y
1
,then
()
‖
‖≤ ≤‖‖yρyy
qp
;
(iv)if
‖
‖>y
1
,then
()
‖
‖≤ ≤‖‖yρyy
pq
;
(v)
‖
‖→y
0
if and only if ()→
ρ
y
0
;
(vi)
‖
‖→+∞y
if and only if
()→+∞
ρ
y.
We suppose that
() ()∈>
∞∈
a
LaxΩsuch thatinf 0
.
xΩ
(2.5)
Next, we introduce the nonlinear operator
→
∗
F
VV:
given by
() (()∣ ∣ ()∣ ∣ ) (∣∣ ()∣∣ )
∫∫
⟨
⟩≔ ∇ ∇+ ∇ ∇⋅∇ + +
−− −−
Fuv axu uμxu uvx u uμxuuvx,dd
,
pq pq
Ω
22
Ω
22 (2.6)
for ∈
u
v
V
,with
⟨
⋅⋅
⟩
,being the duality pairing between
V
and its dual space ∗
V. The following proposition
states the main properties of
→
∗
F
VV:
. We refer to Liu and Dai [31, Proposition 3.1]or Crespo-Blanco et al.
[14, Proposition 3.4]for its proof.
Proposition 2.3. Let hypotheses (2.4)and (2.5)be satisfied. Then, the operator F defined by (2.6)is bounded,
continuous, monotone (hence maximal monotone)and of type (
)
+
S,that is,
⟶⟨−⟩≤
→∞
u
uinVand Fuu ulimsup , 0
,
nw
nnn
imply
→
uu
n
in
V
.
We now recall some notions and results concerning nonsmooth analysis and multivalued analysis.
Throughout the article the symbols “
⟶
w
”and “
→
”stand for the weak and the strong convergence,
respectively, in various spaces. Moreover, let us recall the notions of pseudomonotonicity and generalized
pseudomonotonicity in the sense of Brézis for multivalued operators (see, e.g., Migórski et al. [41,Definition
3.57]), which will be useful in the sequel.
Definition 2.4. Let
X
be a reflexive real Banach space. The operator
→∗
AX:2
Xis called
(a)pseudomonotone (in the sense of Brézis)if the following conditions hold:
(i)the set (
)
Au is nonempty, bounded, closed and convex for all
∈
uX
;
(ii)
A
is upper semicontinuous from each finite-dimensional subspace of
X
to the weak topology
on
∗
X
;
(iii)if
{
}⊂u
X
nwith
⟶
uu
nw
in
X
and (
)
∈
∗
u
Au
nnare such that
⟨−⟩≤
→∞ ∗×
∗
uu ulimsup , 0
,
nnnXX
then to each element ∈
v
X, there exists () (
)
∈
∗
u
vAu
with
()
⟨
−⟩ ≤ ⟨ −⟩
∗×→∞ ∗×
∗∗
uv u v u u v, liminf , ;
XX nnnXX
Identification of discontinuous parameters in double phase obstacle problems 5
(b)generalized pseudomonotone (in the sense of Brézis)if the following holds: Let
{
}⊂u
X
nand
{
}⊂
∗∗
uX
n
with (
)
∈
∗
u
Au
nn.If
⟶
uu
nw
in
X
and ⟶
∗∗
u
u
nwin
∗
X
and
⟨−⟩≤
→∞ ∗×
∗
uu ulimsup , 0
,
nnnXX
then the element ∗
u
lies in (
)
Au and
⟨
⟩→⟨⟩
∗×∗×
∗∗
uu uu,,
.
nnX X X X
It is not difficult to see that every pseudomonotone operator is generalized pseudomonotone, see, e.g.,
Carl et al. [9, Proposition 2.122]. Also, under an additional assumption of boundedness, we obtain the
converse statement, see, e.g., Carl et al. [9, Proposition 2.123].
Proposition 2.5. Let X be a reflexive real Banach space and assume that
→∗
AX:2
Xsatisfies the following
conditions:
(i)for each
∈
uX
we have that (
)
Au is a nonempty, closed and convex subset of
∗
X
.
(ii)
→∗
AX:2
Xis bounded.
(iii)
A
is generalized pseudomonotone, i.e., if
⟶
uu
nw
in
X
and ⟶
∗∗
u
u
nwin
∗
X
with (
)
∈
∗
u
Au
nnand
⟨−⟩≤
→∞ ∗×
∗
uu ulimsup , 0
,
nnnXX
then (
)
∈
∗
u
Au and
⟨
⟩→⟨⟩
∗×∗×
∗∗
uu uu,,
.
nnX X X X
Then the operator
→∗
AX:2
Xis pseudomonotone.
Let us now recall the definition of Kuratowski limits, see, e.g., Papageorgiou and Winkert [47,Defini-
tion 6.7.4].
Definition 2.6. Let (
)
Xτ,be a Hausdorfftopological space and let
{
}⊂
∈
A2
nn X
be a sequence of sets. We
define the
τ
-Kuratowski lower limit of the sets
A
n
by
⎧⎨⎩⎫⎬⎭
-≔∈=-∈ ≥
→∞ →∞
τ
AxXxτxxA nliminf : lim , for all 1 ,
nnnnn n
and the
τ
-Kuratowski upper limit of the sets
A
n
⎧⎨⎩⎫⎬⎭
-≔∈=-∈<<…<<…
→∞ →∞
τ
AxXxτxxAnn nlimsup : lim , , .
nnknn n k12
kk k
If
=- =-
→∞ →∞
Aτ A τ Aliminf limsup
,
nnnn
then
A
is called
τ
-Kuratowski limit of the sets
A
n
.
Finally, we recall the following surjectivity theorem for multivalued mappings, which is formulated by
the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone map-
ping, see Le [29, Theorem 2.2].
Theorem 2.7. Let
X
be a real reflexive Banach space, let
()⊂→
∗
DX:2
X
be a maximal monotone
operator, let
()=→
∗
DX:2
X
be a bounded multivalued pseudomonotone operator, let
∈∗
Xand let
() {
}
≔∈‖‖<
B
uXu R0:
RX
.Assume that there exist ∈
uX
0and ≥‖‖Ru
X
0such that () ()∩≠∅DB0
Rand
⟨
+− −⟩>
×
∗
ξη uu,
0
XX0(2.7)
6Shengda Zeng et al.
for all (
)
∈
u
Dwith
‖
‖=u
R
X,for all (
)
∈ξu
and for all
(
)
∈ηu
.Then the inclusion
() ()+∋uu
has a solution in (
)
D.
Obviously, if
()
⟨+ −⟩
‖‖ =+∞
‖
‖→+∞
∈×
∗
ξηuu
u
lim ,
u
uD
XX
X
0
X(2.8)
is satisfied, then the estimate in (2.7)holds automatically for some
R
large enough.
3 Double phase elliptic obstacle inclusion problem
In this section, we are interested in the study of the existence of a solution to the double phase elliptic
obstacle inclusion problem (1.1)and in deriving some relevant properties of the solution set to problem (1.1).
More precisely, we are going to apply a surjectivity theorem for multivalued mappings, which is formulated
by the sum of a maximal monotone multivalued operator and a multivalued pseudomonotone operator, to
examine the solvability of problem (1.1).
First, we formulate the hypotheses on the data of problem (1.1).
H(f): The multivalued convection mapping
××→
f
:Ω 2
N
has nonempty, bounded, closed and
convex values and
(i)the multivalued mapping ()↦
x
fxsξ,, is measurable in
Ω
for all ()∈×sξ,N;
(ii)the multivalued mapping ()(
)
↦sξ fxsξ,,,
is upper semicontinuous for a.a. ∈
x
Ω;
(iii)there exist ()∈+
−
α
LΩ
fr
r1and
≥
a
b,
0
ff
such that
∣∣ ∣∣ ∣∣ ()
()
≤++
−
−
ηaξ bs αx
,
ff
rf
1
pr
r1
for all
(
)
∈ηfxsξ,,
, for all ∈
s
, for all
∈ξ
N
and for a.a. ∈
x
Ω, where <<∗
rp
1
with the
critical exponent ∗
p
in the domain
Ω
given in (2.1)for =
sp
;
(iv)there exist ()∈+
β
LΩ
f1and constants
c
f
,≥
d0
fsuch that
∣∣ ∣∣ ()≤++ηs c ξ d s β x
,
fpfpf
for all
(
)
∈ηfxsξ,,
, for all ∈
s
, for all
∈ξ
N
and for a.a. ∈
x
Ω.
H(g): The function ×→g:Ω is such that
(i)for all ∈
s
, the function
(
)
↦
x
gxs,
is measurable;
(ii)for a.a. ∈
x
Ω, the function
()↦
s
gxs,
is continuous;
(iii)there exist >
a0
g
and ()∈
b
LΩ
g1such that
() ∣∣ ()≥−gxss a s b x,,
gςg
for all ∈
s
and for a.a. ∈
x
Ω, where <<∗
p
ςp
;
(iv)for any ()∈∗
u
vL,Ω
p, the function
(())(
)
↦
x
gxux vx,
belongs to (
)L
Ω
1.
H(Φ): The function
[
)
→∞Φ:Ω 0,
is measurable, that is, (
)
∈MΦΩ
.
H(U):
×→
U
:Γ 2
3
satisfies the following conditions:
(i)(
)U
xs,is a nonempty, bounded, closed and convex set in
for a.a.
∈
x
Γ
3
and for all ∈
s
;
(ii)()↦
x
Uxs,is measurable on
Γ
3
for all ∈
s
;
(iii)(
)
↦
s
Uxs,is u.s.c.;
Identification of discontinuous parameters in double phase obstacle problems 7
(iv)there exist
()∈
′+
α
LΓ
Uδ3and ≥
a0
Usuch that
∣( )∣ () ∣∣≤+
−
Uxs α x a s,
UU
δ
1
for a.a.
∈
x
Γ
3
and for all ∈
s
, where
<<
∗
δp
1
with the critical exponent
∗
p
on the boundary
Γ
given in (2.1);
(v)there exist ()∈+
β
LΓ
U13and ≥
b
0
Usuch that∣∣ ()≤+ξs b s β x
UpU
for all
(
)
∈ξUxs,
, for all ∈
s
and for a.a.
∈
x
Γ
3
.
H(0):()∈∞
a
LΩis such that ()≥>
∈ax cinf
0
xΩΛ
and ()∈′
hLΓ
p2.
H(1): The inequality holds ()−− >
−
c
cbλ 0
,
fU
p
S
Λ1, 1
where
λ
p
S
1, is the first eigenvalue of the
p
-Laplacian with Steklov boundary condition, see (2.2)
and (2.3).
Remark 3.1. It should be mentioned that if hypotheses H(
f
)(iv)and H(
U
)(v)are replaced by the following
conditions:
H(f)(iv)’: there exist ()∈+
β
LΩ
f1and constants
c
f
,≥
d0
fsuch that
∣∣ ∣∣ ()≤++ηs c ξ d s β x
ff
pf
ϱ1
for all
(
)
∈ηfxsξ,,
, for all ∈
s
, for all
∈ξ
N
and for a.a. ∈
x
Ω, where
<<
p1
ϱ
1;
H(U)(v)’: there exist ()∈+
β
LΓ
U13and ≥
b
0
Usuch that ∣∣ ()≤+ξs b s β x
UU
ϱ
2
for all
(
)
∈ξUxs,
, for all ∈
s
and for a.a.
∈
x
Γ
3
, where
<<
p1
ϱ2
,
then hypothesis H(1)can be removed. Indeed, it follows from Young’s inequality with >ε0that
∣∣ ∣∣ () ∣∣ () ∣∣ ()
∣∣ () ∣∣ () ()
≤++≤+++
≤+≤++
ηs cξ ds βy εξ cε ds βy
ξsbs βx εs cε βx
ff
pfpfpf
UUpU
ϱ1
ϱ2
1
2
for all
()∈ηfysξ,,
, for all
(
)
∈ξUxs,
, for all ∈
s
, for all
∈ξ
N
, for a.a.
∈
y
Ω
and for a.a.
∈
x
Γ
3
with
some () ()>
c
εcε,
0
12 . If we choose
⎛⎝
⎞⎠
()
∈
+−
ε0,
c
λ1p
S
Λ
1, 1, then the inequality in H(1)holds automatically.
Let
K
be a subset of
V
given by {}≔∈ ≤
K
vVv:ΦinΩ
.
(3.1)
Under H(
Φ
)we see that the set
K
is a nonempty, closed and convex subset of
V
. In fact, from H(
Φ
)(i.e.,
()≥xΦ
0
for a.a. ∈
x
Ω), we know that ∈K0, i.e.,
≠∅
K
. Furthermore, it is clear that
K
is convex. For the
closedness, let
{
}⊂
∈
uK
nn be a sequence such that
→
uu
n
in
V
for some
∈
uV
. The continuity of
V
into
()
L
Ω
pimplies that
→
uu
n
in ()
L
Ω
p. Passing to a subsequence if necessary, we may suppose that
() ()→
u
xux
nfor a.a. ∈
x
Ω. Therefore,
() () ()≥= ∈
→∞
xuxux xΦ lim for a.a. Ω
.
nn
Hence,
∈
u
K
and so
K
is closed.
Next, we state the definition of a weak solution to problem (1.1).
8Shengda Zeng et al.
Definition 3.2. A function
∈
u
K
is said to be a weak solution of problem (1.1), if there exist functions
(
)
∈
′
ηLΩ
r
and
(
)
∈
′
ξLΓ
δ3
with () ( () ())∈∇ηx fx ux ux,, for a.a. ∈
x
Ω,() ( ())∈ξx Uxux,for a.a.
∈
x
Γ
3
and the
equality (()∣∣ ()∣∣)() ()() ()∣∣()
()( ) ()( ) ()( )
∫∫∫
∫∫∫
∇∇+∇∇⋅∇−+ −+ −
≥−+−+−
−− −
axu uμxu u vux gxuvux μxu uvux
ηxv ux hxv u ξxv u
d,d d
ddΓdΓ
pq q
Ω
22
ΩΩ
2
ΩΓ Γ
23
is satisfied for all
∈
v
K
, where the set
K
is defined by (3.1).
The following theorem which is the main result in this section shows that for each pair ()∈ah,
() (
)
×
∞+′
L
LΩΓ
p2satisfying H(0), the solution set to problem (1.1), denoted by
()ah,
, is nonempty,
bounded and weakly closed.
Theorem 3.3. Let hypotheses (2.4),H(
f
),H(g),H(
Φ
),H(
U
),H(0)and H(1)be satisfied. Then, the solution set
of problem (1.1)is nonempty, bounded and weakly closed (hence, weakly compact).
Proof. We divide the proof into three parts.
I Existence:
First, we consider the following nonlinear functions
→
∗
F
VV:
,() (
)
⊂→
′
G
VL L:ΩΩ
ςςand
() ()→′
L
LL:Ω Ω
ppdefined by
(()∣ ∣ ()∣ ∣ ) (∣∣ ()∣∣ )
()
∣∣
() ()
() ()
∫∫
∫∫
⟨⟩≔ ∇∇+∇∇⋅∇+ +
⟨
⟩≔
⟨⟩ ≔
−− −−
××−
′′
Fuv axu uμxu uvx u uμxuuvx
Gu w g x u w x
Ly z y yz x
,dd,
,,d
,d
pq pq
LL
LL p
Ω
22
Ω
22
ΩΩ
Ω
ΩΩ
Ω
2
ςς
pp
for all ∈
u
v
V
,, for all (
)
∈
w
LΩ
ςand for all (
)
∈
y
zL,Ω
p.
Let
∈
uV
be fixed. By the Yankov-von Neumann-Aumann selection theorem (see e.g., Papageorgiou
and Winkert [47, Theorem 2.7.25]) and assumptions H(
f
)(i)and (ii), we know that the multivalued func-
tion (()())↦∇
x
fx ux ux,, admits a measurable selection. Let →η:Ω be a measurable selection of
(()())↦∇
x
fx ux ux,, , that is, () ( () ())∈∇ηx fxux ux,, for a.a. ∈
x
Ω. From H(
f
)(iii)and the inequality
(∣∣ ∣∣) (∣∣ ∣∣)+≤ + ∈ ≥
−
rr r r rr s2for all,with1,
ss s s
12 112 12
it follows that there exist constants >
M
M,
0
12 satisfying
∣()∣ ⎛⎝∣∣ ∣∣ ()
⎞⎠
(∣ ∣ ∣∣ ())
()
()
∫∫
∫
≤∇+ +
≤∇++
=‖∇‖+‖‖+‖‖
≤‖‖+‖‖+‖‖
′′−′
′′′
′′
ηx x a u b u α x x
Muuαxx
Mu u α
Mu u α
dd
d
,
rf
p
rfrf
r
pr
fr
p
pr
rfr
r
V
pV
rfr
r
ΩΩ
1
1
Ω
1,Ω ,Ω ,Ω
2,Ω
(3.2)
where we have used the fact that the embeddings of
V
into
(
)
WΩ
p1,
and of
V
into (
)L
Ω
rare continuous.
Hence,
(
)
∈
′
ηLΩ
r
. This permits us to consider the Nemytskij operator
()
(
)
⊂→
′
N
VL:Ω2
frLΩ
rassociated
with the multivalued mapping
f
defined by
Identification of discontinuous parameters in double phase obstacle problems 9
() { () () ( () ()) }≔∈ ∈ ∇ ∈
′
N
uηL ηxfxuxux xΩ : , , for a.a. Ω
fr
for all
∈
uV
. Similarly, because of hypotheses H()U(i),(ii)and (iii), for each ()∈
u
LΓ
δ3
fixed, we are able to
find a measurable function →ξ:Γ
3satisfying () ( ())∈ξx Uxux,for a.a.
∈
x
Γ
3
and
∣()∣
(() ∣∣)
(() ∣∣)
()
∫∫∫
‖
‖=
≤+
≤+
=‖‖+‖‖
′′′
−′
′
′′
ξξx
αx au
Mαx u
Mα u
dΓ
dΓ
dΓ
δ
δδ
UU
δδ
Uδδ
Uδ
δδ
δ
,Γ
Γ
Γ
1
3
Γ
3,Γ ,Γ
3
3
3
3
33
(3.3)
for some >
M0
3. Therefore, in what follows, we denote by () ()
→′
N
L:Γ 2
UδL
3Γ
δ3the Nemytskij operator
corresponding to the multivalued mapping
U
defined by
() { () () ( ()) }≔∈ ∈ ∈
′
N
uηL ηxUxux xΓ : , for a.a. Γ
Uδ33
for all ()∈
u
LΓ
δ3
.
Let ()→
ι
VL:Ω
r,()→
ω
VL:Ω
ςand ()→
β
VL:Ω
pbe the embedding operators of
V
to (
)L
Ω
r,
V
to
()
L
Ω
ςand
V
to ()
L
Ω
p, respectively, with its adjoint operators ()→
∗′∗
ι
LV:Ω
r,()→
∗′∗
ω
LV:Ω
ςand
()→
∗′∗
β
LV:Ω
p
, respectively. Also, we denote by
()→γV L:Γ
δ3
the trace operator of
V
into
(
)L
Γ
δ3
with
its adjoint operator
()→
∗′∗
γL V:Γ
δ3
. Consider the indicator function of the set
K
formulated as
() ⎧⎨⎩
≔∈
+∞ ∉
Iu uK
uK
0if,
if .
K
Under the aforementioned definitions, we could use a standard procedure for variation calculus to obtain
that
∈
u
K
is a weak solution of problem (1.1)if and only if it solves the following nonlinear inclusion
problem:
() () ()+−−− +∂∋
∗∗∗ ∗ ∗
F
uωGuβLuιNu γNu Iu h Vin ,
fUcK
where
∂
I
cKis the convex subdifferential operator of
I
K
.
Observe that the functions
F
,
G
and
L
are bounded. The latter combined with (3.2),(3.3)and hypotheses
(
)
Hf
and (
)
HU implies that for each
∈
uV
the set
() () ()≔+ −− −
∗∗∗ ∗
Hu Fu ωGu βLu ιNu γN u
fU
is nonempty, bounded, closed and convex. We show that His a pseudomonotone operator. Let
{
}⊂
∈
u
V
nn
,
{
}⊂
∈∗
ζV
nnbe sequences and let ()∈×∗
uζ V V,be such that
()∈∈⟶⟨−⟩≤
→∞
ζHu n ζ ζ ζuufor each , and limsup , 0
.
nnnw
nnn
(3.4)
Then, for every ∈n, there are
(
)
∈ηNu
nfn
and (
)
∈ξNu
nUn
such that
=+ −−− ∈
∗∗∗∗
ζFuωGuβLuιηγξ nfor all
.
nnnn
nn
Taking (3.2)and (3.3)into account, we can see that the sequences
{
}(
)
⊂
∈′
ηLΩ
nnrand
{
}()⊂
∈′
ξLΓ
nnδ3
are
both bounded. Without any loss of generality, we may assume that there exist functions
()() (
)
∈×
′′
ηξ L L,ΩΓ
rδ
3
such that
() ()⟶⟶
′′
ηηL ξξLin Ω and in Γ
.
nwrnwδ3
Recall that
V
is embedded compactly into ()
L
Ω
ς,(
)L
Ω
rand ()
L
Ω
p, respectively, and
()→γV L:Γ
δ3
is
compact. Using this we have
10 Shengda Zeng et al.
()
()
()
()
() ()
() ()
() ()
() ()
⟨−⟩=⟨−⟩=
⟨−⟩=⟨−⟩=
⟨−⟩=⟨−⟩=
⟨−⟩=⟨−⟩ =
→∞ ∗→∞ ×
→∞ ∗→∞ ×
→∞ ∗→∞ ×
→∞ ∗→∞ ×
′′
′′
ωGuu u Guωu u
βLuu u Luβu u
ιηuu ηιuu
γξuu ξγuu
lim , lim , 0,
lim , lim , 0,
lim , lim , 0,
lim , lim , 0.
nnn nnn LL
nnn nnn LL
nnnnnnLL
nnnnnnLL
ΩΩ
ΩΩ
ΩΩ
ΓΓ
ςς
pp
rr
δδ
33
(3.5)
Inserting (3.5)into the inequality in (3.4)yields
≥⟨−⟩
≥ ⟨ −⟩+ ⟨ −⟩− ⟨ −⟩+ ⟨ −
⟩
+⟨−⟩
≥⟨−⟩
→∞
→∞ →∞ ∗→∞ ∗→∞ ∗
→∞ ∗
→∞
ζu u
Fuu u ωGuu u βLuuu ιηuu
γξ u u
Fu u u
0limsup,
limsup , liminf , limsup , liminf ,
liminf ,
limsup , .
nnn
nnn nnn nnn
nnn
nnn
nnn
From Proposition 2.3 we know that
F
is of type (
)
+
S. Therefore,
→
u
uVin
.
n
Passing to a subsequence if necessary, we may assume that
() () () ()→∇→∇ ∈
u
xux ux ux xand for a.a. Ω
.
nn (3.6)
Applying Mazur’s theorem there exists a sequence
{
}∈
χnnof convex combinations to
{
}
∈
η
nn
satisfying
()→
′
χη Lin Ω
.
nr
Therefore, we can suppose that () (
)
→χx ηx
nfor a.a. ∈
x
Ω. Due to the convexity of
f
we see that
() ( () ())∈∇ ∈χx fxux ux x, , for a.a. Ω
.
nnn
Recall that
f
is u.s.c. and has nonempty, bounded, closed and convex values (see hypotheses H(
f
)(i)and
(ii)). So, we can use Proposition 4.1.9 of Denkowski et al. [15]to infer that the graph of ()(
)
↦sξ fxsξ,,,
is
closed for a.a. ∈
x
Ω. Taking the convergence properties in (3.6)and () (
)
→χx ηx
nfor a.a. ∈
x
Ωinto account,
we obtain () ( () ())∈∇ ∈ηx fx ux ux x, , for a.a. Ω
.
This shows that
(
)
∈ηNu
f
. Applying the same arguments as we did before, we conclude that (
)
∈ξNu
U.
Recall that
F
,
G
and
L
are continuous. So we can use the convergence (3.4)in order to obtain
=+ −−−⟶+−−−=
∗∗∗∗ ∗∗∗∗ ∗
ζFuωGuβLuιηγξ FuωGuβLuιηγξζ Vin
.
nnnn
nn
w
This implies that ()∈ζHu
. Hence, we have
() () () ()
⟨⟩=⟨+ −−− ⟩
=⟨+ − ⟩−⟨⟩ −⟨⟩
=⟨ + − − − ⟩=⟨ ⟩
→∞ →∞ ∗∗∗∗
→∞ ∗∗ →∞ ×→∞ ×
∗∗∗∗
′′
ζu Fu ωGu βLu ιη γξu
Fu ω Gu β Lu u η ιu ξ γu
Fu ω Gu β Lu ι η γ ξ u ζ u
lim , lim ,
lim , lim , lim ,
,,.
nnnnnnn
nn
n
nnnnn
nnnLL nnnLLΩΩ Γ Γ
rr δδ
33
This shows that His a generalized pseudomonotone operator. Employing Proposition 2.5, we conclude that
His pseudomonotone.
Next, we show the coercivity of H. To this end, we introduce a subspace Wof
(
)
WΩ
p1,
defined by
{() }≔∈ =WuW uΩ: 0 onΓ
.
p1, 1(3.7)
Because
Γ
1
has positive measure, it is not difficult to prove that Wendowed with the norm
‖
‖≔‖∇‖ ∈uu u
W
for all
Wp,Ω
Identification of discontinuous parameters in double phase obstacle problems 11
is a reflexive and separable Banach space. Moreover, since the embedding of
V
into Wis continuous, there
exists a constant >
C0
VW such that
‖
‖≤‖‖ ∈uCu uVfor all
.
WVWV
Let
∈
uV
and ()∈ζHu
be arbitrary. Then, we can find functions
(
)
∈ηNu
f
and (
)
∈ξNu
Usuch that
=+ −−−
∗∗∗∗
ζFuωGuβLuιηγξ
and
∣∣ ∣∣ ()
∣∣ () ∣∣ ()
()
() () () ()
∫
∫∫
⟨
⟩=⟨ ⟩+⟨ − ⟩−⟨ ⟩ −⟨ ⟩
≥‖∇‖ +‖∇‖+‖‖− ∇ + +
−++−
≥ − ‖∇‖ +‖∇‖ + ‖‖ +‖‖ −‖‖ − ‖‖ −‖‖ − ‖‖
−‖‖
∗∗ ××
′′
ζu Fuu ωGu βLuu ηu ξu
cu u u cu du βxx
bu βx au bxx
ccu u au u b du β bu
β
,, ,, ,
d
dΓ d
.
LL LL
p
pqμ
qqμ
qfpfpf
UpUgςg
fp
pqμ
qgς
ςqμ
qgf
p
pfUp
p
U
ΩΩ Γ Γ
Λ,Ω ,,
Ω
ΓΩ
Λ,Ω ,,Ω ,1,Ω ,Ω 1,Ω ,Γ
1,Γ
rr δδ
33
3
3
3
(3.8)
We set
(( ) )
=++
−
εa
λbd21
.
g
p
SUf
1, 1
Keeping in mind that >ς
p
, it follows from Young’s inequality and the eigenvalue problem of the
p
-Laplacian with Steklov boundary condition (see (2.2)and (2.3)) that the following inequalities hold
()( )‖‖ ≤ ‖∇‖ +‖‖
−
b
ubλ u u
Up
pUp
Sp
pp
p
,Γ 1, 1,Ω ,Ω
3(3.9)
and
∣∣ ∣∣ () (
)
∫∫
‖
‖= ≤ +=‖‖+u u x ε u x cε εu cεdd
p
ppς ς
ς
,Ω
ΩΩ
,Ω (3.10)
with some ()>
c
ε
0
. Using (3.9)and (3.10)in (3.8), we obtain
(())
()
() ()
() ()
{} ()
⟨
⟩≥−− ‖∇‖+‖∇‖+‖‖+‖‖+‖‖−‖‖−‖‖
−‖‖ −
≥ ‖∇‖ +‖∇‖ +‖‖ +‖‖ + ‖‖ −‖‖ −‖‖ −‖‖ −
= + ‖‖ −‖‖ −‖‖ −‖‖ −
≥ ‖‖‖‖+ ‖‖ −‖‖ −‖‖ −‖‖ −
−
ζu c c bλ u u auuub β
βcε
Mu u u u aub β βcε
Mu
aub β βcε
Muuaub β βcε
,2
ˆ2
ˆϱ2
ˆmin , 2,
fU
p
Sp
pqμ
qgς
ςp
pqμ
qgf
U
p
pqμ
qp
pqμ
qgς
ςgfU
gς
ςgfU
V
pV
qgς
ςgfU
Λ1, 1,Ω ,,Ω ,Ω ,1,Ω 1,Ω
1,Γ
0,Ω ,,Ω ,,Ω 1,Ω 1,Ω 1,Γ
0,Ω 1,Ω 1,Ω 1,Γ
0,Ω 1,Ω 1,Ω 1,Γ
3
3
3
3
(3.11)
where >
M
ˆ
0
0
is defined by {()}≔−−−
M
ccbλ
ˆmin , 1
.
fU
p
S
0Λ 1, 1
Since
()−− >
−
c
cbλ
0
fU
p
S
Λ1, 1, we deduce that His coercive.
It is well-known that
I
K
is a proper, convex and l.s.c. function. Note that (see, e.g., Proposition 1.10 of
Brézis [6])
()≥‖‖ ∈ <Iu αu u V αfor all with some 0
.
KKV K
So, we have
() () ()
⟨
⟩≥ − ≥ ‖‖ ∈∂ ∈κu Iu I αu κ Iu u K, 0 for all and for all ,
KK KV cK
where we have used the fact that ∈K0. Combining the inequality above and (3.11)gives
12 Shengda Zeng et al.
{} ()∣∣⟨+− ⟩≥ ‖‖‖‖+ ‖‖ −‖‖ −‖‖ −‖‖ − − ‖‖
−‖‖‖‖
′
ζκhu M u u aub β βcεαu
Mh u
,ˆmin , 2
V
pV
qgς
ςgfU KV
pV
0,Ω 1,Ω 1,Ω 1,Γ
4,Γ
3
2
for all ()∈ζHu
and for all ()∈∂
κ
Iu
cK with some >
M0
4. Therefore, we infer that (2.8)is satisfied with
=
u0
0
,
=∂I
cK and
=H
. Thus, all conditions of Theorem 2.7 are verified. Using this theorem, we
conclude that problem (1.1)has at least one weak solution
∈
u
K
. Recalling that
(){}≠
f
x,0,0 0
,
u
turns
out to be a nontrivial weak solution of problem (1.1).
II Boundedness:
Suppose that the solution set
()ah,
is unbounded. Then, without loss of generality, there exists a
sequence
{
}()⊂
∈
uah,
nn such that
‖
‖→+∞ →∞unas
.
nV
Employing the same arguments as in the proof of the first part, we obtain the estimate
()≥ ‖‖‖‖+ ‖‖ −‖‖ −‖‖ −‖‖ − ‖‖ ‖‖−
′
MuuaubββMhuM0ˆmin , 2
nV
pnV
qgnς
ςgfU pnV0,Ω 1,Ω 1,Ω 1,Γ 5 ,Γ 5
32 (3.12)
for all ∈nand for some
>
M0
5
. Letting
→∞n
in the inequality above, we obtain a contradiction.
Therefore, the solution set
()ah,
is bounded in
V
.
III Closedness:
Let
{
}()⊂
∈
uah,
nn be a sequence such that
⟶
u
u
V
in
nw
for some
∈
u
K
. Then, there exist functions
(
)
∈ηNu
nfn
and (
)
∈ξNu
nUn
such that
() ()() ()
∫∫∫
⟨
+−−⟩≥−+−+−
∗∗
Fu ωGu βLuvu ηvux hxvu ξvu, d dΓ dΓ
nnnn
nnn
nn
ΩΓ Γ
23
(3.13)
for all
∈
v
K
. Due to the boundedness of the operators
N
fand
NU
we may suppose that there are functions
(
)
∈
′
ηLΩ
r
and
(
)
∈
′
ξLΓ
δ3
satisfying
() ()⟶⟶
′′
ηηL ξξLin Ω and in Γ
.
nwrnwδ3
Taking =
vu
in (3.13)and passing to the upper limit as
→∞n
in the resulting inequality, we obtain
( ∣ ∣ ( ))( ) ()( )
()
∫∫
∫
⟨−⟩≤ + + −+ −
+−≤
→∞ →∞ −→∞
→∞
Fuuu η u ugxuuux hxuu
ξu u
limsup , lim , d lim d
Γ
lim dΓ 0.
nnn nnnpnnn
nn
nnn
Ω
2
Γ
Γ
2
3
Applying Proposition 2.3 we obtain that
→
uu
n
in
V
. Using the upper semicontinuity of
f
and
U
, one has
(
)
∈ηNu
f
and (
)
∈ξNu
U. Passing to the upper limit as
→∞n
in equality (3.13), we derive that
(
)
∈
u
ah,
and so,
()ah,
is weakly closed. This completes the proof. □
4 An inverse problem for double phase elliptic obstacle inclusion
systems
This section is concerned with the study of an inverse problem to identify a discontinuous parameter in the
domain and a discontinuous boundary datum for the double phase elliptic obstacle problem given in (1.1).
For any ()∈gLΩ
1
fixed, in what follows, we denote by (
)
g
T
Vthe total variation of the function g
given by
Identification of discontinuous parameters in double phase obstacle problems 13
() ⎧⎨⎩() () ∣()∣ ⎫⎬⎭
()
∫
≔≤∈
∈
ggxφxxφxx
T
Vsup divd:1for allΩ
.
φCΩ; Ω
N1
By (
)B
VΩ, we denote the function space of all integrable functions with bounded variation, namely,
() { () () }≔∈ <+∞gL g
B
VΩ Ω :TV
.
1
It is well-known that (
)B
VΩ endowed with the norm() ()
()
‖
‖≔‖‖+ ∈ggg gTV for all BV Ω
BV Ω 1,Ω
is a Banach space.
In the sequel, let Hbe a nonempty, closed and convex subset of ()
′
L
Γ
p3. Given positive constants
cΛ
and
d
Λ
, we denote by
Λ
the set of all admissible parameters for the double phase differential operator given in
(1.2)defined by
{() () }≔∈ <≤≤ ∈acaxdxΛ BV Ω : 0 for a.a. Ω
.
ΛΛ
Obviously, we see that the admissible set
Λ
is a closed and convex subset of both (
)B
VΩ and ()
∞
L
Ω.
Given two regularization parameters >
κ0
and >
τ0
and the known observed or measured datum
(
)
∈zLΩ;
pN
, we consider the inverse problem formulated in the following regularized optimal control
framework:
Problem 4.1. Find
∈
∗
aΛ
and
∈
∗
h
H
such that ()( )=
∈∈ ∗∗
Cah Ca hinf , , ,
a
hHΛand (4.1)
where the cost functional
×→
C
H:Λ
is given by
() ()
() ()
≔‖∇−‖++‖‖
∈′
C
ah u z κ a τh,min TV
,
uah Lp
,Ω; ,Γ
pN 2(4.2)
and
()ah,
stands for the solution set of the double phase elliptic obstacle problem (1.1)with respect to
()∈∞
a
LΩand ()∈′
hLΓ
p2.
The main result in this section is the following existence result for the regularized optimal control
problem given in Problem 4.1.
Theorem 4.2. Assume that all conditions of Theorem 3.3 are satisfied. Then the solution set of Problem 4.1
is nonempty and weakly compact.
Proof. The proof of this theorem is divided into four steps.
Step 1: The functional
C
defined in (4.2)is well-defined.
We only need to verify that for
()∈×ah H,Λ
fixed, the optimal problem
() ()
‖∇−‖
∈
uzmin
u
ah L
,Ω;
pN
is solvable. Suppose that
{
}()⊂
∈
uah,
nn is a minimizing sequence of the problem
() (
)
‖∇−‖
∈
uzinf
uah L
,Ω;
pN
,
that is,
() () ()
‖∇−‖ = ‖∇ −‖
∈→∞
uz u zinf lim
.
u
ah LnnL
,Ω; Ω;
pN pN
From Theorem 3.3, we know that
{
}
∈
u
nn
is bounded in
V
. Passing to a subsequence if necessary, we can
assume that
⟶
∗
u
u
nwin
V
for some
∈
∗
uV
. This fact along with the weak closedness of
()ah,
ensures that
()∈
∗
u
ah,. On the other hand, the weak lower semicontinuity of the norm
(
)
‖
⋅‖
LΩ;
pN
implies that
() () () () () ()
‖∇−‖ = ‖∇ −‖ ≥‖∇ −‖ ≥ ‖∇−‖
∈→∞∗∈
uz u z u z uzinf liminf inf
.
u
ah LnnLL
uah L
,Ω; Ω; Ω; ,Ω;
pN pN pN pN
14 Shengda Zeng et al.
This means that for each
()∈×ah H,Λ
there exists ()∈
∗
u
ah,such that
() () ()
‖∇−‖ =‖∇ −‖
∈∗
uz u zinf
.
u
ah LL
,Ω; Ω;
pN pN
Hence,
C
is well-defined.
For any
()∈×ah H,Λ
and
(
)
∈
u
ah,
fixed, it follows from (3.12)that
{}≥ ‖‖‖‖+ ‖‖ −‖‖ −‖‖ −‖‖ − ‖‖ ‖‖−
′
Muuaub β βMhuM0ˆmin , 2
V
pV
qgς
ςgfU pV0,Ω 1,Ω 1,Ω 1,Γ 6 ,Γ 6
32
for some >
M0
6. Therefore, we conclude that
maps bounded sets of () ()×⊂ ×′
HLΛBVΩΓ
p2into
bounded sets of
K
.
Step 2: If
{
()}⊂×
∈
ah
H
,Λ
nnn is a sequence such that
{
}
∈
a
nn is bounded in (
)B
VΩ,→
aa
nin (
)L
Ω
1and
⟶h
h
nwin Hfor some
()()∈×ah L H,Ω
1, then ∈
aΛ
and one has
()()
∅
≠- ⊂
→∞
wahahlimsup , ,
.
nnn (4.3)
Let
{
()}⊂×
∈
ah
H
,Λ
nnn be a sequence such that →
aa
nin (
)L
Ω
1and ⟶h
h
nwin Hfor some
()()∈×ah L H,Ω
1. By the properties of
Λ
(i.e.,
Λ
is nonempty, closed and convex in (
)B
VΩ and (
)L
Ω
1),
one has
()∈×ah H,Λ
. Moreover, the boundedness of
{
}()()⊂∩
∈∞
aLBV Ω Ω
nn and the map
implies that
(
)∪
≥
ah,
nnn
1is bounded in
K
. Also, the reflexivity of
V
guarantees that the set
()-
→∞
w
ahlimsup ,
nnn
is
nonempty.
For any
()∈-
→∞
u
wahlimsup ,
nnn
, passing to a subsequence if necessary, there exists a sequence
{
}⊂
∈
uK
nn such that
()∈⟶
u
ah u u V,and in
.
nnn n
w
Hence, for every ∈n, we are able to find functions
(
)
∈ηNu
nfn
and (
)
∈ξNu
nUn
such that
( ()∣ ∣ ()∣ ∣ ) ( )
()∣∣() ()()
()( ) ()( ) ()( )
∫∫∫
∫∫∫
∇∇+∇∇⋅∇−
+−+−
≥−+−+−
−−
−
ax u u μx u u v u x
μxu uvux gxuvux
ηxv u x hxv u ξxv u
d
d, d
ddΓdΓ
nn
pnn
qnn
nqnn n n
nnnnnn
Ω
22
Ω
2
Ω
ΩΓ Γ
23
(4.4)
for all
∈
v
K
. Taking =
vu
in (4.4)gives
( ()∣ ∣ ()∣ ∣ ) ( ) ()∣ ∣ ( )
()( )
()( ) ()( ) ()( )
∫∫
∫∫∫∫
∇∇+∇∇⋅∇−+ −
+−
≥−+−+−
−− −
ax u u μx u u u u x μxu uu u x
gxu u u x
ηxu u x hxu u ξxu u
dd
,d
ddΓdΓ.
nn
pnn
qnn n
qnn
nn
nnnnnn
Ω
22
Ω
2
Ω
ΩΓ Γ
23
(4.5)
Hypotheses
(
)
Hf
(iii)and (
)
HU(iv)imply that the sequences
{
}
∈
η
nn
and
{
}
∈
ξ
nn
are bounded in
(
)
′
L
Ω
rand
(
)
′
L
Γ
δ3
, respectively. Since the embeddings of
V
to ()
L
Ω
ςand (
)L
Ω
rare compact, we obtain
Identification of discontinuous parameters in double phase obstacle problems 15
()∣∣ ( )
()( )
()( )
()( )
()( )
∫∫∫∫∫
−=
−=
−=
−=
−=
→∞ −
→∞
→∞
→∞
→∞
μx u u u u x
gxu u u x
ηxu u x
hxu u
ξxu u
lim d 0,
lim , d 0,
lim d 0,
lim dΓ 0,
lim dΓ 0,
nnqnn
nnn
nnn
nnn
nnn
Ω
2
Ω
Ω
Γ
Γ
2
3
(4.6)
where we have also used the compactness of ()↪VLΓ
p2and (
)
↪VLΓ
δ3
.
From Simon [50, formula (2.2)] we have the well-known inequalities
∣∣(∣∣∣∣)()−≤ − ⋅− ≥
−−
M
ξη ξ ξ η ηξη s,if 2
,
sss s22 (4.7)
∣ ∣ (∣∣ ∣∣ )( )(∣∣ ∣∣)−≤ − ⋅− + ≤≤
−−
−
ξη ξ ξ η ηξηξ η s,if1 2
,
sss ss222
s
s
2
(4.8)
for all ξ,
∈η
N
with some constants
M
s
,
>
0
sindependent of ξ,
∈η
N
.
Next, we consider the following cases: <<p
1
2and ≥
p
2.If ≥
p
2, then we use (4.7)in order to obtain
()(∣ ∣ ∣ ∣ ) ( )
∫
∇∇−∇∇⋅∇−≥‖−‖
−−
ax u u u u u ux cMu ud,
nn
pnpnpn
W
p
Ω
22 Λ
where the function space Wis given in (3.7). Consider the sets
{}{}
{}
=∈∇≠∪∈∇≠
=∈∇=∇=
xu xu
xuu
ΩΩ:0Ω:0,
ΣΩ: 0.
nn
nn
We observe that
=∪ΩΩ Σ
nn
and
∩=∅ΩΣ
nn
.
By the absolute continuity of the Lebesgue integral, one has
()(∣ ∣ ∣ ∣ ) ( )
∫
∇∇−∇∇⋅∇−=
−−
ax u u u u u uxd0
.
nn
pnpn
Σ
22
n
Hence, we have
()(∣ ∣ ∣ ∣ ) ( )
()(∣ ∣ ∣ ∣ ) ( ) ()(∣ ∣ ∣ ∣ ) ( )
()(∣ ∣ ∣ ∣ ) ( )
∫∫∫
∫∇∇−∇∇⋅∇−
=∇∇−∇∇⋅∇−+∇∇−∇∇⋅∇−
=∇∇−∇∇⋅∇−
−−
−− −−
−−
ax u u u u u ux
ax u u u u u ux ax u u u u u ux
ax u u u u u ux
d
dd
d.
nn
pnpn
nn
pnpnnn
pnpn
nn
pnpn
Ω
22
Ω
22
Σ
22
Ω
22
n n
n
When <<p
1
2, we can apply (4.8)and obtain
()(∣ ∣ ∣ ∣ ) ( )
()(∣ ∣ ∣ ∣ ) ( )(∣ ∣ ∣ ∣)
(∣ ∣ ∣ ∣)
()∣ ∣(∣ ∣ ∣ ∣)
∣∣(∣∣∣∣)
∫∫∫∫
∇∇−∇∇⋅∇−
=∇∇−∇∇⋅∇−
∇+∇
∇+∇
≥∇−∇∇+∇
≥∇−∇∇+∇
−−
−− −−
−
−
ax u u u u u ux
ax u u u u u u uu
uux
ax u u u u x
cuuuux
d
d
d
d.
nn
pnpn
nn
pnpnnpnp
npnp
pn n n
pnp
pn n
pnp
Ω
22
Ω
22
Ω
2
Λ
Ω
2
n
p
p
p
p
n
pp
n
pp
2
2
2
2
(4.9)
Due to <<p
1
2, one has >
1
p
2. Using this and Hölder’s inequality yields
16 Shengda Zeng et al.
∣∣∣∣
⎛⎝∣∣(∣∣∣∣)
⎞⎠(∣ ∣ ∣ ∣)
⎛⎝⎜⎜∣∣(∣∣∣∣)
⎞⎠⎟⎟⎛⎝⎜⎜(∣ ∣ ∣ ∣) ⎞⎠⎟⎟
∫∫
∫∫∫
∇−∇ =∇−∇
=∇−∇∇+∇ ∇+∇
≤∇−∇∇+∇ ×∇+∇
⋅−
−
−−
uux uux
uuuu uux
uuu u x u ux
dd
d
dd.
npn
nn
pp
ppnpp
nn
pp
ppnpp
ΩΩ
2
Ω
22
Ω
22
Ω
p
pp
pp
2
222
222
This means that
∣∣(∣∣∣∣)
⎛⎝⎜⎜∣∣
⎞⎠⎟⎟⎛⎝⎜⎜(∣ ∣ ∣ ∣) ⎞⎠⎟⎟
∫
∫∫
∇−∇∇+∇ ≥∇−∇ ∇+∇ −
−−
uuuux uux uuxdd d
.
nn
pnpnpnpnp
Ω
2
ΩΩ
pp
pp
p
2
22
Combining the inequality above and (4.9), we obtain
()(∣ ∣ ∣ ∣ ) ( )
⎛⎝⎜⎜∣∣
⎞⎠⎟⎟⎛⎝⎜⎜(∣ ∣ ∣ ∣) ⎞⎠⎟⎟
⎛⎝⎜⎜∣∣
⎞⎠⎟⎟
∫∫∫
∫
∇∇−∇∇⋅∇−
≥∇−∇∇+∇
≥∇−∇
−− −−
ax u u u u u ux
cuuxuux
Mc u u x
d
dd
d,
nn
pnpn
pnpnpp
pnp
Ω
22
Λ
ΩΩ
7Λ
Ω
pp
p
p
22
2
(4.10)
where
>
M0
7
is such that ()‖‖+‖‖ ≥
−−
uu M
nW
pW
p7
p
p
2owing to the boundedness of
{
}
∈
u
nn
in
V
and the
continuity of embedding from
V
to W.
Next, we apply Hölder’s inequality to obtain
(( () ())∣ ∣ ) ( )
∣ () ()∣∣ ∣ ∣( )∣
∣ () ()∣ ∣ ∣ ∣ () ()∣∣( )∣
⎛⎝⎜⎜∣() ()∣∣∣ ⎞⎠⎟⎟⎛⎝⎜⎜∣ () ()∣∣( )∣ ⎞⎠⎟⎟
() ⎛⎝⎜⎜∣() ()∣∣∣ ⎞⎠⎟⎟
∫∫∫∫∫
∫
−∇∇⋅∇−
≥− − ∇ ∇−
=− − ∇ − ∇−
≥− −∇ −∇−
≥− ‖−‖ − ∇
−−−
−−
−
ax ax u u u ux
ax ax u u u x
ax ax u ax ax u u x
ax ax u x ax ax u u x
cuu axaxux
d
d
d
dd
2d.
npn
npn
npnn
npnn
p
nW n p
Ω
2
Ω
1
Ω
1
ΩΩ
Λ
Ω
ppp
ppp
p
pp
11
11
1
1
Since →
aa
nin (
)L
Ω
1, without loss of generality, we may assume that () (
)
→
a
xax
nfor a.a. ∈
x
Ω. Passing to
the limit as
→∞n
in the last estimate and using Lebesgue’s dominated convergence theorem as well as the
boundedness of
{
}
∈
u
nn
in Wyields
(( () ())∣ ∣ ) ( ) ⎡⎣⎢⎢⎢⎛⎝⎜⎜∣() ()∣∣∣ ⎞⎠⎟⎟⎤⎦⎥⎥⎥
∫∫
−∇∇⋅∇−≥−‖−‖ −∇ =
→∞ −→∞ −
axaxu uuux cuu axaxuxlim d lim 2 d 0
.
nnpnnnW n p
Ω
2Λ
Ω
pp1
(4.11)
Identification of discontinuous parameters in double phase obstacle problems 17
Therefore, we have
( ()∣ ∣ ()∣ ∣ ) ( )
()(∣ ∣ ∣ ∣ ) ( ) (( () ())∣ ∣ ) ( )
(()∣ ∣ ) ( ) (()∣ ∣ ) ( )
∫∫∫
∫∫
∇∇+∇∇⋅∇−
=∇∇−∇∇⋅∇−+−∇∇⋅∇−
+ ∇∇⋅∇− + ∇∇⋅∇−
−−
−− −
−−
ax u u μx u u u ux
ax u u u u u ux ax ax u u u ux
ax u u u u x μx u u u u x
d
dd
dd.
nn
pnn
qnn
nn
pnpnnpn
pnn
qnn
Ω
22
Ω
22
Ω
2
Ω
2
Ω
2
Passing to the upper limit as
→∞n
in (4.5)and using (4.6),(4.10),(4.11)as well as
(()∣ ∣ ()∣ ∣ ) ( )
(()(∣ ∣ ∣ ∣ )) ( )
∫∫
∇∇+∇∇⋅∇−=
∇∇−∇∇⋅∇−≥
→∞ −−
−−
ax u u μx u u u u x
μx u u u u u u x
lim d 0,
d0,
n
pq
n
nqnqn
Ω
22
Ω
22
we obtain for ≥
p
2‖−‖≤
→∞cM u ulimsup
0
npn W
p
Λ
and for <<p
1
2
‖−‖≤
→∞
uulimsup 0
.
npn
W
2
We conclude that
→
uu
n
in W.
Moreover, the boundedness of
{
}
∈
η
nn
and
{
}
∈
ξ
nn
, as well as the reflexivity of
(
)
′
L
Ω
rand
(
)
′
L
Γ
δ3
permit us
to find functions
(
)
∈
′
ηLΩ
r
and
(
)
∈
′
ξLΓ
δ3
such that, by passing to a subsequence if necessary,
() ()⟶⟶
′′
ηηL ξξLin Ω and in Γ
.
nwrnwδ3
Arguing as in the proof of Theorem 3.3, we obtain
(
)
∈ηNu
f
and (
)
∈ξNu
U. Without loss of generality,
we may assume that () (
)∇
→∇ux ux
nfor a.a. ∈
x
Ω. Applying Lebesgue’s dominated convergence theorem,
we have ( ()∣ ∣ ()∣ ∣ ) ( )
( ()∣ ∣ ()∣ ∣ ) ( )
(()∣ ∣ ()∣ ∣ ) ( )
∫∫∫∇∇+∇∇⋅∇−
=∇∇+∇∇⋅∇−
=∇∇+∇∇⋅∇−
→∞ −−
→∞ −−
−−
ax u u μx u u v u x
ax u u μx u u v u x
ax u u μx u u v u x
lim d
lim d
d.
nnn
pnn
qnn
nnn
pnn
qnn
pq
Ω
22
Ω
22
Ω
22
Letting
→∞n
in equality (4.4)and using the aforementioned convergence properties we obtain
(()∣∣ ()∣∣)() ()∣∣() ()()
()( ) ()( ) ()( )
∫∫∫
∫∫∫
∇∇+∇∇⋅∇−+ −+ −
≥−+−+−
−− −
ax u u μx u u v ux μxu uv ux gxuv ux
ηxv ux hxv u ξxv u
dd,d
ddΓdΓ
pq q
Ω
22
Ω
2
Ω
ΩΓ Γ
23
for all
∈
v
K
. Therefore, we can observe that
∈
u
K
is a solution of problem (1.1)corresponding to
()∈×ah H,Λ
, that is,
(
)
∈
u
ah,
. Hence, ()()-⊂
→∞
w
ah ahlimsup , ,
nnn and so we have proved (4.3).
Step 3: If
{
()}⊂×
∈
ah
H
,Λ
nnn is such that
{
}
∈
a
nn is bounded in (
)B
VΩ,→
aa
nin (
)L
Ω
1and ⟶h
h
nwin
()
′
L
Γ
p2
for some
()()∈×ah L H,Ω
1, then the inequality
18 Shengda Zeng et al.
() ( )≤→∞
C
ah Ca h, liminf ,
nnn (4.12)
holds.
Let
{
()}⊂×
∈
ah
H
,Λ
nnn be such that →
aa
nin (
)L
Ω
1and ⟶h
h
nwin
()
′
L
Γ
p2
for some
()()∈×ah L H,Ω
1. From Step 2 one has ∈
aΛ
. Let
{
}⊂
∈
uK
nn be a sequence such that
()
() () (
)
∈‖∇−‖=‖∇−‖
∈
u
ah u z u z,andinf
nnn uah LnL
,Ω; Ω;
nn
pN pN (4.13)
for each ∈n.
Recalling that
(
)∪
≥
ah,
nnn
1is bounded, passing to a subsequence if necessary, we have
⟶
∗
u
u
nwin
V
for some
∈
∗
u
K
, that is,
()∈-
∗→∞
u
wahlimsup ,
nnn
. Applying again Step 2, we conclude that ()∈
∗
u
ah,.
Therefore, from the lower semicontinuity of the function
() ()∋↦ ∈
L
aaΩTV
1
and the weak lower semi-
continuity of
()
∋↦‖∇−‖ ∈Wu uz
LΩ;
pN
and
()∋↦‖‖∈
′′
L
hhΓ
pp2,Γ
2
, it follows that
() [ () ]
()
() ()
()
()
()
()
() ()
=‖∇−‖++‖‖
≥‖∇−‖+ +‖‖
≥‖∇ −‖ + +‖‖
≥‖∇−‖++‖‖
=
→∞ →∞ ′
→∞ →∞ →∞ ′
∗′
∈′
Ca h u z κ a τh
uz κa τh
uz κaτh
uz κ a τh
Cah
liminf , liminf TV
liminf liminf TV liminf
TV
inf TV
,.
nnn nnLnnp
nnLnnnnp
Lp
uah Lp
Ω; ,Γ
Ω; ,Γ
Ω; ,Γ
,Ω; ,Γ
pN
pN
pN
pN
2
2
2
2
Hence (4.12)follows.
Step 4: The solution set of Problem 4.1 is nonempty and weakly compact.
By the definition of
C
, we see that
C
is bounded from below. Let
{
()}⊂×
∈
ah
H
,Λ
nnn be a minimizing
sequence of 4.1, namely, () ( )=
∈∈ →∞
Cah Ca hinf , lim ,
.
a
hH n nn
Λand (4.14)
This indicates that the sequences
{
}⊂
∈
a
Λ
nn and
{
}()⊂
∈′
hLΓ
nn p2are bounded in (
)B
VΩ and
()
′
L
Γ
p2
,
respectively. Passing to a subsequence if necessary we have
() ()→⟶
∗∗
′
a
aL hhLin Ω and in Γ
nn
wp
12(4.15)
for some () (
)
∈×
∗∗ ′
ah L,ΛΓ
p2, where we have used the closedness of
Λ
in (
)L
Ω
1and the compactness of the
embedding (
)B
VΩ to (
)L
Ω
1.
Let us consider a sequence
{
}⊂
∈
uK
nn satisfying (4.13). Employing the convergence (4.15)and the
boundedness of
, it implies that
{
}
∈
u
nn
is bounded in
V
. So, we are able to select a subsequence of
{
}
∈
u
nn
,
not relabeled, such that
⟶
∗
u
u
nwin
V
for some
∈
∗
u
K
. From Step 2 it is clear that (
)
∈
∗∗∗
u
ah,. Therefore,
we have
() [ () ]
()
() ()
()
()
()
()
()
() ()
=‖∇−‖++‖‖
≥‖∇−‖+ +‖‖
≥‖∇ −‖ + +‖‖
≥‖∇−‖++‖‖
=≥
→∞ →∞ ′
→∞ →∞ →∞ ′
∗∗∗
′
∈∗∗
′
∗∗
∈∈
∗∗
Ca h u z κ a τh
uz κ a τ h
uz κa τh
uz κ a τh
Ca h Cah
liminf , liminf TV
liminf liminfTV liminf
TV
inf TV
,
inf , .
nnn nnLnnp
nnLnnnnp
Lp
uah Lp
ahH
Ω; ,Γ
Ω; ,Γ
Ω; ,Γ
,Ω; ,Γ
Λand
pN
pN
pN
pN
2
2
2
2
(4.16)
The latter combined with (4.14)implies that ()∈×
∗∗
ah H,Λ
is a solution of Problem 4.1.
Finally, we prove the weak compactness of the solution set to Problem 4.1. To this end, let
{
()}
∈
ah,
nnn
be a sequence of solutions to Problem 4.1. It is obvious that
{
}⊂
∈
a
Λ
nn is bounded in (
)B
VΩ and
{
}
∈
h
nn
is bounded in
()
′
L
Γ
p2
. Using the same arguments, we may assume that (4.15)holds with some
() (
)
∈×
∗∗ ′
ah L,ΛΓ
p2. Similarly, there exists a sequence
{
}
∈
u
nn
such that (4.13)is fulfilled and
⟶
∗
u
u
nw
Identification of discontinuous parameters in double phase obstacle problems 19
in
V
for some ()∈
∗∗∗
u
Sa h,. As done before, we can prove the validity of (4.16). This means that
()∈×
∗∗
ah H,Λ
is a solution of Problem 4.1. Therefore, the solution set of Problem 4.1 is weakly compact.
This completes the proof. □
Remark 4.3. The results of this section remain valid if the functional (4.2)is replaced by the following
regularized cost functional:
() ()
() ()
=‖∇−‖++‖‖
∈′
C
ah u z κ a τh, min TV ,
uah L
ωp
,Ω; ,Γ
pN
ω
12
2
where
<≤ω
p1
1
and <≤′ωp
1
2. The latter for ==
ω
ω2
12is the most popular and commonly used the
output least-squares objective functional utilized in the numerical approaches.
Acknowledgments: The authors acknowledge support by the German Research Foundation and the Open
Access Publication Fund of TU Berlin.
Funding information: This project has received funding from the NNSF of China Grant Nos. 12001478 and
12101143, the European Union’s Horizon 2020 Research and Innovation Programme under the Marie
Sklodowska-Curie grant agreement No. 823731 CONMECH, National Science Center of Poland under
Preludium Project No. 2017/25/N/ST1/00611 and the Startup Project of Doctor Scientific Research of Yulin
Normal University No. G2020ZK07. It is also supported by Natural Science Foundation of Guangxi Grants
Nos. 2021GXNSFFA196004, 2020GXNSFBA297137 and GKAD21220144 and the Ministry of Science and
Higher Education of Republic of Poland under Grants Nos. 4004/GGPJII/H2020/2018/0 and 440328/
PnH2/2019.
Conflict of interest: The authors declare that they have no conflict of interest.
References
[1]A. Bahrouni, V. D. Rădulescu, and P. Winkert, Double phase problems with variable growth and convection for the
Baouendi-Grushin operator, Z. Angew. Math. Phys. 71 (2020), no. 6, Paper no. 183, 15 pp.
[2]P. Baroni, M. Colombo, and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal. 121 (2015),
206–222.
[3]P. Baroni, M. Colombo, and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differ.
Equ. 57 (2018), no. 2, 62, 48 pp.
[4]O, Benslimane, A. Aberqi, and J. Bennouna, Existence and uniqueness of entropy solution of a nonlinear elliptic equation in
anisotropic Sobolev-Orlicz space, Rend. Circ. Mat. Palermo (2)70 (2021), no. 3, 1579–1608.
[5]S. Biagi, F. Esposito, and E. Vecchi, Symmetry and monotonicity of singular solutions of double phase problems,J.Differ.
Equ. 280 (2021), 435–463.
[6]H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
[7]S.-S. Byun and J. Oh, Regularity results for generalized double phase functionals, Anal. PDE 13 (2020), no. 5, 1269–1300.
[8]S. Carl and V. K. Le, Multi-valued Variational Inequalities and Inclusions, Springer, Cham, 2021.
[9]S. Carl, V. K. Le, and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities, Springer, New York, 2007.
[10]C. Clason, A. A. Khan, M. Sama, and C. Tammer, Contingent derivatives and regularization for noncoercive inverse
problems, Optimization 68 (2019), no. 7, 1337–1364.
[11]F. Colasuonno and M. Squassina, Eigenvalues for double phase variational integrals, Ann. Mat. Pura Appl. (4)195 (2016),
no. 6, 1917–1959.
[12]M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218
(2015), no. 1, 219–273.
[13]M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215 (2015),
no. 2, 443–496.
[14]Á. Crespo-Blanco, L. Gasiński, P. Harjulehto, and P. Winkert, A new class of double phase variable exponent problems:
Existence and uniqueness.J.Differ. Equ. 322 (2022), 182–228.
20 Shengda Zeng et al.
[15]Z. Denkowski, S. Migórski, and N. S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic
Publishers, Boston, MA, 2003.
[16]S. El Manouni, G. Marino, and P. Winkert, Existence results for double phase problems depending on Robin and Steklov
eigenvalues for the p-Laplacian, Adv. Nonlinear Anal. 11 (2022), no. 1, 304–320.
[17]F. Faraci, D. Motreanu, and D. Puglisi, Positive solutions of quasi-linear elliptic equations with dependence on the
gradient, Calc. Var. Partial Differ. Equ. 54 (2015), no. 1, 525–538.
[18]F. Faraci and D. Puglisi, A singular semilinear problem with dependence on the gradient,J.Differ. Equ. 260 (2016), no. 4,
3327–3349.
[19]C. Farkas and P. Winkert, An existence result for singular Finsler double phase problems,J.Differ. Equ. 286 (2021),
455–473.
[20]G. M. Figueiredo and G. F. Madeira, Positive maximal and minimal solutions for non-homogeneous elliptic equations
depending on the gradient,J.Differ. Equ. 274 (2021), 857–875.
[21]A. Fiscella, A double phase problem involving Hardy potentials, Appl. Math. Optim. 85 (2022), 45.
[22]L. Gasiński and N. S. Papageorgiou, Constant sign and nodal solutions for superlinear double phase problems, Adv. Calc.
Var. 14 (2021), no. 4, 613–626.
[23]L. Gasiński and N. S. Papageorgiou, Positive solutions for nonlinear elliptic problems with dependence on the gradient,
J. Differ. Equ. 263 (2017), 1451–1476.
[24]L. Gasiński and P. Winkert, Constant sign solutions for double phase problems with superlinear nonlinearity, Nonlinear
Anal. 195 (2020), 111739.
[25]L. Gasiński and P. Winkert, Existence and uniqueness results for double phase problems with convection term,J.Differ.
Equ. 268 (2020), no. 8, 4183–4193.
[26]L. Gasiński and P. Winkert, Sign changing solution for a double phase problem with nonlinear boundary condition via the
Nehari manifold,J.Differ. Equ. 274 (2021), 1037–1066.
[27]J. Gwinner, An optimization approach to parameter identification in variational inequalities of second kind, Optim. Lett. 12
(2018), no. 5, 1141–1154.
[28]J. Gwinner, B. Jadamba, A. A. Khan and M. Sama, Identification in variational and quasi-variational inequalities, J. Convex
Anal. 25 (2018), no. 2, 545–569.
[29]V. K. Le, A range and existence theorem for pseudomonotone perturbations of maximal monotone operators, Proc. Amer.
Math. Soc. 139 (2011), no. 5, 1645–1658.
[30]A. Lê, Eigenvalue problems for the p-Laplacian, Nonlinear Anal. 64 (2006), no. 5, 1057–1099.
[31]W. Liu and G. Dai, Existence and multiplicity results for double phase problem,J.Differ. Equ. 265 (2018), no. 9, 4311–4334.
[32]Z. Liu, D. Motreanu, and S. Zeng, Positive solutions for nonlinear singular elliptic equations of
p
-Laplacian type with
dependence on the gradient, Calc. Var. Partial Differ. Equ. 58 (2019), no. 1, Paper no. 28, 22 pp.
[33]Z. Liu and N. S Papageorgiou, Positive solutions for resonant (
pq,
)-equations with convection, Adv. Nonlinear Anal. 10
(2021), no. 1, 217–232.
[34]W. Liu and P. Winkert, Combined effects of singular and superlinear nonlinearities in singular double phase problems in
N
, J. Math. Anal. Appl. 507 (2022), no. 2, 125762, 19 pp.
[35]S. A. Marano and P. Winkert, On a quasilinear elliptic problem with convection term and nonlinear boundary condition,
Nonlinear Anal. 187 (2019), 159–169.
[36]P. Marcellini, Regularity and existence of solutions of elliptic equations with
pq,
-growth conditions,J.Differ. Equ. 90
(1991), no. 1, 1–30.
[37]P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch.
Rational Mech. Anal. 105 (1989), no. 3, 267–284.
[38]S. Migórski, A. A. Khan, and S. Zeng, Inverse problems for nonlinear quasi-hemivariational inequalities with application to
mixed boundary value problems, Inverse Problems 36 (2020), no. 2, 024006, 20 pp.
[39]S. Migórski, A. A. Khan, and S. Zeng, Inverse problems for nonlinear quasi-variational inequalities with an application to
implicit obstacle problems of
p
-Laplacian type, Inverse Problems 35 (2019), no. 3, 035004, 14 pp.
[40]S. Migórski and A. Ochal, An inverse coefficient problem for a parabolic hemivariational inequality, Appl. Anal. 89 (2010),
no. 2, 243–256.
[41]S. Migórski, A. Ochal, and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities, Springer, New York, 2013.
[42]P. D. Panagiotopoulos, Nonconvex problems of semipermeable media and related topics, Z. Angew. Math. Mech. 65
(1985), no. 1, 29–36.
[43]P. D. Panagiotopoulos, Hemivariational Inequalities, Springer-Verlag, Berlin, 1993.
[44]N. S. Papageorgiou, V. D. Rădulescu, and D. D. Repovš,Positive solutions for nonlinear Neumann problems with singular
terms and convection, J. Math. Pures Appl. (9)136 (2020),1–21.
[45]N. S. Papageorgiou and C. Vetro, Existence and relaxation results for second order multivalued systems, Acta Appl. Math.
173 (2021), Paper No. 5, 36 pp.
[46]N. S. Papageorgiou, C. Vetro, and F. Vetro, Solutions for parametric double phase Robin problems, Asymptot. Anal. 121
(2021), no. 2, 159–170.
[47]N. S. Papageorgiou and P. Winkert, Applied Nonlinear Functional Analysis: An Introduction, De Gruyter, Berlin, 2018.
Identification of discontinuous parameters in double phase obstacle problems 21
[48]K. Perera and M. Squassina, Existence results for double-phase problems via Morse theory, Commun. Contemp. Math.
20 (2018), no. 2, 1750023, 14 pp.
[49]M. A. Ragusa and A. Tachikawa, Regularity for minimizers for functionals of double phase with variable exponents,
Adv. Nonlinear Anal. 9(2020), no. 1, 710–728.
[50]J. Simon, Régularité de la solution d’une équation non linéaire dans
N
, Journées d’Analyse Non Linéaire 665 (1978),
205–227, (Proc. Conf. Besançon, 1977, Springer, Berlin).
[51]R. Stegliński, Infinitely many solutions for double phase problem with unbounded potential in
N
, Nonlinear Anal. 214
(2022), Paper No. 112580.
[52]S. Zeng, Y. Bai, L. Gasiński, and P. Winkert, Convergence analysis for double phase obstacle problems with multivalued
convection term, Adv. Nonlinear Anal. 10 (2021), no. 1, 659–672.
[53]S. Zeng, Y. Bai, L. Gasiński, and P. Winkert, Existence results for double phase implicit obstacle problems involving
multivalued operators, Calc. Var. Partial Differ. Equ. 59 (2020), no. 5, 176, 18pp.
[54]S. Zeng, L. Gasiński, P. Winkert, and Y. Bai, Existence of solutions for double phase obstacle problems with multivalued
convection term, J. Math. Anal. Appl. 501 (2021), no. 1, 123997.
[55]S. Zeng and N. S. Papageorgiou, Positive solutions for (
pq,
)-equations with convection and a sign-changing reaction,
Adv. Nonlinear Anal. 11 (2022), no. 1, 40–57.
[56]V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat.
50 (1986), no. 4, 675–710.
[57]V. V. Zhikov, On Lavrentiev’s phenomenon, Russian J. Math. Phys. 3(1995), no. 2, 249–269.
[58]V. V. Zhikov, On variational problems and nonlinear elliptic equations with nonstandard growth conditions, J. Math. Sci.
173 (2011), no. 5, 463–570.
22 Shengda Zeng et al.
