- Research Article
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Regularity of Parabolic Hemivariational Inequalities with Boundary Conditions
Journal of Inequalities and Applications volume 2009, Article number: 207873 (2009)
Abstract
We prove the regularity for solutions of parabolic hemivariational inequalities of dynamic elasticity in the strong sense and investigate the continuity of the solution mapping from initial data and forcing term to trajectories.
1. Introduction
In this paper, we deal with the existence and a variational of constant formula for solutions of a parabolic hemivariational inequality of the form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ1_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ2_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ3_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ4_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ5_HTML.gif)
where is a bounded domain in
with sufficiently smooth boundary
Let
,
,
The boundary
is composed of two pieces
and
, which are nonempty sets and defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ6_HTML.gif)
where is the unit outward normal vector to
. Here
,
is the displacement,
is the strain tensor,
,
is a multi-valued mapping by filling in jumps of a locally bounded function
,
. A continuous map
from the space
of
symmetric matrices into itself is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ7_HTML.gif)
where is the identity of
,
denotes the trace of
, and
,
. For example, in the case
,
, where
is Young's modulus,
is Poisson's ratio and
is the density of the plate.
Let and
be two complex Hilbert spaces. Assume that
is a dense subspace in
and the injection of
into
is continuous. Let
be a continuous linear operator from
into
which is assumed to satisfy GÃ¥rding's inequality. Namely, we formulated the problem (1.1) as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ8_HTML.gif)
The existence of global weak solutions for a class of hemivariational inequalities has been studied by many authors, for example, parabolic type problems in [1–4], and hyperbolic types in [5–7]. Rauch [8] and Miettinen and Panagiotopoulos [1, 2] proved the existence of weak solutions for elliptic one. The background of these variational problems are physics, especially in solid mechanics, where nonconvex and multi-valued constitutive laws lead to differential inclusions. We refer to [3, 4] to see the applications of differential inclusions. Most of them considered the existence of weak solutions for differential inclusions of various forms by using the Faedo-Galerkin approximation method.
In this paper, we prove the existence and a variational of constant formula for strong solutions of parabolic hemivariational inequalities. The plan of this paper is as follows. In Section 2, the main results besides notations and assumptions are stated. In order to prove the solvability of the linear case with we establish necessary estimates applying the result of Di Blasio et al. [9] to (1.1)–(1.5) considered as an equation in
as well as
. The existence and regularity for the nondegenerate nonlinear systems has been developed as seen in [10, Theorem 4.1] or [11, Theorem 2.6], and the references therein. In Section 3, we will obtain the existence for solutions of (1.1)–(1.5) by converting the problem into the contraction mapping principle and the norm estimate of a solution of the above nonlinear equation on
. Consequently, if
is a solution asociated with
, and
, in view of the monotonicity of
, we show that the mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ9_HTML.gif)
is continuous.
2. Preliminaries and Linear Hemivariational Inequalities
We denote for
,
and
for
. Throughout this paper, we consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ10_HTML.gif)
We denote the dual space of
,
the dual pairing between
and
.
The norms on ,
, and
will be denoted by
,
and
, respectively. For the sake of simplicity, we may consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ11_HTML.gif)
We denote by
. Let
be the operator associated with a sesquilinear form
which is defined GÃ¥rding's inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ12_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ13_HTML.gif)
Then is a symmetric bounded linear operator from
into
which satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ14_HTML.gif)
and its realization in which is the restriction of
to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ15_HTML.gif)
is also denoted by . Here, we note that
is dense in
. Hence, it is also dense in
. We endow the domain
of
with graph norm, that is, for
, we define
. So, for the brevity, we may regard that
for all
. It is known that—
generates an analytic semigroup
in both
and
.
From the following inequalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ16_HTML.gif)
it follows that there exists a constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ17_HTML.gif)
So, we may regard as where
is the real interpolation space between
and
.
Consider the following initial value problem for the abstract linear parabolic type equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ18_HTML.gif)
A continuous map from the space
of
symmetric matrices into itself is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ19_HTML.gif)
It is easily known that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ20_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ21_HTML.gif)
Note that the map is linear and symmetric and it can be easily verified that the tensor
satisfies the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ22_HTML.gif)
Let be the smallest positive constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ23_HTML.gif)
Simple calculations and Korn's inequality yield that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ24_HTML.gif)
and hence is equivalent to the
norm on
Then by virtue of [9, Theorem 3.3], we have the following result on the linear parabolic type equation (LE).
Proposition 2.1.
Suppose that the assumptions stated above are satisfied. Then the following properties hold.
(1)For any and
, there exists a unique solution
of (LE) belonging to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ25_HTML.gif)
and satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ26_HTML.gif)
where is a constant depending on
.
(2)Let and
for any
. Then there exists a unique solution
of (LE) belonging to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ27_HTML.gif)
and satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ28_HTML.gif)
where is a constant depending on
.
Proof.
-
(1)
Let
be a bounded sesquilinear form defined in
by
(2.19)
Noting that by(2.10)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ30_HTML.gif)
and by (2.12), (2.14), and (1.6),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ31_HTML.gif)
it follows that there exist and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ32_HTML.gif)
Let be the operator associated with this sesquilinear form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ33_HTML.gif)
Then is also a symmetric continuous linear operator from
into
which satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ34_HTML.gif)
So we know that— generates an analytic semigroup
in both
and
. Hence, by applying [9, Theorem 3.3] to the regularity for the solution of the equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ35_HTML.gif)
in the space , we can obtain a unique solution
of (LE) belonging to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ36_HTML.gif)
and satisfying the norm estimate (2.16).
-
(2)
It is easily seen that
(2.27)
for the time . Therefore, in terms of the intermediate theory we can see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ38_HTML.gif)
and follow the argument of (1) term by term to deduce the proof of (2) results.
3. Existence of Solutions in the Strong Sense
This Section is to investigate the regularity of solutions for the following parabolic hemivariational inequality of dynamic elasticity in the strong sense:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ39_HTML.gif)
Now, we formulate the following assumptions.
(Hb) Let be a locally bounded function verifying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ40_HTML.gif)
where We denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ41_HTML.gif)
The multi-valued function is obtained by filling in jumps of a function
by means of the functions
as follows.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ42_HTML.gif)
We denote ,
for
. We will need a regularization of
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ43_HTML.gif)
where and
It is easy to show that
is continuous for all
and
satisfy the same condition (Hb) with possibly different constants if
satisfies (Hb). It is also known that
is locally Lipschitz continuous in
, that is for any
, there exists a number
such that
(Hb-1)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ44_HTML.gif)
holds for all with
We denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ45_HTML.gif)
The following lemma is from [[12]; Lemma A.5].
Lemma 3.1.
Let satisfying
for all
and
be a constant. Let
be a continuous function on
satisfying the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ46_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ47_HTML.gif)
Proof.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ48_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ49_HTML.gif)
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ50_HTML.gif)
Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ51_HTML.gif)
Since is absolutely continuous and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ52_HTML.gif)
for all , it holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ53_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ54_HTML.gif)
Therefore, combining this with (3.11), we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ55_HTML.gif)
for arbitrary .
From now on, we establish the following results on the local solvability of the following equation,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ56_HTML.gif)
Lemma 3.2.
Let be a solution of (HIE-1) and
. Then, the following inequality holds, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ57_HTML.gif)
where .
Proof.
We remark that from (2.11), (2.12), it follows that there is a constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ58_HTML.gif)
Consider the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ59_HTML.gif)
Multipying on both sides of , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ60_HTML.gif)
and integrating this over , by (1.6), (2.5), (3.18) and (Hb-1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ61_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ62_HTML.gif)
Applying Gronwall lemma, the proof of the lemma is complete.
Theorem 3.3.
Assume that ,
and (Hb). Then, there exists a time
such that (HIE-1) admits a unique solution
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ63_HTML.gif)
Proof.
Assume that (2.5) holds for . Let the constant
satisfy the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ64_HTML.gif)
Let us fix such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ65_HTML.gif)
where is given by (Hb).
Invoking Proposition 2.1, for a given , the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ66_HTML.gif)
has a unique solution . To prove the existence and uniqueness of solutions of semilinear type (HIE-1), by virtue of Lemma 3.2, we are going to show that the mapping defined by
maps is strictly contractive from
into itself if the condition (3.25) is satisfied.
Lemma 3.4.
Let be the solutions of (HIE-2) with
replaced by
where
is the ball of radius
centered at zero of
, respectively. Then the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ67_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ68_HTML.gif)
Proof.
Let be the solutions of (HIE-2) with
replaced by
, respectively. Then, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ69_HTML.gif)
Multiplying on both sides of and by (2.8), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ70_HTML.gif)
and so, by (3.18), (2.5), (Hb), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ71_HTML.gif)
Putting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ72_HTML.gif)
and integrating (3.30) over , this yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ73_HTML.gif)
From (3.32) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ74_HTML.gif)
Integrating (3.33) over we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ75_HTML.gif)
thus, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ76_HTML.gif)
From (3.32) and (3.35) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ77_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ78_HTML.gif)
By using Lemma 3.1, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ79_HTML.gif)
The proof of lemma is complete.
From (3.26) and (3.36) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ80_HTML.gif)
Starting from the initial value , consider a sequence
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ81_HTML.gif)
Then from (3.39) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ82_HTML.gif)
So by virtue of the condition (3.25) the contraction principle gives that there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ83_HTML.gif)
and hence, from (3.41) there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ84_HTML.gif)
Now, we give a norm estimation of the solution (HIE) and establish the global existence of solutions with the aid of norm estimations.
Theorem 3.5.
Let the assumption (Hb) be satisfied. Assume that and
for any
. Then, the solution
of (HIE) exists and is unique in
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ85_HTML.gif)
Furthermore, there exists a constant depending on
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ86_HTML.gif)
Proof.
Let be the solution of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ87_HTML.gif)
Then, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ88_HTML.gif)
by multiplying by , from (Hb), (3.18) and the monotonicity of
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ89_HTML.gif)
By integrating on (3.48) over we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ90_HTML.gif)
By the procedure similar to (3.39) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ91_HTML.gif)
Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ92_HTML.gif)
Then it holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ93_HTML.gif)
and hence, from (2.16) in Proposition 2.1, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ94_HTML.gif)
for some positive constant . Noting that by (Hb)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ95_HTML.gif)
and by Proposition 2.1
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ96_HTML.gif)
it is easy to obtain the norm estimate of in
satisfying (3.45).
Now from Theorem 3.3 it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ97_HTML.gif)
So, we can solve the equation in and obtain an analogous estimate to (3.53). Since the condition (3.25) is independent of initial values, the solution of (HIE-1) can be extended the internal
for a natural number
, that is, for the initial
in the interval
, as analogous estimate (3.53) holds for the solution in
. Furthermore, the estimate (3.45) is easily obtained from (3.53) and (3.56).
We show that is a solution of the problem (HIE). Lemma 3.4 and (Hb) give that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ98_HTML.gif)
and for , there exists a unique solution
of (HIE) belonging to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ99_HTML.gif)
and satisfying (3.44).
From (3.44)and (3.57), we can extract a subsequence from , still denoted by
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ100_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ101_HTML.gif)
Here, we remark that if is compactly embedded in
and
, the following embedding
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ102_HTML.gif)
is compact in view [13, Theorem 2]. Hence, the mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ103_HTML.gif)
By a solution of (HIE-1), we understand a mild solution that has a form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ104_HTML.gif)
so letting and using the convergence results above, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ105_HTML.gif)
Now, we show that a.e. in
. Indeed, from (3.62) we have
strongly in
and hence
a.e. in
for each
Let
and
Using the theorem of Lusin and Egoroff, we can choose a subset
such that
and
uniformly on
Thus, for each
there is an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ106_HTML.gif)
Then, if we have
for all
and
Therefore we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ107_HTML.gif)
Let Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ108_HTML.gif)
Letting in this inequality and using (3.60), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ109_HTML.gif)
where Letting
in this inequality, we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ110_HTML.gif)
and letting we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ111_HTML.gif)
This implies that a.e. in
This completes the proof of theorem.
Remark 3.6.
In terms of Proposition 2.1, we remark that if and
for any
then the solution
of (HIE) exists and is unique in
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ112_HTML.gif)
Futhermore, there exists a constant depending on
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ113_HTML.gif)
Theorem 3.7.
Let the assumption (Hb) be satisfied
(1)if , then the solution
of (HIE) belongs to
and the mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ114_HTML.gif)
is continuous.
-
(2)
let
. Then the solution
of (HIE) belongs to
and the mapping
(3.74)
is continuous.
Proof.
-
(1)
It is easy to show that if
and
, then
belongs to
. let
and
be the solution of (HIE) with
in place of
for
Then in view of Proposition 2.1, we have
(3.75)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ117_HTML.gif)
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ118_HTML.gif)
Hence, arguing as in (2.8), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ119_HTML.gif)
Combining (3.75) with (3.78), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ120_HTML.gif)
Suppose that in
and let
and
be the solutions (HIE) with
and
, respectively. Let
be such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ121_HTML.gif)
Then by virtue of (3.79) with replaced by
we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ122_HTML.gif)
This implies that in
. Hence the same argument shows that
in
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ123_HTML.gif)
Repeating this process we conclude that in
-
(2)
If
then
belongs to
from Theorem 3.5. Let
and
be the solution of (HIE) with
in place of
for
. Multiplying (HIE) by
, we have
(3.83)
Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ125_HTML.gif)
Then, by the similar argument in (3.32), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ126_HTML.gif)
and we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ127_HTML.gif)
thus, arguing as in (3.35) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ128_HTML.gif)
Combining this inequality with (3.85) it holds that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ129_HTML.gif)
By Lemma 3.1 the following inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ130_HTML.gif)
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ131_HTML.gif)
Hence, from (3.88) and (3.90) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ132_HTML.gif)
The last term of (3.91) is estimated as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ133_HTML.gif)
Let be such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ134_HTML.gif)
Hence, from (3.91) and (3.92) it follows that there exists a constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ135_HTML.gif)
Suppose in
, and let
and
be the solutions (HIE) with
and
, respectively. Then, by virtue of (3.94), we see that
in
. This implies that
in
. Therefore the same argument shows that
in
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F207873/MediaObjects/13660_2008_Article_1917_Equ136_HTML.gif)
Repeating this process, we conclude that in
.
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The authors wish to thank the referees for careful reading of manuscript, for valuable suggestions and many useful comments.
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Park, DG., Jeong, JM. & Park, S.H. Regularity of Parabolic Hemivariational Inequalities with Boundary Conditions. J Inequal Appl 2009, 207873 (2009). https://doi.org/10.1155/2009/207873
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DOI: https://doi.org/10.1155/2009/207873
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