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Existence of weak solutions for Cahn-Hilliard systems coupled
with elasticity and damage
Christian Heinemann∗
, Christiane Kraus∗
June 11, 2010
Abstract
A typical phase field approach for describing phase separation and coarsening phenomena in alloys
is the Cahn-Hilliard model. This model has been generalized to the so-called Cahn-Larch´e system by
combining it with elasticity to capture non-neglecting deformation phenomena, which occur during
phase separation and coarsening processes in the material. In order to account for damage effects,
we extend the existing framework of Cahn-Hilliard and Cahn-Larch´e systems by incorporating an
internal damage variable of local character. This damage variable allows to model the effect that
damage of a material point is influenced by its local surrounding. The damage process is described
by a unidirectional rate-dependent evolution inclusion for the internal variable. For the introduced
Cahn-Larch´e systems coupled with rate-dependent damage processes, we establish a suitable notion
of weak solutions and prove existence of weak solutions.
AMS Subject classifications: 35K85, 49J40, 74C10, 82C26, 35J50 35K35, 35K55.
Keywords: Cahn-Hilliard systems, phase separation, damage, elliptic-parabolic systems, energetic solu-
tion, weak solution, doubly nonlinear differential inclusions, existence results, rate-dependent systems.
This project is supported by the DFG Research Center “Mathematics for Key Technologies” Matheon in
Berlin.
1 Introduction
Due to the ongoing miniaturization in the area of micro-electronics the demands on strength and lifetime
of the materials used is considerably rising, while the structural size is continuously being reduced.
Materials, which enable the functionality of technical products, change the microstructure over time.
Phase separation and coarsening phenomena take place and the complete failure of electronic devices like
motherboards or mobile phones often results from micro–cracks in solder joints.
Solder joints, for instance, are essential components in electronic devices since they form the electrical
and the mechanical bond between electronic components like micro–chips and the circuit–board. The
Figures 1 and 2 illustrate the typical morphology in the interior of solder materials. At high temperatures,
one homogeneous phase consisting of different components of the alloy is energetically favourable. If the
temperature is decreased below a critical value a fine microstructure of two or more phases (different
compositions of the components of the material) arises on a very short time scale. The formation of
microstructures, also called phase separation or spinodal decomposition, take place to reduce the bulk
chemical free energy. Then coarsening phenomena occur, which are mainly driven by decreasing interfacial
energy. Due to the misfit of the crystal lattices, the different heat expansion coefficients and the different
elastic moduli of the components, very high mechanical stresses occur preferably at the interfaces of the
∗Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117 Berlin (Germany). E-mail:
christian.heinemann@wias-berlin.de and christiane.kraus@wias-berlin.de
1
arXiv:1502.05826v1 [math.AP] 20 Feb 2015
phases. These stress concentrations initiate the nucleation of micro–cracks, whose propagation can finally
lead to the failure of the whole electronic device.
Figure 1: Left: Solder ball and micro–structural coarsening in eutectic Sn–Pb; Right: a) directly after
solidification, b) after 3 hours, and c) after 300 hours [HCW91];
Figure 2: Initiation and propagation of cracks along the phase boundary [FBFD06].
The knowledge of the mechanisms inducing phase separation, coarsening and damage phenomena is of
great importance for technological applications. A uniform distribution of the original materials is aimed
to guarantee evenly distributed material properties of the sample. For instance, mechanical properties,
such as the strength and the stability of the material, depend on how finely regions of the original
materials are mixed. The control of the evolution of the microstructure and therefore of the lifetime of
materials relies on the ability to understand phase separation, coarsening and damage processes. This
shows the importance of developing reliable mathematical models to describe such effects.
In the mathematical literature, coarsening and damage processes are treated in general separately.
Phase separation and coarsening phenomena are usually described by phase–field models of Cahn-Hilliard
type. The evolution is modeled by a parabolic diffusion equation for the phase fractions. To include elastic
effects, resulting from stresses caused by different elastic properties of the phases, Cahn-Hilliard systems
are coupled with an elliptic equation, describing the quasi-static balance of forces. Such coupled Cahn-
Hilliard systems with elasticity are also called Cahn-Larch´e systems. Since in general the mobility, stiffness
and surface tension coefficients depend on the phases (see for instance [BDM07] and [BDDM07] for the
explicite structure deduced by the embedded atom method), the mathematical analysis of the coupled
problem is very complex. Existence results were derived for special cases in [Gar00, CMP00, BP05]
(constant mobility, stiffness and surface tension coefficients), in [BCD+02] (concentration dependent
mobility, two space dimensions) and in [PZ08] in an abstract measure-valued setting (concentration
dependent mobility and surface tension tensors). For numerical results and simulations we refer [Wei01,
Mer05, BM10].
Damage models for elastic materials have been analytically investigated for the last ten years. In
the simplest case, the damage variable is a scalar function and describes the local accumulation of
damage in the body. The damage process is typically modeled as a unidirectional evolution, which
means that damage can increase, but not decrease. Based on the model developed in [FN96], the damage
evolution is described by an equation of balance for forces which is coupled with a unidirectional parabolic
[BSS05, FK09, Gia05] or rate–independent [MR06, MRZ10] evolution inclusion for the damage variable.
The models studied in [FK09, MR06, Gia05] also include the effect that the applied forces have to pass
over a threshold before the damage starts to increase.
2
In this work, we introduce a mathematical model describing both phenomena, phase separation/coars-
ening and damage processes, in a unifying model. We focus on the analytical modeling on the meso– and
macroscale. To this end, we couple phase–field models of Cahn-Larch´e type with damage models. The
evolution system consists of an equation of balance for forces which is coupled with a parabolic evolution
equation for the phase fractions and a unidirectional evolution inclusion for the damage variable. The
evolution inclusion also comprises the phenomenon that a threshold for the loads has to be passed before
the damage process increases.
The main aim of the present work is to show existence of weak solutions of the introduced model
for rate-dependent damage processes. A crucial step has been to establish a suitable notion of weak
solutions. We first study the model with regularization terms and prove existence of weak solutions for
the regularized model based on a time–incremental minimization problem with constraints due to the
unidirectionality of the damage. The regularization allows us to prove an energy inequality which occurs
in the weak notion of our coupled system. The major task has been to prove convergence of the time
incremental solutions for the regularized model when the discretization fineness tends to zero. In this
context, several approximation results have been established to handle the damage evolution inclusion
and the unidirectionality of damage processes. More precisely, the internal variable z, describing damage
effects, is bounded with values in [0,1] and monotonically decreasing with respect to the time variable.
The main results are stated in Sections 4.1 and 4.2, see Theorems 4.4 and 4.6.
To the best of our knowledge, phase separation processes coupled with damage are not studied yet
in the mathematical literature. However, promising simulations were carried out in the context of phase
field models of Cahn-Hilliard and Cahn-Larch´e type with damage, see [USG07, GUaMM+07].
The paper is organized as follows: We start with introducing a phase field model of Cahn-Larch´e type
coupled with damage, cf. Section 2. Then we state some assumptions for this model, see Section 3. In
Section 4, we establish a suitable notation for weak formulations of solutions for the introduced model
and a regularized version of the model and state the main results. Section 5.2 is devoted to the existence
proof for the regularized Cahn-Larch´e system coupled with damage. Finally, we pass to the limit in the
regularized version, which shows the existence of weak solutions of the original model, see Section 5.3.
2 Model
We consider a material of two components occupying a bounded Lipschitz domain Ω ⊆R3. The state of
the system at a fixed time point is specified by a triple q= (u, c, z). The displacement field u: Ω →R3
determines the current position x+u(x) of an undeformed material point x. Throughout this paper,
we will work with the linearized strain tensor e(u) = 1
2(∇u+ (∇u)T), which is an adequate assumption
only when small strains occur in the material. However, this assumption is justified for phase-separation
processes in alloys since the deformation usually has a small gradient. The function c: Ω →Ris a phase
field variable describing a scaled concentration difference of the two components. To account for damage
effects, we choose an isotropic damage variable z: Ω →R, which models the reduction of the effective
volume of the material due to void nucleation, growth, and coalescence. The damage process is modeled
unidirectional, i.e. damage may only increase. Self-healing processes in the material are forbidden. No
damage at a material point x∈Ω is described by z(x) = 1, whereas z(x) = 0 stands for a completely
damaged material point x∈Ω. We require that even a damaged material can store a small amount of
elastic energy. Plastic effects are not considered in our model.
2.1 Energies and evolutionary equations
Here, we qualify our model formally and postpone a rigorous treatment to Section 4. The presented
model is based on two functionals, i.e. a generalized Ginzburg-Landau free energy functional Eand a
damage dissipation potential R. The free energy density ϕof the system is given by
ϕ(e, c, ∇c, z, ∇z) := γ
2|∇c|2+δ
p|∇z|p+Wch(c) + Wel(e, c, z), γ, δ > 0,(1)
3
where the gradient terms penalize spatial changes of the variables cand z,Wch denotes the chemical
energy density and Wel is the elastically stored energy density accounting for elastic deformations and
damage effects. For simplicity of notation, we set γ=δ= 1.
The chemical free energy density Wch has usually the form of a double well potential for a two phase
system. For a rigorous treatment, we need the assumptions (A1)-(A6), see Section 3. Hence, in particular,
classical ansatzes such as
Wch = (1 −c2)2
fit in our framework.
The elastically stored energy density ˆ
Wel due to stresses and strains, which occur in the material, is
typically of quadratic form, i.e.
ˆ
Wel(c, e) = 1
2e−e∗(c):C(c)e−e∗(c).(2)
Here, e∗(c) denotes the eigenstrain, which is usually linear in c, and C(c)∈ L(Rn×n
sym ) is a fourth order
stiffness tensor, which is symmetric and positive definite. If the stiffness tensor does not depend on the
concentration, i. e. C(c) = C, we refer to homogeneous elasticity.
To incorporate the effect of damage on the elastic response of the material, ˆ
Wel is replaced by
Wel = (Φ(z) + ˜η)ˆ
Wel,(3)
where Φ : [0,1] →R+is a continuous and monotonically increasing function with Φ(0) = 0 and ˜η > 0 is
a small value. The small value ˜η > 0 in (3) is introduced for analytical reasons, see for instance (A1).
Rigorous results in the present work are obtained under certain growth conditions for the elastic
energy density Wel, see Section 3. These conditions are, for instance, satisfied for Wel as in (3) in the
case of homogeneous elasticity.
The overall free energy Eof Ginzburg-Landau type has the following structure:
E(u, c, z) := ˜
E(u, c, z) + ZΩ
I[0,∞)(z) dx,
˜
E(u, c, z) := ZΩ
ϕ(e(u), c, ∇c, z, ∇z) dx.
(4)
Here, I[0,∞)signifies the indicator function of the subset [0,∞)⊆R, i.e. I[0,∞)(x) = 0 for x∈[0,∞) and
I[0,∞)(x) = ∞for x < 0. We assume that the energy dissipation for the damage process is triggered by
a dissipation potential Rof the form
R( ˙z) := ˜
R( ˙z) + ZΩ
I(−∞,0]( ˙z) dx,
˜
R( ˙z) := ZΩ
−α˙z+1
2β˙z2dxfor α > 0 and β > 0.
(5)
Due to β > 0, the dissipation potential is referred to as rate-dependent. In the case β= 0, which is
not considered in this work, Ris called rate-independent. We refer for rate-independent processes to
[EM06, MT99, MR06, MRZ10, Rou10] and in particular to [Mie05] for a survey.
The governing evolutionary equations for a system state q= (u, c, z) can be expressed by virtue of the
functionals (4) and (5). The evolution is driven by the following elliptic-parabolic system of differential
equations and differential inclusion:
Diffusion :∂tc= ∆µ(u, c, z),(6a)
Mechanical equilibrium : div(σ(e(u), c, z)) = 0,(6b)
Damage evolution : 0 ∈∂zE(u, c, z) + ∂˙zR(∂tz),(6c)
where σ=σ(e, c, z) := ∂eϕ(e, c, ∇c, z , ∇z) denotes the Cauchy stress tensor and µis the chemical poten-
tial given by µ=µ(u, c, z) := ∂cϕ(e, c, ∇c, z, ∇z)−div(∂∇cϕ(e, c, ∇c, z , ∇z)). Equation (6a) is a fourth
4
order quasi-linear parabolic equation of Cahn-Hilliard type and describes phase separation processes for
the concentration cwhile the elliptic equation (6b) constitutes a quasi-static equilibrium for u. This
means physically that we neglect kinetic energies and instead assume that mechanical equilibrium is at-
tained at any time. The doubly nonlinear differential inclusion (6c) specifies the flow rule of the damage
profile according to the constraints 0 ≤z≤1 and ∂tz≤0 (in space and time). The inclusion (6c) has to
be read in terms of generalized sub-differentials.
We choose Dirichlet conditions for the displacements uon a subset Γ of the boundary ∂Ω with
Hn−1(Γ) >0. Let b: [0, T ]×Γ→Rnbe a function which prescribes the displacements on Γ for a fixed
chosen time interval [0, T ]. The imposed boundary and initial conditions and constraints are as follows:
Boundary displacements :u(t) = b(t) on Γ for all t∈[0, T ],(IBC1)
Initial concentration :c(0) = c0in Ω,(IBC2)
Initial damage : 0 ≤z(0) = z0≤1 in Ω,(IBC3)
Damage constraints : 0 ≤z≤1 and ∂tz≤0 in ΩT.(IBC4)
Moreover, we use homogeneous Neumann boundary conditions for the remaining variables on (parts of)
the boundary:
σ·ν= 0 on ∂Ω\Γ,(IBC5)
∇µ(t)·ν= 0 on ∂Ω,(IBC6)
∇c(t)·ν= 0 on ∂Ω,(IBC7)
∇z(t)·ν= 0 on ∂Ω,(IBC8)
where νstands for the outer unit normal to ∂Ω.
We like to mention that mass conservation of the system follows from the diffusion equation (6a) and
(IBC6), i.e. ZΩ
c(t)−c0dx= 0 for all t∈[0, T ].
3 Assumptions and Notation
In the following, we collect all assumptions and constants which are used for a rigorous analysis in the
subsequent sections.
(i) Setting. Ω⊆Rnis a bounded domain with Lipschitz boundary, n∈ {1,2,3},p > n,β > 0,
Wel ∈C1(Rn×n×R×R;R+), Wch ∈C1(R;R+), Wel(e, c, z) = Wel (et, c, z) for all e∈Rn×nand
c, z ∈R. Furthermore, C > 0 always denotes a constant, which may vary from estimate to estimate,
and [0, T ] is the time interval of interest.
(ii) Convexity and growth assumptions. The function Wel is assumed to satisfy for some constants
η > 0 and C > 0 the following estimates:
η|e1−e2|2≤(∂eWel(e1, c, z )−∂eWel(e2, c, z)) : (e1−e2),(A1)
Wel(e, c, z )≤C(|e|2+|c|2+ 1),(A2)
|∂eWel(e1+e2, c, z )| ≤ C(Wel(e1, c, z) + |e2|+ 1),(A3)
|∂cWel(e, c, z )| ≤ C(|e|+|c|2+ 1),(A4)
|∂zWel(e, c, z )| ≤ C(|e|2+|c|2+ 1) (A5)
for arbitrary c∈R,z∈[0,1] and symmetric e, e1, e2∈Rn×n.
The chemical energy density function Wch satisfies
|∂cWch(c)| ≤ ˆ
C(|c|2?/2+ 1) (A6)
5
for some constant ˆ
C > 0. For dimension n= 3, the constant 2?denotes the Sobolev critical
exponent given by 2n
n−2. In the two dimensional case n= 2, the constant 2?can be an arbitrary
positive real number and in one space dimension (A6) can be dropped.
(iii) Boundary displacements. We assume that Γ is a Hn−1-measurable subset of ∂Ω with Hn−1(Γ) >0
and that the boundary displacement b: [0, T ]×Γ→Rnmay be extended by ˆ
b∈W1,1([0, T ]; W1,∞
(Ω; Rn)) such that b(t)|Γ=ˆ
b(t)|Γin the sense of traces for a.e. t∈[0, T ]. In the following, we write
binstead of ˆ
b.
Remark 3.1 Conditions (A1),(A2) and (A3) imply the fol lowing estimates
|∂eWel(e, c, z )| ≤ C(|e|+|c|2+ 1),(11a)
η|e|2−C(|c|4+ 1) ≤Wel (e, c, z) (11b)
for some appropriate constants η > 0and C > 0, cf. [Gar00, Section 3.2] for (11b).
We introduce some auxiliary spaces to shorten the notation for the construction of solution curves of
the evolutionary problem. First of all, we define the trajectory space Qfor the limit problem (6a)-(6c)
as
Q:=
q= (u, c, z) with
u∈L∞([0, T ]; H1(Ω; Rn)),
c∈L∞([0, T ]; H1(Ω)) ∩H1([0, T ],(H1(Ω))?),
z∈L∞([0, T ]; W1,p(Ω)) ∩H1([0, T ]; L2(Ω))
.
Based on Q, the set of admissible functions of the viscous problem (see Section 4) is
Qv:= q= (u, c, z)∈ Q | c∈H1([0, T ]; L2(Ω)) and u∈L∞([0, T ]; W1,4(Ω; Rn)).
It will be convenient for the variational formulation to define Sobolev spaces with functions taking
only non-negative and non-positive values, respectively, and Sobolev spaces consisting of functions with
vanishing traces on the boundary Γ:
W1,r
+(Ω) := ζ∈W1,r(Ω) ζ≥0 a.e. in Ω,
W1,r
−(Ω) := ζ∈W1,r(Ω) ζ≤0 a.e. in Ω,
W1,r
Γ(Ω; Rn) := ζ∈W1,r(Ω; Rn)ζ|Γ= 0 in the sense of traces
for r∈[1,∞]. In this context, IW1,r
±(Ω) :W1,r(Ω) → {0,∞} denote the indicator functions given by
IW1,r
±(Ω)(ζ) := (0,if ζ∈W1,r
±(Ω),
∞,else.
Since Cahn-Hilliard systems can be expressed as H−1-gradient flows, we introduce the following spaces
in order to apply the direct method in the time-discrete version (see Section 5):
V0:= ζ∈H1(Ω) ZΩ
ζdx= 0,
˜
V0:= nζ∈(H1(Ω))∗hζ, 1i(H1)∗×H1= 0o.
This permits us to define the operator (−∆)−1:˜
V0→V0as the inverse of the operator −∆ : V0→˜
V0,u7→
h∇u, ∇·iL2(Ω). The space ˜
V0will be endowed with the scalar product hu, vi˜
V0:= h∇(−∆)−1u, ∇(−∆)−1viL2(Ω).
We end this section by introducing some notation which is frequently used for some approximation
features in this paper. The expression BR(K) denotes the open neighborhood with width R > 0 of a
subset K⊆Rn. Whenever we consider the zero set of a function ζ∈W1,p(Ω) for p>nabbreviated in
the following by {ζ= 0}we mean {x∈Ω|ζ(x) = 0}by taking the embedding W1,p(Ω) →C0(Ω) into
account. We adapt the convention that for two given functions ζ, ξ ∈L1([0, T ]; W1,p (Ω)) the inclusion
{ζ= 0}⊇{ξ= 0}is an abbreviation for {ζ(t)=0}⊇{ξ(t)=0}for a.e. t∈[0, T ].
6
4 Weak formulation and existence theorems
Existence results for multi-phase Cahn-Larch´e systems without considering damage phase fields are shown
in [Gar00] provided that the chemical energy density Wch can be decomposed into W1
ch +W2
ch with convex
W1
ch and linear growth behavior of ∂cW2
ch (see [Gar00, Section 3.2] for a detailed explanation). Logarithmic
free energies Wch are also studied in [Gar00] as well as in [Gar05b]. Further variants of Cahn-Larch´e
systems are investigated in [CMP00], [BP05], [BCD+02] and [Gar05a].
Purely mechanical systems with rate-independent damage processes are analytically considered and
reviewed for instance in [MR06] and [MRZ10]. The rate-independence enables the concept of the so-called
global energetic solutions (see Remark 4.2 (i)) to such systems.
Coupling rate-independent systems with other (rate-dependent) processes (such as with inertial or
thermal effects) may lead, however, to serious mathematical difficulties as pointed out in [Rou10].
In our situation where the Cahn-Larch´e system is coupled with rate-dependent damage, we will treat
our model problem analytically by a regularization method that gives better regularity property for c
and integrability for uin the first instance. A passage to the limit will finally give us solutions to
the original problem. In doing so, the notion of a weak solution consists of variational equalities and
inequalities as well as an energy estimate, inspired by the concept of energetic solutions in the framework
of rate-independent systems.
4.1 Regularization
The regularization, we want to consider here, is achieved by adding the term ε∆∂tcto the Cahn-Hilliard
equation (referred to as viscous Cahn-Hilliard equation [BP05]) and the 4-Laplacian εdiv(|∇u|2∇u) to
the quasi-static equilibrium equation of the model problem. The classical formulation of the regularized
problem for ε > 0 now reads as
∂tc= ∆(−∆c+∂cWch(c) + ∂cWel(e(u), c, z) + ε∂tc),(12a)
div(σ(e(u), c, z)) + εdiv(|∇u|2∇u) = 0,(12b)
0∈∂zEε(u, c, z) + ∂˙zR(∂tz) (12c)
with the regularized energies
Eε(u, c, z) := E(u, c, z ) + εZΩ
1
4|∇u|4dx,
˜
Eε(u, c, z) := ˜
E(u, c, z) + εZΩ
1
4|∇u|4dx.
In the following, we motivate a formulation of weak solutions of the system (12a)-(12b) admissible for
curves q= (u, c, z)∈ Qv. For every t∈[0, T ], equation (12a) can be translated with the boundary
conditions in a weak formulation as follows:
ZΩ
(∂tc(t))ζdx=−ZΩ
∇µ(t)· ∇ζdx(13)
for all ζ∈H1(Ω) and
ZΩ
µ(t)ζdx=ZΩ
∇c(t)· ∇ζ+∂cWch(c(t))ζ+∂cWel(e(u(t)), c(t), z(t))ζ+ε(∂tc(t))ζdx(14)
for all ζ∈H1(Ω). In the same spirit, we rewrite (12b) as
ZΩ
∂eWel(e(u(t)), c(t), z (t)) : e(ζ) + ε|∇u(t)|2∇u(t) : ∇ζdx= 0 (15)
for all ζ∈W1,4
Γ(Ω; Rn) by using the symmetry condition
∂eWel(e, c, z ) = (∂eWel(e, c, z))tfor e∈Rn×n
sym , c, z ∈R,
7
following from the assumptions in Section 3 (i). The differential inclusion (12c) is equivalent to
0=dz˜
Eε(u(t), c(t), z(t)) + r(t)+d˙z˜
R(∂tz(t)) + s(t)
with some r(t)∈∂IW1,p
+(Ω)(z(t)) and s(t)∈∂IW1,p
−(Ω)(∂tz(t)) (see (4) and (5) for the definitions of ˜
Eand
˜
R). This can be expressed to the following system of variational inequalities:
IW1,p
−(Ω)(∂tz(t)) −Ddz˜
Eε(q(t)) + r(t)+d˙z˜
R(∂tz(t)), ζ −∂tz(t)E≤IW1,p
−(Ω)(ζ) for ζ∈W1,p(Ω),
IW1,p
+(Ω)(z(t)) + hr(t), ζ −z(t)i ≤ IW1,p
+(Ω)(ζ) for ζ∈W1,p(Ω).
Here, h·,·i denotes the dual pairing between (W1,p(Ω))?and W1,p(Ω). This system is, in turn, equivalent
to the inequality system
z(t)≥0 and ∂tz(t)≤0,(16a)
−Ddz˜
Eε(q(t)) + r(t)+d˙z˜
R(∂tz(t)), ∂tz(t)E≥0,(16b)
Ddz˜
Eε(q(t)) + r(t)+d˙z˜
R(∂tz(t)), ζE≥0 for ζ∈W1,p
−(Ω),(16c)
hr(t), ζ −z(t)i ≤ 0 for ζ∈W1,p
+(Ω).(16d)
Due to the lack of regularity of q, (16b) cannot be justified rigorously. To overcome this difficulty, we use
a formal calculation originating from energetic formulations introduced in [MT99].
Proposition 4.1 (Energetic characterization) Let q∈ Qv∩C2(ΩT;Rn×R×R)be a smooth solution
of (13)-(15) with (IBC1)-(IBC8). Then the following two conditions are equivalent:
(i) (16b) with r(t)∈∂IW1,p
+(Ω)(z(t)) for all t∈[0, T ],
(ii) for all 0≤t1≤t2≤T:
Eε(q(t2)) + Zt2
t1
hd˙z˜
R(∂tz), ∂tzids+Zt2
t1ZΩ
|∇µ|2+ε|∂tc|2dxds− Eε(q(t1))
≤Zt2
t1ZΩ
∂eWel(e(u), c, z ) : e(∂tb) dxds+εZt2
t1ZΩ
|∇u|2∇u:∇∂tbdxds. (17)
Proof. We first show for all t∈[0, T ]:
hr, ∂tz(t)i= 0 for all r∈∂IW1,p
+(Ω)(z(t)).(18)
The inequality 0 ≤ hr, ∂tz(t)ifollows directly from (16d) by putting ζ=z(t)−∂tz(t). The ’≥’ - part
can be shown by an approximation argument. Applying Lemma 5.1 with fM=z(t) and f=z(t)
and ζ=−∂tz(t), we obtain a sequence {ζM} ⊆ W1,p
+(Ω) and constants νM>0 such that −ζM→
∂tz(t) in W1,p(Ω) as M→ ∞ and 0 ≤z(t)−νMζMa.e. in Ω for all M∈N. Testing (16d) with ζ=
z(t)−νMζMshows hr, −ζMi ≤ 0. Passing to M→ ∞ gives hr, ∂tz(t)i ≤ 0.
To (ii)⇒(i) : We remark that (14) and (15) can be written in the following form:
ZΩ
µ(t)ζ1−ε(∂tc(t))ζ1dx=hdc˜
Eε(q(t)), ζ1i,(19a)
hdu˜
Eε(q(t)), ζ2i= 0,(19b)
for all t∈[0, T ], all ζ1∈H1(Ω) and all ζ2∈W1,4
Γ(Ω; Rn).
8
Let t0∈[0, T ). It follows
Eε(q(t0+h)) − Eε(q(t0))
h+−
Zt0+h
t0
hd˙z˜
R(∂tz), ∂tzidt+−
Zt0+h
t0ZΩ
|∇µ|2+ε|∂tc|2dxdt
≤ −
Zt0+h
t0ZΩ
∂eWel(e(u), c, z ) : e(∂tb) dxdt+ε−
Zt0+h
t0ZΩ
|∇u|2∇u:∇∂tbdxdt.
Letting h&0 gives
d
dt˜
Eε(q(t0)) + hd˙z˜
R(∂tz(t0)), ∂tz(t0)i+ZΩ
|∇µ(t0)|2+ε|∂tc(t0)|2dx
≤ZΩ
∂eWel(e(u(t0)), c(t0), z (t0)) : e(∂tb(t0))dx+εZΩ
|∇u(t0)|2∇u(t0) : ∇∂tb(t0) dx
=hdu˜
Eε(q(t0)), ∂tb(t0)i.
Using the chain rule and (13)-(15) yield
d
dt˜
Eε(q(t0)) = hdu˜
Eε(q(t0)), ∂tu(t0)i
|{z }
apply (19b)
+hdc˜
Eε(q(t0)), ∂tc(t0)i
|{z }
apply (19a) and (13)
+hdz˜
Eε(q(t0)), ∂tz(t0)i
=hdu˜
Eε(q(t0)), ∂tb(t0)i+ZΩ
−|∇µ(t0)|2−ε|∂tc(t0)|2dx+hdz˜
Eε(q(t0)), ∂tz(t0)i.
In consequence, property (i) follows together with (18). The case t0=Tcan be derived similarly
by considering the difference quotient of t0and t0−h.
To (i)⇒(ii) : This implication follows from the relation Eε(q(t2)) − Eε(q(t1)) = Rt2
t1
d
dt˜
Eε(q(t)) dtas
well as the equations (13)-(15) and (18).
Remark 4.2
(i) In the rate-independent case β= 0 and for convex Eεwith respect to z, condition (16c) can be
characterized by a stability condition which reads as
Eε(u(t), c(t), z(t)) ≤ Eε(u(t), c(t), ζ ) + R(ζ−z(t)) (20)
for all t∈[0, T ]and all test-functions ζ∈W1,p
+(Ω). Thereby, (17) and (20) give an equivalent
description of the differential inclusion (12c) for smooth solutions. This concept of solutions is
referred to as global energetic solutions and was introduced in [MT99]. We emphasize that the
damage variable zin the rate-independent case β= 0 is a function of bounded variation and is
allowed to exhibit jumps. For a comprehensive introduction, we refer to [AFP00]. To tackle rate-
dependent systems and non-convexity of Eεwith respect to z, we can not use formulation (20) (cf.
[MRS09, MRZ10]).
(ii) For smooth solutions q, satisfying (13)-(15), the energy inequality (17) and the variational inequality
(16c), we even obtain the following energy balance:
Eε(q(t2)) + Zt2
t1
hd˙z˜
R(∂tz), ∂tzids+Zt2
t1ZΩ
|∇µ|2+ε|∂tc|2dxds
=Eε(q(t1)) + Zt2
t1ZΩ
∂eWel(e(u), c, z ) : e(∂tb) dxds+εZt2
t1ZΩ
|∇u|2∇u:∇∂tbdxds
for all 0≤t1≤t2≤T.
9
This motivates the definition of a solution in the following sense:
Definition 4.3 (Weak solution - viscous problem) A triple q= (u, c, z )∈ Qvwith c(0) = c0,
z(0) = z0,z≥0and ∂tz≤0a.e. in ΩTis called a weak solution of the viscous system (12a)-(12c)
with initial-boundary data and constraints (IBC1)-(IBC8) if it satisfies the following conditions:
(i) for all ζ∈L2([0, T ]; H1(Ω))
ZΩT
(∂tc)ζdxdt=−ZΩT
∇µ· ∇ζdxdt, (21)
where µ∈L2([0, T ]; H1(Ω)) satisfies for all ζ∈L2([0, T ]; H1(Ω))
ZΩT
µζ dxdt=ZΩT
∇c· ∇ζ+∂cWch(c)ζ+∂cWel(e(u), c, z)ζ+ε(∂tc)ζdxdt, (22)
(ii) for all ζ∈L4([0, T ]; W1,4
Γ(Ω; Rn))
ZΩT
∂eWel(e(u), c, z ) : e(ζ) + ε|∇u|2∇u:∇ζdxdt= 0,(23)
(iii) for al l ζ∈Lp([0, T ]; W1,p
−(Ω)) ∩L∞(ΩT)
0≤ZΩT
|∇z|p−2∇z· ∇ζ+ (∂zWel(e(u), c, z )−α+β(∂tz))ζdxdt+ZT
0
hr(t), ζ(t)idt, (24)
where r∈L1(ΩT)⊂L1[0, T ]; (W1,p(Ω))∗satisfies for all ζ∈W1,p
+(Ω) and for a.e. t∈[0, T ]
hr(t), ζ −z(t)i ≤ 0,(25)
(iv) for a.e. 0≤t1≤t2≤T
Eε(q(t2)) + ZΩ
α(z(t1)−z(t2)) dx+Zt2
t1ZΩ
β|∂tz|2dxds+Zt2
t1ZΩ
|∇µ|2+ε|∂tc|2dxds
≤ Eε(q(t1)) + Zt2
t1ZΩ
∂eWel(e(u), c, z ) : e(∂tb) dxds+εZt2
t1ZΩ
|∇u|2∇u:∇∂tbdxds. (26)
Theorem 4.4 (Existence theorem - viscous problem) Let the assumptions in Section 3 be satisfied
and let c0∈H1(Ω),z0∈W1,p(Ω) with 0≤z0≤1a.e. in Ωand a viscosity factor ε∈(0,1] be given.
Then there exists a weak solution q∈ Qvof the viscous system (12a)-(12c) in the sense of Definition 4.3.
In addition:
r=−χ{z=0}[∂zWel(e(u), c, z )]+,(27)
where [·]+is defined by max{0,·}.
4.2 Limit problem
Our main aim in this work is to establish an existence result for the system (12a)-(12c) with vanishing
ε-terms, i.e. with ε= 0. In the same fashion as in Section 4.1 we introduce a weak notion of (6a)-(6c) as
follows.
Definition 4.5 (Weak solution - limit problem) A triple q= (u, c, z )∈ Q with z(0) = z0,z≥0
and ∂tz≤0a.e. in ΩTis called a weak solution of the system (6a)-(6c) with boundary and initial
conditions (IBC1)-(IBC8) if it satisfies the following conditions:
10
(i) for all ζ∈L2([0, T ]; H1(Ω)) with ∂tζ∈L2(ΩT)and ζ(T)=0
ZΩT
(c−c0)∂tζdxdt=ZΩT
∇µ· ∇ζdxdt,
where µ∈L2([0, T ]; H1(Ω)) satisfies for all ζ∈L2([0, T ]; H1(Ω))
ZΩT
µζ dxdt=ZΩT
∇c· ∇ζ+∂cWch(c)ζ+∂cWel(e(u), c, z)ζdxdt,
(ii) for all ζ∈L2([0, T ]; H1
Γ(Ω; Rn))
ZΩT
∂eWel(e(u), c, z ) : e(ζ) dxdt= 0,
(iii) for al l ζ∈Lp([0, T ]; W1,p
−(Ω)) ∩L∞(ΩT)
0≤ZΩT
|∇z|p−2∇z· ∇ζ+∂zWel(e(u), c, z )ζ−αζ +β(∂tz)ζdxdt+ZT
0
hr(t), ζ(t)idt,
where r∈L1(ΩT)satisfies for all ζ∈W1,p
+(Ω) and for a.e. t∈[0, T ]
hr(t), ζ −z(t)i ≤ 0,
(iv) for a.e. 0≤t1≤t2≤T
E(q(t2)) + ZΩ
α(z(t1)−z(t2)) dx+Zt2
t1ZΩ
β|∂tz|2dxds+Zt2
t1ZΩ
|∇µ|2dxds
≤ E(q(t1)) + Zt2
t1ZΩ
∂eWel(e(u), c, z ) : e(∂tb)dxds.
Theorem 4.6 (Existence theorem - limit problem) Let the assumptions in Section 3 be satisfied
and let c0∈H1(Ω),z0∈W1,p(Ω) with 0≤z0≤1a.e. in Ωbe given. Then there exists a weak solution
q∈ Q of the system (6a)-(6c) in the sense of Definition 4.5.
5 Proof of the existence theorems
5.1 Preliminaries
The proof of Theorem 4.4 is based on recursive functional minimization that comes from an implicit Euler
scheme of the system (12a)-(12c) with respect to the time variable. To obtain from the time-discrete model
the time-continuous model (12a)-(12c), we need some preliminary results on approximation schemes for
test-functions, which will be presented in this section.
Lemma 5.1 (Approximation of test-functions) Let p>nand f , ζ ∈W1,p
+(Ω) with {ζ= 0}⊇{f=
0}. Furthermore, let {fM}M∈N⊆W1,p
+(Ω) be a sequence with fM f in W1,p(Ω) as M→ ∞. Then,
there exist a sequence {ζM}M∈N⊆W1,p
+(Ω) and constants νM>0,M∈N, such that
(i) ζM→ζin W1,p(Ω) as M→ ∞,
(ii) ζM≤ζa.e. in Ωfor all M∈N,
(iii) νMζM≤fMa.e. in Ωfor all M∈N.
11
Proof. Without loss of generality we may assume ζ6≡ 0 on Ω.
Let {δk}be a sequence with δk&0 as k→ ∞ and δk>0. Define for every k∈Nthe approximation
function ˜
ζk∈W1,p
+(Ω) as
˜
ζk:= [ζ−δk]+,
where [·]+stands for max{0,·}. Let 0 < α < 1−n
pbe a fixed constant. Then ˜
ζk∈C0,α(Ω) due to
W1,p(Ω) →C0,α(Ω). Furthermore, set the constant Rk,k∈N, to
Rk:= δk/kζkC0,α(Ω) 1/α >0.
It follows {˜
ζk= 0} ⊇ Ω∩BRk({ζ= 0})⊇Ω∩BRk({f= 0}). Without loss of generality we may assume
Ω\BRk({f= 0})6=∅for all k∈N. Furthermore, there exists a strictly increasing sequence {Mk} ⊆ N
such that we find for all k∈N:
fM≥ηk/2 a.e. on Ω \BRk({f= 0}) for all M≥Mk
with ηk:= inf{f(x)|x∈Ω\BRk({f= 0})}>0, k∈N, (note that fM→fin C0,α(Ω) as M→ ∞).
This implies ˜νk˜
ζk≤fMa.e. on Ω for all M≥Mkby setting ˜νk:= ηk/(2kζkL∞(Ω))>0. The claim
follows with ζM:= 0 and νk:= 1 for M∈ {1, . . . , M1−1}and ζM:= ˜
ζδkand νM:= ˜νkfor each
M∈ {Mk, . . . , Mk+1 −1},k∈N.
Lemma 5.2 (Approximation of time-dependent test-functions) Let p > n,q≥1and f, ζ ∈
Lq([0, T ]; W1,p
+(Ω)) with {ζ= 0} ⊇ {f= 0}. Furthermore, let {fM}M∈N⊆Lq([0, T ]; W1,p
+(Ω)) be a
sequence with fM(t) f(t)in W1,p(Ω) as M→ ∞ for a.e. t∈[0, T ]. Then, there exist a sequence
{ζM}M∈N⊆Lq([0, T ]; W1,p
+(Ω)) and constants νM,t >0such that
(i) ζM→ζin Lq([0, T ]; W1,p(Ω)) as M→ ∞,
(ii) ζM≤ζa.e. in ΩTfor all M∈N(in particular {ζM= 0} ⊇ {ζ= 0}),
(iii) νM,tζM(t)≤fM(t)a.e. in Ωfor a.e. t∈[0, T ]and for all M∈N.
If, in addition, ζ≤fa.e. in ΩTthen condition (iii) can be refined to
(iii)’ ζM≤fMa.e. in ΩTfor all M∈N.
Proof. Let {δk}with δk&0 as k→ ∞ and δk>0 be a sequence and 0 < α < 1−n
pbe a fixed constant.
We construct the approximation functions ζM∈Lq([0, T ]; W1,p
+(Ω)), M∈N, as follows:
ζM(t) :=
M
X
k=1
χAk
M(t)[ζ(t)−δk]+,(28)
where χAk
M: [0, T ]→ {0,1}is defined as the characteristic function of the measurable set Ak
Mgiven by
Ak
M:= (Pk
M\SM
i=k+1 Pi
Mif k < M,
PM
Mif k=M,
with
Pk
M:= nt∈[0, T ]Ω\BRk(t)({f(t)=0})6=∅
and fM(t)≥ηk(t)/2 a.e. on Ω \BRk(t)({f(t)=0})o,(29)
12
where the functions Rk, ηk: [0, T ]→R+are defined by
Rk(t) = δk/kζ(t)kC0,α(Ω) 1/α ,
ηk(t) = inf{f(t, x)|x∈Ω\BRk(t)({f(t) = 0})}.
Here, we use the convention Rk(t) := ∞for ζ(t)≡0. Note that Ak
M, 1 ≤k≤M, are pairwise disjoint
by construction.
Consider a t∈[0, T ] with fM(t) f (t) in W1,p (Ω) and ζ(t)6≡ 0 with {ζ(t) = 0} ⊇ {f(t) = 0}. Let
K∈Nbe arbitrary but large enough such that Ω \BRK(t)({f(t)=0})6=∅holds. It follows the existence
of an ˜
M≥Kwith t∈PK
Mfor all M≥˜
M. Therefore, for each M≥˜
Mexists a k≥Ksuch that t∈Ak
M,
i.e. ζM(t)=[ζ(t)−δk]+. Thus ζM(t)→ζ(t) in W1,p(Ω) as K→ ∞. Lebesgue’s convergence theorem
shows (i).
Property (ii) follows immediately from (28). It remains to show (iii). Let M∈Nbe arbitrary. If
ζM(t)≡0 we set νM,t = 1. Otherwise we find a unique 1 ≤k≤Mwith t∈Ak
Mand ζM(t)=[ζ(t)−δk]+.
This, in turn, implies the existence of a νM,t >0 with νM,t ζM≤fM(see proof of Lemma 5.1).
In the case ζ≤f, we use instead of (29) the set:
Pk
M:= nt∈[0, T ]kfM(t)−f(t)kC0(Ω) ≤δko.
With a similar argumentation, {ζM}fulfills (i), (ii) and (iii)’.
Lemma 5.3 Let p>nand f∈Lp/(p−1)(Ω; Rn),g∈L1(Ω),z∈W1,p
+(Ω) with f· ∇z≥0and
{f= 0}⊇{z= 0}a.e.. Furthermore, we assume that
ZΩ
f· ∇ζ+gζ dx≥0for all ζ∈W1,p
−(Ω) with {ζ= 0}⊇{z= 0}.
Then
ZΩ
f· ∇ζ+gζ dx≥Z{z=0}
[g]+ζdxfor all ζ∈W1,p
−(Ω).
Proof. We assume z6≡ 0 on Ω. Let ζ∈W1,p
−(Ω) be a test-function. For δ > 0 small enough such that
Ω\Bδ({z= 0})6=∅, we define
ζδ:= max ζ , −zkζkL∞C−1
δ
with the constant
Cδ:= inf z(x)|x∈Ω\Bδ({z= 0})>0.
We consider the following partition of Ω:
Ω=Σ1∪Σ≤
2∪Σ>
2
with
Σ1:= Ω \Bδ({z= 0}),
Σ≤
2:= Ω ∩Bδ({z= 0})∩ {ζ≤ −zkζkL∞C−1
δ},
Σ>
2:= Ω ∩Bδ({z= 0})∩ {ζ > −zkζkL∞C−1
δ}.
By construction, the sequence {ζδ}δ∈(0,1] satisfies
ζδ(x) = (ζ(x),if x∈Σ1∪Σ>
2,
−z(x)kζkL∞C−1
δ,if x∈Σ≤
2.
13
In particular, ζδ= 0 on {z= 0}for every δ∈(0,1] and ζδ
?
ζ in L∞({z > 0}) as δ&0. By using the
assumptions, we estimate
ZΩ
f· ∇ζ+gζ dx−Z{z=0}
[g]+ζdx
=ZΩ
f· ∇(ζ−ζδ) + g(ζ−ζδ) dx−Z{z=0}
[g]+ζdx+ZΩ
f· ∇ζδ+gζδdx
| {z }
≥0
≥ZΩ
f· ∇(ζ−ζδ) dx+Z{z>0}
g(ζ−ζδ) dx
=ZΣ1
f· ∇(ζ−ζδ) dx
| {z }
=0
+ZΣ≤
2
f· ∇(ζ−ζδ) dx+ZΣ>
2
f· ∇(ζ−ζδ) dx
| {z }
=0
+Z{z>0}
g(ζ−ζδ) dx
=kζkL∞C−1
δZΣ≤
2
f· ∇z
|{z }
≥0
dx+ZΣ≤
2
f· ∇ζdx
| {z }
=RΣ≤
2\{z=0}f·∇ζdx
+Z{z>0}
g(ζ−ζδ) dx
≥ZΣ≤
2\{z=0}
f· ∇ζdx+Z{z>0}
g(ζ−ζδ) dx.
The terms on the right hand side converge to 0 as δ&0.
5.2 Viscous case
This section is aimed to prove Theorem 4.4. The initial displacement u0
εis chosen to be a minimizer
of the functional u7→ Eε(u, c0, z0) defined on the space W1,4(Ω) with the constraint u|Γ=b(0)|Γ(the
existence proof is based on direct methods in the calculus of variations - see the proof of Lemma 5.4
below). We now apply an implicit Euler scheme of the system (12a)-(12c). The discretization fineness is
given by τ:= T
M, where M∈N. We set q0
M,ε := (u0
M,ε, c0
M,ε, z0
M,ε) := (u0
ε, c0, z0) and construct qm
M,ε for
m∈ {1, . . . , M }recursively by considering the functional
Em
M,ε(u, c, z) := ˜
Eε(u, c, z) + ˜
R z−zm−1
M,ε
τ!τ+1
2τkc−cm−1
M,ε k2
˜
V0+ε
2τkc−cm−1
M,ε k2
L2(Ω).
The set of admissible states for Em
M,ε is
Qm
M,ε := q= (u, c, z)∈W1,4(Ω; Rn)×H1(Ω) ×W1,p(Ω)
with u|Γ=b(mτ)|Γ,ZΩ
c−c0dx= 0 and 0 ≤z≤zm−1
M,ε a.e. in Ω.
A minimization problem for the functional Em
M,ε(u, c, z) = Em
M,ε(u, c) = RΩ
1
2|∇c|2+Wch(c)+Wel(e(u), c) dx+
1
2τkc−cm−1
M,ε k2
Lcontaining a weighted (H1(Ω,Rn))?-scalar product h·,·iLhas been considered in [Gar00].
However, due to the additional internal variable z, the passage to M→ ∞ becomes much more involved.
In the following, we will omit the ε-dependence in the notation since ε∈(0,1] is fixed until Section
5.3.
Lemma 5.4 The functional Em
Mhas a minimizer qm
M= (um
M, cm
M, zm
M)∈ Qm
M.
14
Proof. The existence is shown by direct methods in the calculus of variations. We can immediately see
that Qm
Mis closed with respect to the weak topology in W1,4(Ω; Rn)×H1(Ω) ×W1,p(Ω). Furthermore,
we need to show coercivity and sequentially weakly lower semi-continuity of Em
Mdefined on Qm
M.
(i) Coercivity. We have the estimate
Em
M(q)≥1
2k∇ck2
L2(Ω) +1
pk∇zkp
Lp(Ω) +ε
4k∇uk4
L4(Ω).
Therefore, given a sequence {qk}k∈Nin Qm
Mwith the boundedness property Em
M(qk)< C for all
k∈N, we obtain the boundedness of ukin W1,4(Ω) by Poincar´e’s inequality (ukhas fixed boundary
data on Γ), the boundedness of ckin H1(Ω) by Poincar´e’s inequality (RΩckdxis conserved) and
the boundedness of zkin W1,p(Ω) by also considering the restriction 0 ≤zk≤1 a.e. in Ω.
(ii) Sequentially weakly lower semi-continuity. All terms in Em
Mexcept RΩWch(c) dxand RΩWel(e(u), c, z) dx
are convex and continuous and therefore sequentially weakly l.s.c.. Now let (uk, ck, zk)(u, c, z)
be a weakly converging sequence in Qm
M. In particular, zk→zin Lp(Ω), zk→za.e. in Ω and
ck→cin Lr(Ω) as k→ ∞ for all 1 ≤r < 2?and ck→ca.e. in Ω for a subsequence. Lebesgue’s
generalized convergence theorem yields RΩWch(ck) dx→RΩWch(c) dxusing (A6). The remaining
term can be treated by employing the uniform convexity of Wel(·, c, z) (see (A1)):
ZΩ
Wel(e(uk), ck, zk)−Wel (e(u), c, z) dx
=ZΩ
Wel(e(u), ck, zk)−Wel (e(u), c, z) dx+ZΩ
Wel(e(uk), ck, zk)−Wel (e(u), ck, zk) dx
≥ZΩ
Wel(e(u), ck, zk)−Wel (e(u), c, z) dx
| {z }
→0 by Lebesgue’s gen. conv. theorem and (A2)
+ZΩ
∂eWel(e(u), ck, zk)(e(uk)−e(u)) dx.
The second term also converges to 0 because of ∂eWel(e(u), ck, zk)→∂eWel(e(u), c, z ) in L2(Ω) (by
Lebesgue’s generalized convergence theorem and (11a)) and e(uk)−e(u)0 in L2(Ω).
Thus there exists qm
M= (um
M, cm
M, zm
M)∈ Qm
Msuch that Em
M(qm
M) = infq∈Qm
M
Em
M(q).
The minimizers qm
Mfor m∈ {0, . . . , M }are used to construct approximate solutions qMand ˆqMto our
viscous problem by a piecewise constant and linear interpolation in time, respectively. More precisely,
qM(t) := qm
M,
ˆqM(t) := βqm
M+ (1 −β)qm−1
M
with t∈((m−1)τ, mτ ] and β=t−(m−1)τ
τ. The retarded function q−
Mis set to
q−
M(t) := (qM(t−τ),if t∈[τ, T ],
q0
ε,if t∈[0, τ ).
The functions bMand b−
Mare analogously defined adapting the notation bm
M:= b(mτ). Furthermore, the
discrete chemical potential is given by (note that ∂tˆcM(t)∈V0)
µM(t) := −(−∆)−1(∂tˆcM(t)) + λM(t) (30)
with the Lagrange multiplier λMoriginating from mass conservation:
λM(t) := −
ZΩ
∂cWch(cM(t)) + ∂cWel(e(uM(t)), cM(t), zM(t)) dx. (31)
15
The discretization of the time variable twill be expressed by the functions
dM(t) := min{mτ |m∈N0and mτ ≥t},
d−
M(t) := min{(m−1)τ|m∈N0and mτ ≥t}.
The following lemma clarifies why the functions qM,q−
Mand ˆqMare approximate solutions to our problem.
Lemma 5.5 (Euler-Lagrange equations and energy estimate) The tuples qM,q−
Mand ˆqMsatisfy
the following properties:
(i) for all t∈(0, T )and all ζ∈H1(Ω)
ZΩ
(∂tˆcM(t))ζdx=−ZΩ
∇µM(t)· ∇ζdx, (32)
(ii) for all t∈(0, T )and all ζ∈H1(Ω)
ZΩ
µM(t)ζdx=ZΩ
∇cM(t)· ∇ζ+∂cWch(cM(t))ζdx
+ZΩ
∂cWel(e(uM(t)), cM(t), zM(t))ζ+ε(∂tˆcM(t))ζdx, (33)
(iii) for al l t∈[0, T ]and for all ζ∈W1,4
Γ(Ω; Rn)
0 = ZΩ
∂eWel(e(uM(t)), cM(t), zM(t)) : e(ζ) + ε|∇uM(t)|2∇uM(t) : ∇ζdx, (34)
(iv) for al l t∈(0, T )and all ζ∈W1,p (Ω) such that there exists a constant ν > 0with 0≤νζ +zM(t)≤
z−
M(t)a.e. in Ω
0≤ZΩ
|∇zM(t)|p−2∇zM(t)· ∇ζ+∂zWel(e(uM(t)), cM(t), zM(t))ζ−αζ +β(∂tˆzM(t))ζdx, (35)
(v) for all t∈[0, T ]
Eε(qM(t)) + ZdM(t)
0
R(∂tˆzM) ds+ZdM(t)
0ZΩ
ε
2|∂tˆcM|2+1
2|∇µM|2dxds
≤ Eε(q0
ε) + ZdM(t)
0ZΩ
∂eWel(e(u−
M+b−b−
M), c−
M, z−
M) : e(∂tb) dxds
+εZdM(t)
0ZΩ
|∇u−
M+∇b− ∇b−
M|2∇(u−
M+b−b−
M) : ∇∂tbdxds. (36)
Proof. Using Lebesgue’s generalized convergence theorem, the mean value theorem of differentiability
and growth conditions (11a), (A4)-(A6), we obtain the variational derivatives of ˜
Eεwith respect to u,c
and z:
hdu˜
Eε(q), ζi=ZΩ
∂eWel(e(u), c, z ) : e(ζ) + ε|∇u|2∇u:∇ζdxfor ζ∈W1,4(Ω; Rn),(37a)
hdc˜
Eε(q), ζi=ZΩ
∇c· ∇ζ+∂cWch(c)ζ+∂cWel(e(u), c, z)ζdxfor ζ∈H1(Ω),(37b)
hdz˜
Eε(q), ζi=ZΩ
|∇z|p−2∇z· ∇ζ+∂zWel(e(u), c, z )ζdxfor ζ∈W1,p(Ω).(37c)
To (i)-(v):
16
(i) This follows from (30).
(ii) qm
Mfulfills hdcEm
M(qm
M), ζ1i= 0 for all ζ1∈V0and all m∈ {1, . . . , M}. Therefore,
0 = hdc˜
Eε(qM(t)), ζ1i+h∂tˆcM(t), ζ1i˜
V0+εh∂tˆcM(t), ζ1iL2(Ω).
On the one hand, definition (30) implies
h∂tˆcM(t), ζ1i˜
V0=h(−∆)−1(∂tˆcM(t)) , ζ1iL2(Ω)
=h−µM(t) + λM(t), ζ1iL2(Ω)
=−hµM(t), ζ1iL2(Ω)
and consequently
0 = hdc˜
Eε(qM(t)), ζ1i−hµM(t), ζ1iL2(Ω) +εh∂tˆcM(t), ζ1iL2(Ω) for all ζ1∈V0.(38)
On the other hand, definitions (30) and (31) yield for ζ2≡˜
Cwith constant ˜
C∈R:
hdc˜
Eε(qM(t)), ζ2i−hµM(t), ζ2iL2(Ω) +εh∂tˆcM(t), ζ2iL2(Ω)
=˜
CLn(Ω)λM(t) + h(−∆)−1(∂tˆcM(t)) , ζ2iL2(Ω)
| {z }
=0
− hλM(t), ζ2iL2(Ω)
| {z }
˜
CLn(Ω)λM(t)
+ 0
= 0.(39)
Setting ζ1=ζ−−
Rζand ζ2=−
Rζ, inserting (37b) into (38) and (39), and adding (38) to (39) shows
finally (ii) (cf. [Gar00, Lemma 3.2]).
(iii) This property follows from (37a) and 0 = hduEm
M(qm
M), ζi=hdu˜
Eε(qm
M), ζifor all ζ∈W1,4
Γ(Ω; Rn).
(iv) By construction, zm
Mminimizes Em
M(um
M, cm
M,·) in the space W1,p(Ω) with the constraints 0 ≤zand
z−zm−1
M≤0 a.e. in Ω. This implies
− hdz˜
Eε(qm
M), ζ −zm
Mi − d˙z˜
Rzm
M−zm−1
M
τ, ζ −zm
ML2(Ω)
≤0 (40)
for all ζ∈W1,p(Ω) with 0 ≤ζ≤zm−1
Ma.e. in Ω. Now, let the functions ζ∈W1,p (Ω) and ν > 0
with 0 ≤νζ +zM(t)≤z−
M(t) a.e. in Ω be given. Since ν > 0, we obtain from (40):
−hdz˜
Eε(qM(t)), ζ(t)i−hd˙z˜
R(∂tˆzM(t)) , ζ(t)iL2(Ω) ≤0.
This and (37c) gives (iv).
(v) Testing Em
Mwith q= (um−1
M+bm
M−bm−1
M, cm−1
M, zm−1
M) and using the chain rule yields:
Eε(qm
M) + Rzm
M−zm−1
M
ττ+1
2τkcm
M−cm−1
Mk2
˜
V0+ε
2τkcm
M−cm−1
Mk2
L2(Ω)
≤ Eε(um−1
M+bm
M−bm−1
M, cm−1
M, zm−1
M)
=Eε(qm−1
M) + Eε(um−1
M+bm
M−bm−1
M, cm−1
M, zm−1
M)− Eε(qm−1
M)
=Eε(qm−1
M) + Zmτ
(m−1)τ
d
dsEε(um−1
M+b(s)−bm−1
M, cm−1
M, zm−1
M) ds
=Eε(qm−1
M)
+Zmτ
(m−1)τZΩ
∂eWel(e(um−1
M+b(s)−bm−1
M), cm−1
M, zm−1
M) : e(∂tb) dxds
17
+εZmτ
(m−1)τZΩ
|∇um−1
M+∇b(s)− ∇bm−1
M|2∇(um−1
M+b(s)−bm−1
M) : ∇∂tbdxds.
Summing this inequality for k= 1, . . . , m one gets:
Eε