Content uploaded by Christian Heinemann
Author content
All content in this area was uploaded by Christian Heinemann on Jan 20, 2016
Content may be subject to copyright.
Shape optimisation for a class of semilinear variational
inequalities with applications to damage models
Christian Heinemann and Kevin Sturm
January 15, 2016
Abstract
The present contribution investigates shape optimisation problems for a class of
semilinear elliptic variational inequalities with Neumann boundary conditions. Sen-
sitivity estimates and material derivatives are firstly derived in an abstract operator
setting where the operators are defined on polyhedral subsets of reflexive Banach
spaces. The results are then refined for variational inequalities arising from minimi-
sation problems for certain convex energy functionals considered over upper obstacle
sets in H1. One particularity is that we allow for dynamic obstacle functions which
may arise from another optimisation problems. We prove a strong convergence
property for the material derivative and establish state-shape derivatives under reg-
ularity assumptions. Finally, as a concrete application from continuum mechanics,
we show how the dynamic obstacle case can be used to treat shape optimisation
problems for time-discretised brittle damage models for elastic solids. We derive a
necessary optimality system for optimal shapes whose state variables approximate
desired damage patterns and/or displacement fields.
Keywords: shape optimisation, semilinear elliptic variational inequalities, optimisation
problems in Banach spaces, obstacle problems, damage phase field models, elasticity;
AMS subject classification: 49J27, 49J40, 49Q10, 35J61, 49K20, 49K40, 74R05, 74B99.
Contents
1 Introduction 2
2 Preliminaries 4
2.1 Notation and basic relations . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Polyhedricity of upper obstacle sets in H1(Ω) ................ 6
2.3 Eulerian semi and shape derivative . . . . . . . . . . . . . . . . . . . . . . 11
3 Abstract sensitivity analysis 13
3.1 Sensitivity result for minimisers of energy functionals . . . . . . . . . . . . 13
3.2 Sensitivity result for uniformly monotone operators . . . . . . . . . . . . . 16
3.3 Variational inequality for the material derivative . . . . . . . . . . . . . . 17
1
4 A semilinear dynamic obstacle problem 19
4.1 Setting and state system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Perturbed problem and sensitivity estimates . . . . . . . . . . . . . . . . . 21
4.3 Limiting system for the transformed material derivative . . . . . . . . . . 25
4.4 Limiting system for the material derivative . . . . . . . . . . . . . . . . . 29
4.5 Limiting system for the state-shape derivative . . . . . . . . . . . . . . . . 31
4.6 Eulerian semi-derivative of certain shape functions . . . . . . . . . . . . . 34
5 Applications to damage phase field models 34
5.1 Physicalmodel ................................. 34
5.2 Setting up time-discretisation scheme and shape optimisation problem . . 36
5.3 Material derivative and necessary optimality system . . . . . . . . . . . . 39
1 Introduction
Finding optimal shapes such that a physical system exhibits an intended behaviour is
of great interest for plenty of engineering applications. For example design questions
arise in the construction of air- and spacecrafts, wind and combustion turbines, wave
guides and inductor coils. More examples can be found in [5] and references therein.
The physical system is usually modelled by a pde or a coupled pde system supplemented
with suitable boundary conditions. In certain cases the state is given as a minimiser of
an energy, e.g., an equilibrium state of an elastic membrane, which has to be in some
set of admissible states. The solution is then characterised by a variational inequality
holding for test-functions on the sets of admissible states.
The treatment of optimal shape and control problems for variational inequalities is
substantially more difficult as without constraints, where the sets of admissible states
is a linear space. For optimal control problems there exist a rapidly growing litera-
ture exploring different types of stationarity conditions and their approximations (see,
for instance, [14, 18]). However shape optimisation problems for systems described by
variational inequalities are less explored and reveal additional difficulties due to the in-
tricated structure of the set of admissible domains. Some results following the paradigm
first optimise—then discretise can be found in [22, 21, 15, 19] and for the first discretise–
then optimise approach we refer to [3, 1, 10].
The main aim of this paper is to establish sensitivity estimates and material deriva-
tives for certain nonlinear elliptic variational inequalities with respect to the domain.
Our approach is based on the paradigm first optimise then discretise, thus the sen-
sitivity is derived in the infinite dimensional setting. In order to highlight the main
arguments needed in the proof of these main results and to increase their applicability,
we investigate the optimisation problems firstly on an abstract operator level formulated
over a polyhedric subset Kof some reflexive Banach space V. The domain-to-state map
is there replaced by a parametrised family of operators (At) and sensitivity estimates
are shown in Theorem 3.2 and Theorem 3.3 under general assumption (see Assump-
tion (E) and Assumption (O1)). By strengthen the assumptions (see Assumption (O2))
2
differentiability with respect to the parameter thas been shown in Theorem 3.5. One
crucial requirement is the polyhedricity of the closed convex set Kon which the oper-
ators are defined. The results are applicable for optimal shape as well as for optimal
control problems.
Equipped with the proven abstract results we resort to shape optimisation problems
where the state system is a variational inequality of semilinear elliptic type given by
u∈KψΩand ∀ϕ∈KψΩ:dE(Ω, u;ϕ−u)≥0
with the energy
E(Ω, u) = ZΩ
1
2|∇u|2+λ
2|u|2+WΩ(x, u) dx(λ > 0)
and the upper obstacle set
KψΩ=v∈H1(Ω) : v≤ψΩa.e. in Ω.
In the classical theory of VI-constrained shape optimisation problems established in [23],
linear variational inequalities with constant obstacle and WΩ(x, u) = f(x) for some given
fixed function f:D→Rdefined on a “larger set” D⊃Ω have been investigated by
means of conical derivatives of projection operators in Hilbert spaces dating back to [18].
For results on topological sensitivity analysis for variational inequalities and numerical
implementations we refer to [13] as well as [2].
In our paper we allow for semilinear terms in the variational inequality by including
convex contributions to WΩwith respect to uand also consider a dependence of WΩ
and ψΩon Ω in a quit general sense. As presented in the last section of this work ψΩ
may itself be a solution of a variational inequality. Such general Ω-dependence of the
obstacle will be referred to as “dynamic obstacle” in constrast to the case of a “static
obstacle” where ψΩ(x) = g(x) for some fixed function g:D→R.
To apply the abstract sensitivity results we perform the transformation u7→ y:=
u−ψΩsuch that the transformed problem is formulated over the cone H1
−(Ω), i.e.,
the non-positive half space of H1(Ω). Existence of the material derivative ˙ywhich
turns out to be the unique solution of a variational inequality considered over the cone
Ty(H1
−(Ω)) ∩kern(dE(u;·)) and strong convergence of the corresponding difference quo-
tients are established in Theorem 4.8 and Corollary 4.9. The variational inequality
characterising the material derivative ˙uis then established in Corollary 4.11. Moreover
in the case of a static obstacle and H2(Ω)-regularity for uwe derive relations for the
state-shape derivative u0in Theorem 4.15 and Corollary 4.16.
The theorems for the abstract semilinear case are then applied to a specific model
problem from continuum damage mechanics. There one considers an elastic solid which
undergoes deformation and damage processes in a small strain setting. The state of
damage is modelled by a phase field variable χwhich influences the material stiffness and
which is described by an evolution inclusion forcing the variable χto be monotonically
decreasing in time. We consider a time-discretised version of the evolution system (but
3
we stay continuous in the spatial components) where the damage variable fulfills for all
time steps the constraints
χN≤χN−1≤. . . ≤χ0≤1 a.e. in Ω.
Such constraints lead to N-coupled variational inequalities with dynamic obstacle sets
of the type
Kk−1(Ω) = v∈H1(Ω) : v≤χk−1a.e. in Ω, k = 1, . . . , N.
Our objective is to find an optimal shape Ω such that the associated displacement fields
(uk)N
k=1 and damage phase fields (χk)N
k=1 minimise a given tracking type cost functional.
We derive relations for the material derivative and establish necessary optimality condi-
tions for optimal shapes which are summarised in Proposition 5.3.
Structure of the paper
In Section 2 we recall some basics notions from convex analysis and shape optimisation
theory. For reader’s convenience and for the sake of clarity we derive tangential and
normal cones of KψΩand prove polyhedricity of KψΩby using arguments from [18, 4, 12].
In Section 3 we establish sensitivity and material derivative results in an abstract
operator setting (see Theorem 3.2, Theorem 3.3 and Theorem 3.5). Some results are
even applicable to quasi-linear problems such as to p-Laplace equations. The advantage
of this approach is that the theorems can be applied to a large class of optimisation
problems including shape optimisation and optimal control problems.
This flexibility is demonstrated in Section 4 where semilinear VI-constrained shape
optimisation problems with an energy and obstacle of type E(Ω, u) and KψΩfrom above
are treated. By applying the abstract results from Section 3 we derive sensitivity esti-
mates for the shape-perturbed problem in Proposition 4.5, material derivatives in The-
orem 4.8 and state-shape derivatives in Theorem 4.15.
Finally, in Section 5, we apply the still abstract results from Section 4 to a particular
problem in continuum damage mechanics where dynamic obstacles occur.
2 Preliminaries
2.1 Notation and basic relations
For the treatment of variational inequalities we recall certain well-known cones from
convex analysis (the definitions can, for instance, be found in [4, Chapter 2.2.4] and [23,
Chapter 4.1]). Let K⊆Vbe a subset of a real Banach space Vand denote by V∗its
topological dual space.
The radial cone at y∈Kof the set Kis defined by
Cy(K) := {w∈V:∃t > 0, y +tw ∈K},(1)
the tangent cone at yas
Ty(K) := Cy(K)V(2)
4
and the normal cone at yas
Ny(K) := {w∗∈V∗:∀v∈K, hw∗, v −yiV≤0}.(3)
Furthermore we introduce the polar cone of a set Kas
[K]◦:= {w∗∈V∗:∀v∈K, hw∗, viV≤0},(4)
and the orthogonal complements of elements y∈Vand y∗∈V∗
[y]⊥:= {w∗∈V∗:hw∗, yiV= 0},
kern(y∗) := [y∗]⊥:= {w∈V:hy∗, wiV= 0}.
The normal cone may also be written as
Ny(K)=[Ty(K)]◦= [Cy(K)]◦.(5)
In combination with the bipolar theorem (see [4, Prop. 2.40]) we obtain
Ty(K) = [[Ty(K)]◦]◦= [Ny(K)]◦.(6)
We recall that a closed convex set K⊆Vis polyhedric if (cf. [14])
∀y∈K, ∀w∈Ny(K), Cy(K)∩[w]⊥V=Ty(K)∩[w]⊥.(7)
Note that the inclusion “⊆” is always satisfied above. Due to Mazur’s lemma and the
convexity of the involved sets, the closure in Vcan also be taken in the weak topology.
The following lemma shows a useful implication of (7) involving variational inequal-
ities arising from (possibly non-)linear operators.
Lemma 2.1. Let K⊆Vbe a polyhedric subset.
(i) Let A:K→V∗be an operator and let ybe a solution of the following variational
inequality
y∈Kand ∀ϕ∈K:hA(y), ϕ −yiV≥0.(8)
Then it holds
Cy(K)∩kern(A(y)) = Ty(K)∩kern(A(y)).(9)
(ii) For all v∈Vit holds
Cy(K)∩[v−y]⊥=Ty(K)∩[v−y]⊥,
where ydenotes the projection of von K.
Proof. To (i): We infer from (8) that −A(y)∈Ny(K). Thus definition (7) implies
Cy(K)∩kern(−A(y)) = Ty(K)∩kern(−A(y)).
The identity kern(−A(y)) = kern(A(y)) completes the proof.
To (ii): This follows from v−y∈Ny(K).
5
2.2 Polyhedricity of upper obstacle sets in H1(Ω)
Let us consider an important class of polyhedral subsets which will be utilized in Section
4 where semilinear obstacle problems are treated. Let Ω ⊆Rdbe a Lipschitz domain
and V=H1(Ω). Moreover let ψ∈Vbe a given function. We define the upper obstacle
set as
Kψ:= {w∈H1(Ω) : w≤ψa.e. in Ω}.(10)
In the remaining part of this subsection we will sketch the proofs for the characterisation
of the tangential and normal cones as well as of the polyhedricity of Kψfor reader’s
convenience since such obstacles sets are usually considered in the space
◦
H1(Ω) in the
literature. The adaption to H1(Ω) requires some careful modifications in the proofs.
Furthermore we denote with M+(Ω) the Radon measures on Ω. The Riesz repre-
sentation theorem for local compact Hausdorff spaces (see [6, Theorem VIII.2.5]) states
that for each non-negative functional I:C(Ω) →Rthere exists a unique Radon measure
µ∈M+(Ω) such that for all f∈C(Ω)
I(f) = ZΩ
fdµ. (11)
In the sequel we will use the following notation for the half space
H1
+(Ω) := v∈H1(Ω) : v≥0 a.e. in Ω.
With the help of the Riesz representation theorem we are now in the position to give a
characterisation of (cf. [4, Chapter 6.4.3] for
◦
H1(Ω) instead of H1(Ω))
H1(Ω)∗
+:= I∈H1(Ω)∗:hI, viH1(Ω) ≥0 for all v∈H1
+(Ω).
Lemma 2.2. We have
H1(Ω)∗
+=nI∈H1(Ω)∗:∃!µI∈M+(Ω),∀v∈H1(Ω) ∩C(Ω),hI, viH1(Ω) =ZΩ
vdµIo.
(12)
Proof. Let I:H1(Ω) →Rbe a non-negative functional. Then the restriction I|H1(Ω)∩C(Ω)
is a non-negative functional on the space H1(Ω) ∩C(Ω) =: Y.
Now let y∈Ybe arbitrary. Then y+:= max{0, y}and y−:= min{0, y}(defined in
a pointwise sense) are also in Yand we find by non-negativity of L:
|Ly|=|L(y++y−)|=|L(y+)
|{z }
≥0
+L(y−)
|{z }
≤0
| ≤ | L(y+)
|{z }
≥0
−L(y−)
| {z }
≥0
|
≤ |L(y+−y−)|=L(|y|)
=L(|y| − 1kyk∞)
| {z }
≤0
+kyk∞L(1)
6
≤ kyk∞L(1).
Thus I|Yis continuous in the C(Ω)-topology. Since Yis also dense in C(Ω) the functional
I|Yhas a unique continuous and non-negative extension ˜
I:C(Ω) →Rover C(Ω). By the
Riesz representation theorem (see (11)) we find a µ∈M+(Ω) such that I(v) = RΩvdµ
for all v∈C(Ω).
Conversely, let Ibe in the set on the right-hand side of (12). Then we know
hI, viH1(Ω) =RΩvdµI≥0 for all v∈Y+:= {v∈Y:v≥0 pointwise in Ω}. So
by density of Y+in H1
+(Ω) we obtain I∈H1(Ω)∗
+.
Remark 2.3. Note that, by an abuse of notation, the right-hand side of (12) is some-
times written as H1(Ω)∗∩M+(Ω) (see, e.g., [4, Chapter 6]).
For the notion of capacity of a set,quasi-everywhere (q.e.) and quasi-continuous
representant where refer to [12, Chapter 3.3]. The following result is an extension of
(11) valid for elements from H1(Ω)∗
+.
Lemma 2.4. For all I∈H1(Ω)∗
+and all f∈H1(Ω) we have ˜
f∈L1(Ω, µI)and
hI, f iH1(Ω) =ZΩ
˜
fdµI,(13)
where ˜
f(defined on Ω) denotes a quasi-continuous representative of fand µIthe measure
from (12) of Lemma 2.2.
Proof. The proof of this lemma requires some modifications of [4, Lemma 6.56] and
references therein which were designed to the situation V=
◦
H1(Ω). In our case we will
need the following auxiliary results:
(a) For an arbitrary D⊆Rdthe capacity of Dcalculates as
cap(D) = inf kvk2
H1(Rd):v∈H1(Rd) and v≥1 a.e. in a neighborhood of D.
See [12, Proposition 3.3.5] for a proof.
(b) Any function f∈H1(Ω) can be approximated by a sequence {fn} ⊆ C∞
c(Rd) in
the sense that fn→fin H1(Rd) as n→ ∞ by extending fto Rdwith compact
support and then uses an approximation argument via Friedrichs mollifiers.
The proof carried out in the following steps on the basis of [4, Lemma 6.56] and the
references therein (see also [12, Th´eor`eme 3.3.29] for the case V=H1(Rd)):
Claim 1: There exists a sequence {fn} ⊆ C∞
c(Rd)s.t. fn|Ω→˜
fin H1(Ω) and q.e. in Ω
Let {fn}be given by (b). By resorting to a subsequence (we omit the subscript) we may
find kfn−fkH1(Rd)≤2−nn−1and therefore
∞
X
n=1
4n+1kfn+1 −fnk2
H1(Rd)≤
∞
X
n=1
4n+1(kfn+1 −fkH1(Rd)+kfn−fkH1(Rd)2<+∞.
(14)
7
We define
Bn:= x∈Rd:|fn+1(x)−fn(x)| ≥ 2−n.
Since |fn+1 −fn|is a continuous with compact support in Rd, the set Bnis compact and
2n+1|fn+1 −fn| ≥ 1 holds in a neighborhood of Bn.
Thus by (a)
cap(Bn)≤4n+1kfn+1 −fnk2
H1(Rd).
Using this estimate, the sub-additivity of the capacity (see [12, Remarque 3.3.10]) and
(14), we obtain:
cap∞
[
k=n
Bk≤
∞
X
k=n
cap(Bk)≤
∞
X
k=n
4n+1kfn+1 −fnk2
H1(Rd)→0 as n→ ∞.(15)
Now let n∈Nand x∈Ω\S∞
k=nBkbe arbitrary. Then {fk(x)}k≥nis a Cauchy sequence
since for all m≥n:
|fm(x)−fn(x)| ≤
m−1
X
k=n
|fk+1(x)−fk(x)| ≤
m−1
X
k=n
2−k.
We denote the limit with ˜
f(x) and gain for all N, K ≥n:
|˜
f(x)−fN(x)| ≤ | ˜
f(x)−fK+1(x)|
| {z }
→0 as K→∞
+
K
X
k=N
|fk+1(x)−fk(x)|
| {z }
≤2−ksince x∈Ω\S∞
k=nBk
A limit passage K→ ∞ then shows
|˜
f(x)−fN(x)| ≤
∞
X
k=N
2−k.
This estimate implies that {fN}N≥nconverges uniformly to ˜
fon the set Ω \S∞
k=nBk.
Due to (15) we obtain Claim 1.
Claim 2: If cap(A)=0for a Borel set A⊆Ωthan µI(A)=0.
Let ε > 0 be arbitrary. By (a) we find a function u∈H1(Rd) such that kukH1(Ω) < ε
and u≥1 a.e. on Aεwhere Aεis a neighborhood of A. Thus there exists a Lipschitz
function fε:Rd→[0,1] such that
fε(x) =
0 if x∈Rd\Aε,
∈(0,1) if x∈Aε\A,
1 if x∈A.
8
Then fε−u≤0 a.e. in Ω and by Lemma 2.2
µI(A) = ZA
1dµI≤ZΩ
fεdµI=hI, fεiH1(Ω) =hI, uiH1(Ω) +hI, fε−uiH1(Ω)
| {z }
≤0 since fe≤ua.e. in Ω
≤ hI, uiH1(Ω)
≤εkIkH1(Ω)∗.
The limit passage ε&0 yields to claim.
Claim 3: fn→˜
fin L1(Ω, µI)
Lemma 2.2 implies for every n, m ∈N
ZΩ
|fn−fm|dµI=hI, |fn−fm|iH1(Ω) ≤ kIkH1(Ω)∗kfn−fmkH1(Ω),(16)
where fnis the approximation sequence from Claim 1. Since fn→fin H1(Ω) we obtain
from (16) that {fn}is a Cauchy sequence in L1(Ω, µI). Thus there exists a limit element
˜g∈L1(Ω, µI) and a subsequence (we omit the subscript) such that fn→˜gin L1(Ω, µI)
and pointwise µI-a.e. on Ω. However, by Claim 1, we already know that fnconverges
q.e. to ˜
fon Ω and, by Claim 2, we find that this covergence is also µI-a.e. Thus ˜
f= ˜g
µI-a.e.
Conclusion:
Finally, Lemma 2.2 shows for every n∈N
hI, fniH1(Ω) =ZΩ
fndµI.
With the properties proven above we can pass to the limit n→ ∞ and obtain (13).
We are now in a position to characterise the tangential and normal cones in Kψ.
The proofs of the following results are based on arguments from [18, Lemme 3.1-3.2,
Th´eor`eme 3.2].
Lemma 2.5. Let y∈Kψand Kψbe as in (10). Then it holds
Ty(Kψ) = u∈H1(Ω) : ˜u≤0q.e. on {˜y=˜
ψ},(17a)
Ny(Kψ) = I∈H1(Ω)∗:I∈H1(Ω)∗
+and µI({˜y < ˜
ψ})=0,(17b)
where ˜ydenotes a quasi-continuous representant of y(the same for ˜uand ˜
ψ) and µI∈
M+(Ω) the measure associated to Iby Lemma 2.2.
Please notice that the sets
{˜y=˜
ψ}:= {x∈Ω : ˜y(x) = ˜
ψ(x)},
{˜y < ˜
ψ}:= {x∈Ω : ˜y(x)<˜
ψ(x)}
are calculated for arguments in Ω(not only in Ω).
9
Proof. From the definitions (1)-(3) we see that
Ty(Kψ) = Ty−ψ(K), Ny(Kψ) = Ny−ψ(K)
with K:= {w∈H1(Ω) : w≤0 a.e. in Ω}. Thus it suffices to prove the assertion for
Kψ=K.
We firstly prove (17b).
“⊆”: Let I∈Ny(K). Then by using definition (3) and choosing v=y+wfor an
arbitrary w∈H1(Ω) with w≤0 a.e. we obtain hI, wiH1(Ω) ≤0. Thus I∈H1(Ω)∗
+
and by Lemma 2.2 we find the associated measure µIfrom (12). On the other hand by
choosing v=ψand v= 2yin (3) yields hI, yiH1(Ω) = 0. From Lemma 2.4 we obtain
ZΩ
˜ydµI= 0 with a quasi-continuous representant ˜yof y. (18)
Since y≤0 a.e. in Ω we find ˜y≤0 q.e. in Ω (see [12, Remarque 3.3.6]). This implies
in combination with (18) that RΩ|˜y|dµI= 0. Thus R{˜y<0}|˜y|dµI= 0 and therefore
µI({˜y < 0}) = 0.
“⊇”: Let I∈H1(Ω)∗
+with µI({˜y < 0}) = 0. Now let v∈Kbe arbitrary. The splitting
v= max{v, y}+ min{0, v −y}implies
hI, v −yiH1(Ω) =hI, max{v, y} − yiH1(Ω) +hI , min{0, v −y}iH1(Ω)
| {z }
≤0
≤Z{˜y=0}
max{˜v, ˜y} − ˜ydµI+Z{˜y<0}
max{˜v, ˜y} − ˜ydµI
| {z }
=0 since µI({˜y<0})=0
≤Z{˜y=0}
max{˜v, 0}
| {z }
=0 since v∈K
dµI= 0.
Hence I∈Ny(K).
Now we prove (17a). By applying the bipolar theorem as in (6) as well as Lemma
2.4, we find
Ty(K) = nu∈H1(Ω) : ZΩ
˜udµI≤0 for all I∈H1(Ω)∗
+with µI({˜y < 0})=0o
=nu∈H1(Ω) : Z{˜y=0}
˜udµI≤0 for all I∈H1(Ω)∗
+with µI({˜y < 0}) = 0o.
From this representation we see that the “⊇”-inclusion in (17a) is fulfilled. Conversely,
let u∈Ty(K). By definition of Ty(K) given in (2) we find a sequence vn∈Kand tn>0
such that tn(vn−y)→uin H1(Ω) as n→ ∞. This implies for a subsequence (we omit
the subindex) tn(˜vn−˜y)→˜uq.e. in Ω. Since vn∈Kwe see that
tn(˜vn−˜y) = tn˜vn≤0 q.e. on {˜y= 0}.
Thus ˜u≤0 q.e. on {˜y= 0}.
10
Theorem 2.6 (cf. [18, Th´eor`eme 3.2]).The set Kψis polyhedric.
Proof. Let yand was in (7) and let v∈Ty(Kψ)∩[w]⊥. Then there exists a sequence
vn→vstrongly in H1(Ω) such that vn∈Cy(Kψ). Define
v0
n:= max{vn, v}.
By resorting to quasi-continuous representants we find by Lemma 2.5
v≤0 q.e. in {y= 0}and vn≤0 q.e. in {y= 0}
and thus
v0
n≤0 q.e. in {y= 0}.
Moreover by definition of v0
n
v−v0
n≤0 q.e. in Ω.
Invoking Lemma 2.5 again yield v0
n∈Ty(Kψ) and v−v0
n∈Ty(Kψ). Since w∈Ny(Kψ)
we see by (5) that
hw, v0
ni ≤ 0 and hw, v −v0
ni ≤ 0.
Taking also hw, vi= 0 into account we obtain from above that hw, v0
ni= 0. Thus
v0
n∈Cy(Kψ)∩[w]⊥.
2.3 Eulerian semi and shape derivative
We recall some preliminaries from shape optimisation theory. For more details we refer
to [5].
Let X:Rd→Rdbe a vector field satisfying a global Lipschitz condition: there is a
constant L > 0 such that
|X(x)−X(y)| ≤ L|x−y|for all x, y ∈Rd.
Then we associate with Xthe flow Φtby solving for all x∈Rd
d
dtΦt(x) = X(Φt(x)) on [−τ, τ ],Φ0(x) = x. (19)
The global existence of the flow Φ : R×Rd→Rdis ensured by the theorem of Picard-
Lindel¨of.
Subsequently, we restrict ourselves to a special class of vector fields, namely Ck-
vector fields with compact support in some fixed set. To be more precise for a fixed
open set D⊆Rd, we consider vector fields belonging to Ck
c(D, Rd). We equip the space
Ck
c(D, Rd) respectively C∞
c(D, Rd) with the topology induced by the following family
of semi-norms: for each compact K⊆Dand muli-index α∈Ndwith |α| ≤ kwe
define kfkK,α := supx∈K|∂αf(x)|.With this familiy of semi-norms the space Ck
c(D, Rd)
becomes a locally convex vector space.
Next, we recall the definition of the Eulerian semi-derivative.
11
Definition 2.7. Let D⊆Rdbe an open set. Let J: Ξ →Rbe a shape function defined
on a set Ξof subsets of Dand fix k≥1. Let Ω∈Ξand X∈Ck
c(D, Rd)be such that
Φt(Ω) ∈Ξfor all t > 0sufficiently small. Then the Eulerian semi-derivative of Jat Ω
in direction Xis defined by
dJ(Ω)(X) := lim
t&0
J(Φt(Ω)) −J(Ω)
t.(20)
(i) The function Jis said to be shape differentiable at Ωif dJ (Ω)(X)exists for all
X∈C∞
c(D, Rd)and X7→ dJ(Ω)(X)is linear and continuous on C∞
c(D, Rd).
(ii) The smallest integer k≥0for which X7→ dJ(Ω)(X)is continuous with respect to
the Ck
c(D, Rd)-topology is called the order of dJ (Ω).
The set Din the previous definition is usually called hold-all domain or hold-all set
or universe.
In the case that the state system is given as a solution of a variational inequality
we cannot expect dJ(Ω)(X) to be linear in X. However we have the following general
result:
Lemma 2.8. Suppose that the Eulerian semi-derivative dJ(Ω)(X)exists for all X∈
Ck
c(D, Rd). Then dJ (Ω)(·)is positively 1-homogeneous.
Proof. Let λ > 0 be arbitrary. We write ΦλX
tfor the flow induced by λX. By definition
(19), we see that ΦλX
tand ΦX
λt solve
d
dtΦλX
t(x) = λX(ΦλX
t(x)),d
dtΦX
λt(x) = λX(ΦX
λt(x))
as well as ΦλX
0(x) = xand ΦX
0(x) = x. Uniqueness of the flow implies ΦλX
t= ΦX
λt.
Finally,
dJ(Ω)(λX) = lim
t&0
J(ΦλX
t(Ω)) −J(Ω)
t= lim
t&0
J(ΦX
λt(Ω)) −J(Ω)
t=λ dJ(Ω)(X).
The following result can be found for instance in [5]:
Lemma 2.9. Let D⊆Rdbe open and bounded and suppose X∈C1
c(D, Rd).
(i) We have
∂Φt−I
t→∂X strongly in C(D, Rd,d)
∂Φ−1
t−I
t→ − ∂X strongly in C(D, Rd,d)
det(∂Φt)−1
t→div(X)strongly in C(D).
(ii) For all open sets Ω⊆Dand all ϕ∈W1
µ(Ω),µ≥1, we have
ϕ◦Φt−ϕ
t→∇ϕ·Xstrongly in Lµ(Ω).(21)
12
3 Abstract sensitivity analysis
In this section we will derive sensitivity estimates and relations for material derivatives
under general conditions. We start in Section 3.1 with minimisers of certain p-coercive
energy functionals and deduce a H¨older-type estimate with exponent 1/p. We present an
example which includes the quasi-linear p-Laplacian −∆p(·) = div(|∇ ·|p−2∇·). Then we
proceed in Section 3.2 with solutions of monotone operators where we are able to improve
the estimates from Subsection 3.1. For the case p= 2 we even establish a Lipschitz type
sensitivity estimate. Finally in Subsection 3.3 we strengthen the assumptions in order to
establish the weak material derivative. A crucial requirement will be the polyhedricity
of the underlying set.
In this whole section Vwill denote a Banach space, K⊆Va closed convex subset
and τ > 0 a fixed constant.
3.1 Sensitivity result for minimisers of energy functionals
Our starting point is a family of energy functionals
E: [0, τ ]×V→R,
where we denote the set of attained infima at t∈[0, τ] by
X(t) := ut∈V: inf
ϕ∈KE(t, ϕ) = E(t, ut).(22)
Our aim is to establish a general result showing the convergence of minimisers of E(t, ·)
to minimisers of E(0,·) as t&0. Before we state our abstract sensitivity result, we
recall [20, Theorem 1] which will be used in a subsequent proof:
Theorem 3.1 ([20, Theorem 1]).Let [·]be a seminorm on V. Let E:V→Rbe an
energy functional such that for all v, w ∈Kthe mapping s7→ γ(s) := E(sw + (1 −s)v))
is C1on [0,1]. Let us denote by A:K→V∗the Gateaux-differential of Ewhich is
supposed to be p-coercive on K:
∃α > 0,∀u, v ∈K, hA(u)− A(v), u −viV≥α[v−w]p.
Then every minimum uof Eon Ksatisfies:
∀v∈K, α
p[u−v]p≤E(u)−E(v).
In what follows let Esatisfy the following assumption:
Assumption (E) Suppose that the energy functionals E(t, ·)satisfies for a given p≥1:
(i) ∃c1>0,∃c2>0,∀ϕ∈K, E(·, ϕ)is differentiable and
∀t∈[0, τ ], ∂tE(t, ϕ)≤c1kϕkp
V+c2;
13
(ii) ∃c > 0,∃Λ>0,∀ϕ∈K,∀t∈[0, τ ],
E(t, ϕ)≥ckϕkp
V−Λ;
(iii) ∀t∈[0, τ ],E(t, ·)is Gateaux-differentiable and
∃α > 0,∀u, v ∈K, hAt(u)− At(v), u −viV≥α[v−w]p,
where hAt(v), wiV:= dE(t, v;w)and [·]is a semi-norm on V;
(iv) ∀v, w ∈K,∀t∈[0, τ ],
s7→ γ(s) := E(t, sv + (1 −s)w)is C1([0,1])
Now we are in the position to state and prove our sensitivity result:
Theorem 3.2. Let E: [0, τ]×V→Rbe a family of energy functionals satisfying
Assumption (E) and let X(t)be non-empty for every t∈[0, τ]. Then X(t) = {ut}is a
singleton and there exists a constant c > 0such that for all t∈[0, τ ]:
[ut−u0]≤ct1/p.
Proof. Let t∈[0, τ ] and ut∈X(t). Let us first show that utis bounded in Vuniformly
in t. According to Assumption (E) (i)-(ii), the definition of utand the mean value
theorem, we obtain ηt∈(0, t) such that
ckutkp
V−Λ≤E(t, ut)
≤E(t, u0)
=E(0, u0) + t∂tE(ηt, u0)
≤E(0, u0) + tc1ku0kp
V+c2.
(23)
This shows that kutkV≤Cfor all t∈[0, τ] for some constant C > 0. Furthermore
applying Theorem 3.1 by using Assumption (E) (iii)-(iv) shows
c[ut−u0]p≤E(t, ut)−E(t, u0),(24)
c[ut−u0]p≤E(0, u0)−E(0, ut).(25)
Adding both inequalities, applying the mean value theorem twice with some ηt, ζt∈(0, t)
and using Assumption (E) (i) and the estimate (23) yields
2c[ut−u0]p≤E(t, ut)−E(t, u0) + E(0, u0)−E(0, ut)
≤t∂tE(ηt, ut)−∂tE(ζt, u0)
≤tC(kutkp
V+ku0kp
V)
(23)
≤tC(1 + ku0kp
V).
(26)
This finishes the proof.
14
Example (p-Laplace equation)
As an application of Theorem 3.2 let us consider the p-Laplace equation
−div(|∇u|p−2∇u) = fin K=V=
◦
W1
p(Ω)
on a bounded Lipschitz domain Ω and the associated energy given by
E(0, ϕ) = 1
pZΩ
|∇ϕ|pdx−ZΩ
f ϕ dx, ϕ ∈
◦
W1
p(Ω).
By applying integration by substitution, the energy of the perturbed equation trans-
ported to Ω via integration by substitution is of the form
E(t, ϕ) = 1
pZΩ
ξ(t)|B(t)∇ϕ|pdx−ZΩ
f(t)ϕdx,
More generally we assume that ξ: [0, τ]→Rand B: [0, τ ]→Rd×dare C1-functions
which satisfy ξ(0) = 1 and B(0) = I. Moreover let f(0) = fand f(·, x) be differentiable
and f0(t)∈Lp0(Ω) be uniformly bounded where p0=p/(p−1) denotes the conjugate of
p. We check that the assumptions in (E) are satisfied:
Indeed, we have
∂tE(t, ϕ) = ZΩ
ξ0(t)1
p|B(t)∇ϕ|p+ξ(t)|B(t)∇ϕ|p−2B(t)∇ϕ·B0(t)∇ϕdx−ZΩ
f0(t)ϕdx.
Thus applying H¨older and Young’s inequalities we verify Assumption (E) (i):
∂tE(t, ϕ)≤ZΩ
ξ0(t)1
p|B(t)∇ϕ|p+ξ(t)|B(t)∇ϕ|p−2B(t)∇ϕ·B0(t)∇ϕdx−ZΩ
f0(t)ϕdx
≤cZΩ
|∇ϕ|p+|f0(t)||ϕ|dx
≤ck∇ϕkp
Lp+ 1/p0kf0(t)kp0
Lp0+1
pkϕkp
Lp.
On the other hand using Young’s and Poincar´e’s inequality with small ε > 0
E(t, ϕ)≥ck∇ϕkp
Lp−1/p0(pε)−1
p−1kf(t)kp0
Lp/(p−1) −εkϕkp
Lp
≥c1kϕkp
W1
p−c2−εkϕkp
Lp.
Thus we have verified Assumption (E) (ii). Assumption (E) (iii) follows from uniform
p-monotonicity of −∆p(·) and Assumption (E) (iv) by direct calculations.
Finally we may use Theorem 3.2 and obtain kut−ukW1
p(Ω) ≤ct1/p for some constant
c > 0 and all sufficiently small t > 0. In the case of the usual Laplace equation, that is
for p= 2, we get kut−ukH1(Ω) ≤ct1/2.
15
3.2 Sensitivity result for uniformly monotone operators
In this section we develop sensitivity results for variational inequalites involving uni-
formly monotone operators. Let Vbe a normed space and K⊆Vbe a closed convex
subset.
Assumption (O1) Suppose that (At) : K→V∗,t∈[0, τ]is a family of operators such
that for a given p≥1:
(i) ∃α > 0,∀t∈[0, τ ],∀u, v ∈K:
αku−vkp
V≤ hAt(u)− At(v), u −viV;
(ii) ∀u∈K,∃c > 0,∀t∈[0, τ ],∀v∈K,
|hAt(u)− A0(u), u −viV| ≤ ctku−vkV.
Theorem 3.3. Suppose that (At) : K→V∗is a family of operators satisfying As-
sumption (O1). For every t > 0we denote by ut∈Ka solution of the variational
inequality
ut∈Kand ∀v∈K, hAt(ut), v −utiV≥0.(27)
Then there exists a c > 0such that
∀t∈[0, τ ] : kut−u0kV≤ct 1
p−1.
Proof. Taking into account Assumption (O1) and (27):
αkut−u0kp
V≤ hAt(ut)− At(u0), ut−u0iV
≤ −hAt(u0), ut−u0iV
=hA0(u0), ut−u0iV+hA0(u0)− At(u0), ut−u0iV
≤ |hA0(u0)− At(u0), ut−u0iV|
≤ctkut−u0kV.
Remark 3.4. In the important case p= 2 Theorem 3.3 yields a Lipschitz type estimates.
Example (p-Laplace equation)
It can be checked that the p-Laplace example from Subsection 3.1 where Atis given by
hAt(u), ϕi◦
W1
p
=ZΩ
ξ(t)|B(t)∇u|p−2B(t)∇u·B(t)∇ϕ−f(t)ϕdx
also fulfills Assumption (O1). Thus in this case Theorem 3.3 gives a sharper estimate
than Theorem 3.2.
16
3.3 Variational inequality for the material derivative
In the previous section we have shown that under certain conditions on (At) satisfied for
p= 2 the quotient (ut−u0)/t stays bounded. In this subsection we additionally assume
that Vis reflexive and that K⊆Vis a polyhedric subset. Then there will be a weakly
converging subsequence of (ut−u0)/t converging to some z∈V. If this zis unique the
whole sequence converges and additionally satisfies some limiting equation which is the
subject of this subsection.
Let (At) be as in Subsection 3.2 and define in accordance with (22) for all t∈[0, τ ]
the solution set of the associated variational inequality as
X(t) := ut∈K:∀ϕ∈K, hAt(ut), ϕ −uti ≥ 0.(28)
We will write u:= u0and A:= A0. The variational inequality for the material derivative
will be deduced from the following assumptions:
Assumption (O2) Suppose that the family (At)satisfies
(i) for all v, w ∈Vand all u∈K,
h∂A(u)w, viV:= lim
t&0A(u+tw)− A(u)
t, vV
and
hA0(u), viV:= lim
t&0At(u)− A(u)
t, vV
exist;
(ii) for all null-sequences (tn), for all sequences (vn)in Vconverging weakly to some
v∈V, for all utn∈X(tn)converging strongly to some u∈K, we have
hA0(u), viV= lim
n→0Atn(utn)− A(utn)
tn
, vnV
;
(iii) for all null-sequences (tn), there exists a subsequence (still indexed the same) such
that utn∈X(tn)converges storngly to u∈Kand (un−u)/tnconverges weakly to
some z∈Vand
h∂A(u)z, ziV≤lim inf
n→0A(utn)− A(u)
tn
,utn−u
tnV
and for all (vn)in Vconverging strongly to v∈V:
h∂A(u)z, viV= lim
n→0A(utn)− A(u)
tn
, vnV
.
17
Theorem 3.5. Let Vbe a reflexive Banach space and K⊆Va polyhedric subset.
Suppose that At:K→V∗,t∈[0, τ]is a family of operators satisfying Assumption (O1)
for p= 2 and (O2). Suppose that ut∈X(t), i.e., utsolves
ut∈K, hAt(ut), ϕ −utiV≥0∀ϕ∈K. (29)
Then the material derivative ˙u:= weak −limt&0(ut−u)/t exists and solves
˙u∈Tu(K)∩kern(A(u)) and (30a)
∀ϕ∈Tu(K)∩kern(A(u)) : h∂A(u) ˙u, ϕ −˙uiV≥ −hA0(u), ϕ −˙uiV.(30b)
Proof. Let us firstly show (30a). We get by (29)
∀ϕ∈K(Ω) : hAt(ut), ϕ −uti ≥ 0,(31)
∀ϕ∈K(Ω) : hA(u), ϕ −ui ≥ 0.(32)
Thus testing (31) with uand (32) with utand dividing by t > 0, we obtain by setting
zt:= (ut−u)/t
hAt(ut), zti ≤ 0,hA(u), zti ≥ 0.(33)
By invoking Theorem 3.3 with p= 2 we know that ut→ustrongly in Vand that ztis
bounded in Vwhich allows us to choose a weakly convergence subsequence with limit
˙u∈V. We find (by omitting the subscript)
hAt(ut), zti − hA(u),˙ui
=hAt(ut)− A(ut), zti
|{z }
→0 by Assumption (O2) (ii)
+DA(ut)− A(u)
t, ut−uE
| {z }
→0 by Assumption (O2) (iii)
+hA(u), zt−˙ui
| {z }
→0
Therefore passing to the limit in (33) gives 0 ≤ hA(u),˙ui ≤ 0 and thus ˙u∈kern(A(u)).
Furthermore we know by the definition of the radial cone that zt∈Cu(K). Taking the
weak convergence zt˙uin Vand Mazur’s Lemma into account we find ˙u∈Tu(K).
Thus (30a) is proven.
Now we will show (30b) by using (29) and obtain for every ϕ∈V:
hA(ut)− A(u), ϕ −uti=hA(ut)− At(ut), ϕ −uti+hAt(ut)− A(u), ϕ −uti
≥ hA(ut)− At(ut), ϕ −uti − hA(u), ϕ −uti.(34)
By definition of the radial cone Cu(K) (see (1)) we find for every ϕ∈Cu(K) a t∗>0
such that for all t∈[0, t∗]: u+tϕ ∈K. Plugging this test-function into (34) we obtain
for all ϕ∈Cu(K)
hA(ut)− A(u), tϕ −(ut−u)i ≥ hA(ut)− At(ut), tϕ −(ut−u)i − hA(u), tϕ −(ut−u)i.
(35)
18
Dividing the previous equation by t2and setting zt:= (ut−u)/t, we obtain
A(ut)− A(u)
t, ϕ −zt≥ − At(ut)− A(ut)
t, ϕ −zt−1
thA(u), ϕ −zti.(36)
Now let ϕ∈Cu(K)∩kern(A(u)). Then because of hA(u), ϕi= 0 and the definition of
u∈X(0) (testing the relation in (28) with ut), we find
−hA(u), ϕ −zti ≥ 0.
Thus (36) reads
A(ut)− A(u)
t, ϕ −zt≥ − At(ut)− A(ut)
t, ϕ −zt.(37)
Using Assumption (O2) we may take the lim sup on both sides to obtain (note that
−lim sup(...) = lim inf −(...))
h∂A(u)z, ϕ −zi ≥ −hA0(u), ϕ −zi ∀ϕ∈Cu(K)∩kern(A(u)).
Via density arguments we obtain the inequality for all ϕ∈Cu(K)∩kern(A(u)). Finally
using polyhedricity of Kand Lemma 2.1 (i) finish the proof.
4 A semilinear dynamic obstacle problem
In this section we are going to apply the theorems from Section 3 to generalised obstacle
problems with convex energies. present a generalised obstacle problem. It also covers
previous results from [23] where the zero obstacle case has been treated as a special
case. A non-trivial example from continuum damage mechanics is presented afterward
in Section 5.
4.1 Setting and state system
Let D⊆Rdbe an open and bounded subset. We consider a convex energy of the
following type
E(Ω, ϕ) := ZΩ
1
2|∇ϕ|2+λ
2|ϕ|2+WΩ(x, ϕ) dx, ϕ ∈H1(Ω),(38)
where Ω ⊆Dis a bounded Lipschitz domain and λ > 0. The energy is minimised over
the convex set
KψΩ(Ω) := ϕ∈H1(Ω) : ϕ≤ψΩa.e. in Ω.
A particularity of this setting is that, besides the density function WΩ, also the obstacle
function ψΩis allowed to depend on the shape variable Ω (the precise assumptions are
stated below in Assumption (A1)):
dynamic density function: Ω7→ WΩ
dynamic obstacle: Ω7→ ψΩ∈H1(Ω)
In the special case ψΩ≡0 we write K(Ω) := K0(Ω).
19
Remark 4.1. (i) An important class which is covered by our setting are static obstacle
problems where ψΩ:= Ψ|Ωwith a given function Ψ∈H2(D).
(ii) The energy E(Ω,·)is motivated by time-discretised parabolic problems, where an
additional λ-convex non-linearity may be included in E. By choosing a small time
step size, the incremental minimisation problem may take the form (38).
In context with time-discretised damage models in Section 5 we are faced with
iterative obstacle problems. In this case the obstacle ψΩitself is a solution of a
variational inequality describing the damage profile from the previous time step.
As we will see it suffices to have H1(Ω)-regularity of the damage profile provided
that the material derivative of the obstacle exists in H1(Ω) and the initial value is
in H2(Ω). We will present this application in the last section.
For later use we recall that the Sobolev exponent 2∗depending on the spatial dimen-
sion dto the space H1(Ω) is defined as
2∗:=
2d
d−2if d > 2,
arbitrary in [1,+∞) if d= 2,
+∞if d= 1.
(39)
Its conjugate (2∗)0is given by 2∗
2∗−1with the convention that (2∗)0:= 1 for 2∗= +∞.
For well-posedness of the state system we require the following assumptions (note that
we restrict ourselves to the convex case which will be exploited in the next sections):
Assumption (A1) For all Lipschitz domains Ω⊆Dit holds:
(i) WΩ(x, ·)is convex and in C1(R)for all x∈Ω;
(ii) the following map H1(Ω) →Ris assumed to be continuous (in particular the
integral exists)
y7→ ZΩ
WΩ(x, y(x)) dx
and bounded from below by
ZΩ
WΩ(x, y(x)) dx≥ −c(kykH1+ 1);
(iii) for all y, ϕ ∈H1(Ω):
ZΩ
WΩ(x, y +tϕ)−WΩ(x, y)
tdx→ZΩ
∂yWΩ(x, y)ϕdxas t&0
(in particular the integral on the right-hand side exists);
(iv) ψΩ∈H1(Ω).
20
Remark 4.2. Assumption (A1) (iii) and the continuity property from (A1) (ii) are
satisfied if, e.g., the following growth condition holds: There exist constants , C > 0and
functions s∈L1(Ω) and r∈L(2∗)0(Ω) such that for all x∈Ωand y∈R:
|WΩ(x, y)| ≤ C|y|2∗−+s(x),
|∂yWΩ(x, y)| ≤ C|y|2∗−1+r(x).
The assumptions in (A1) in combination with the direct method in the calculus of
variations imply unique solvability of the variational inequality fulfilled by the minimisers
of E(Ω,·).
Lemma 4.3. Under Assumption (A1) the energy (38) admits for each Lipschitz domain
Ω⊆Da unique minimum u(depending on Ω) on Kψ(Ω) which is given as the unique
solution of
u∈KψΩ(Ω) and ∀ϕ∈KψΩ(Ω) :
ZΩ
∇u· ∇(ϕ−u) + λu(ϕ−u) + wΩ(x, u)(ϕ−u) dx≥0,(40)
where
wΩ(x, y) := ∂yWΩ(x, y).
In the sequel we will treat the variational inequality (40) by making use of the
transformation for the state variable and its test-function:
y:= u−ψΩand ˜ϕ:= ϕ−ψΩ.
The variation inequality becomes a problem involving the standard obstacle set
K(Ω) := ϕ∈H1(Ω) : ϕ≤0 a.e. on Ω.
Substituting above tranformation into (40) we obtain the following variational inequality:
y∈K(Ω) and ∀ϕ∈K(Ω) :
ZΩ
∇y· ∇(ϕ−y) + λy(ϕ−y) + wΩ(x, y +ψΩ)(ϕ−y) dx
≥ − ZΩ
∇ψΩ· ∇(ϕ−y) + λψΩ(ϕ−y) dx
(41)
Hence it will suffice to investigate the solution yto deduce properties of the function u.
4.2 Perturbed problem and sensitivity estimates
In this subsection we prove a shape sensitivity result for the variational inequality (41).
In what follows let us denote by Φtthe flow generated by a vector field X∈C1
c(D, Rd).
For Ω ⊆Ddenote by Ωt:= Φt(Ω), t≥0, the perturbed domains (see Subsection 2.3 for
more details).
21
Perturbed problem
The solution yt∈H1(Ωt) to the perturbed variational inequality to (41) satisfies
yt∈K(Ωt) and ∀ϕ∈K(Ωt) :
ZΩt
∇yt· ∇(ϕ−yt) + λyt(ϕ−yt) + wΩt(x, yt+ψΩt)(ϕ−yt) dx
≥ − ZΩt
∇ψΩt· ∇(ϕ−yt) + λψΩt(ϕ−yt) dx.
(42)
We will sometimes write yt(X) = ytto emphasise the dependence on X. Please note
that in general y0(X) = yt(X) for all t≥0 and for all vector fields X∈C1
c(D, R2) with
the property X·n= 0 on ∂Ω. This implication will be used in the forthcoming Lemma
4.14. Throughout this work we will adopt the following abbreviations:
wt
X(x, ϕ) := wΩt(Φt(x), ϕ), W t
X(x, ϕ) := WΩt(Φt(x), ϕ), ψt
X:= ψΩt◦Φt,
A(t) := ξ(t)(∂Φt)−1(∂Φt)−T, ξ(t) := det ∂Φt, yt:= yt◦Φt
(43)
and (for t= 0)
ψ(x) := ψΩ(x), w(x, ϕ) := w0
X(x, ϕ).
From Lemma 2.9 we can directly infer the following convergences and estimates
Lemma 4.4. Let X∈C1
c(D, Rd)be given. Then it holds:
(i) the convergences as t&0:
A(t)−I
t→A0(0) = div(X)I−∂X −(∂X )Tstrongly in C(D, Rd,d),(44a)
ξ(t)−1
t→ξ0(0) = div(X)strongly in C(D); (44b)
(ii) there is a constant t∗>0such that
∀t∈[0, t∗],∀x∈D, ∀ζ∈Rd, A(t, x)ζ·ζ≥1/2|ζ|2,
∀t∈[0, t∗],∀x∈D, ξ(t, x)≥1/2.
Performing a change of variables and using (∇y)◦Φt= (∂Φt)−T∇(y◦Φt) it is easy
to check that the transported function yt(which is defined on Ω) satisfies the relation
yt∈K(Ω) and ∀ϕ∈K(Ω) :
ZΩ
A(t)∇yt· ∇(ϕ−yt) + ξ(t)λyt(ϕ−yt) + ξ(t)wt
X(x, yt+ψt
X)(ϕ−yt) dx
≥ZΩ
−A(t)∇ψt
X· ∇(ϕ−yt)−ξ(t)λψt
X(ϕ−yt) dx.
(45)
22
For later usage let us introduce the bilinear form
at(v, w) := ZΩ
A(t)∇v· ∇w+ξ(t)λvw dx,
the operator At:KψΩ(Ω) →H1(Ω)∗by
hAt(v), wiH1(Ω) := at(v, w) + ZΩ
ξ(t)wt
X(x, v)wdx(46)
and the “shifted” operator ˜
At:K(Ω) →H1(Ω)∗by
˜
At(v) := At(v+ψt
X).(47)
By making use of this notation the variational inequality (45) can be recasted as
yt∈K(Ω) and h˜
At(yt), ϕ −ytiH1≥0 for all ϕ∈K(Ω).(48)
In the following we also write A:= A0and ˜
A:= ˜
A0.
Sensitivity estimate
Our goal is to apply Theorem 3.3 designed for abstract operators. For this reason we
make the following assumption in addition to (A1):
Assumption (A2)
(i) ∀X∈C1
c(D, Rd),∃c > 0,∀t∈[0, τ ],∀χ∈H1(Ω),
kwt
X(·, χ)−w(·, χ)kL(2∗)0(Ω) ≤ct;
(ii) ∀X∈C1
c(D, Rd),∃c > 0,∀t∈[0, τ ],∀χ1, χ2∈H1(Ω),
kwt
X(·, χ1)−wt
X(·, χ2)kL(2∗)0(Ω) ≤ckχ1−χ2kH1(Ω);
(iii) ∀X∈C1
c(D, Rd),∃c > 0,∀t∈[0, τ ],
kψt
X−ψkH1(Ω) ≤ct.
We are now in the position to prove the following sensitivity result:
Proposition 4.5. Let the Assumptions (A1)-(A2) be satisfied. Then the family of op-
erators (˜
At)defined by (47) fulfills
(i) ∃α > 0,∃t∗>0,∀t∈[0, t∗],∀v, w ∈K(Ω),
αkv−wk2
H1(Ω) ≤ h ˜
At(v)−˜
At(w), v −wi; (49)
23
(ii) ∀v∈K(Ω),∃c > 0,∃t∗>0,∀t∈[0, t∗],∀w∈K(Ω),
|h ˜
At(v)−˜
A(v), v −wi| ≤ ctkv−wkH1(Ω).(50)
Proof. To (i): We first show the monotonicity estimate (49). With the help of Lemma 4.4
(ii) and monotonicity of wt
Xin the second variable (see Assumption (A1) (i)) we obtain
for all v, w ∈H1(Ω) and all small t≥0
1
2ZΩ
|∇(v−w)|2+λ|v−w|2dx
≤at(v−w, v −w)
+ZΩ
ξ(t)wt
X(x, v +ψt
X)−wt
X(x, w +ψt
X)(v+ψt
X)−(w+ψt
X)dx
(51)
Thus (49) is shown.
To (ii): Let us fix v∈H1(Ω). Then by applying H¨older inequality, Sobolev embed-
dings and the assumptions in (A2) we find for all w∈H1(Ω)
h˜
At(v)−˜
A(v), v −wi
≤ZΩ
(A(t)−I)∇v· ∇(v−w) dx
| {z }
≤kA(t)−IkL∞k∇vkL2k∇(v−w)kL2
+ZΩ
λ(ξ(t)−1)v(v−w) dx
| {z }
≤λkξ(t)−1kL∞kvkL2kv−wkL2
+ZΩ
(A(t)∇ψt
X− ∇ψ)· ∇(v−w) + λ(ξ(t)ψt
X−ψ)(v−w) dx
| {z }
≤kA(t)−IkL∞k∇ψt
XkL2+k∇ψt
X−∇ψkL2+λkξ(t)−1kL∞kψt
XkL2+λkψt
X−ψkL2kv−wkH1
+ZΩ
(ξ(t)−1)wt
X(x, v +ψt
X)(v−w)dx
| {z }
≤kξ(t)−1kL∞kwt
X(x,v+ψt
X)kL(2∗)0kv−wkH1
+ZΩ
(wt
X(x, v +ψt
X)−wt
X(x, v +ψ))(v−w) dx.
| {z }
≤kwt
X(x,v+ψt
X)−wt(x,v+ψ)kL(2∗)0kv−wkH1≤ kψt
X−ψkH1kv−wkH1
+ZΩ
(wt
X(x, v +ψ)−w(x, v +ψ))(v−w) dx.
| {z }
≤kwt
X(x,v+ψ)−w(x,v+ψ)kL(2∗)0kv−wkH1≤ctkv−wkH1
Taking Lemma 4.4 into account and using Young’s inequality, we obtain the assertion.
The desired Lipschitz estimate immediately follows from Theorem 3.3 since Propo-
sition 4.5 proves that Assumption (O1) are satisfied for p= 2.
Corollary 4.6. Under the assumption of Proposition 4.5 there exist t∗>0and c > 0
such that
kyt−ykH1(Ω) ≤ct for all t∈[0, t∗].
24
4.3 Limiting system for the transformed material derivative
In Corollary 4.6 we have established a Lipschitz estimate for the mapping t7→ yt. In
this section we are going to prove that there is a unique element ˙yin H1(Ω) – called
the material derivative – such that (yt−y)/t converges strongly to ˙yin H1(Ω) which is
uniquely determined by a variational inequality.
In order to derive the differentiability of ytwe impose the additional assumptions to
(A1) and (A2):
Assumption (A3)
(i) w(x, ·)is of class C1(R)for all x∈Ω;
(ii) for all X∈C1
c(D, Rd), there exists a function ˙wX: Ω ×R→Rsuch that for all
ϕn→ϕstrongly in H1(Ω) we have ˙wX(·, ϕ)∈L(2∗)0(Ω) and for all tn&0
wtn
X(·, ϕn)−w(·, ϕn)
tn
→˙wX(·, ϕ)strongly in L(2∗)0(Ω) as n→ ∞;
(iii) for any given sequences ϕn→ϕin H1(Ω) and tn&0with (ϕn−ϕ)/tn z weakly
in H1(Ω):
w(·, ϕn)−w(·, ϕ)
tn
→∂yw(·, ϕ)zstrongly in L(2∗)0(Ω) as n→ ∞;
(iv) for all X∈C1
c(D, Rd)there exists a function ˙
ψX∈H1(Ω) such that
ψt
X−ψ
t→˙
ψXstrongly in H1(Ω) as t&0.
Remark 4.7. (i) Property (iii) from Assumption (A3) is satisfied if, e.g., there exist
a constant C > 0and a function s∈L2∗−1
2∗−2
(Ω) such that for all x∈Ωand y∈R:
|∂yw(x, y)| ≤ C|y|α+s(x)
with the exponent α:= 2∗(2∗−1)
2∗−2. The constant αis chosen such that the function
x7→ ∂yw(x, ϕ(x))z(x)and x7→ f0(ϕ(x))z(x)are in L(2∗)0(Ω) for given ϕ, z ∈
H1(Ω).
(ii) A useful consequence of properties (ii) and (iii) is the following continuity
wtn
X(·, ϕn)→w(·, ϕ)strongly in L(2∗)0(Ω) as n→ ∞.
for all ϕn→ϕstrongly in H1(Ω) and tn&0.
25
(iii) Let X∈C1
c(D, Rd)be given. Then we have by using property (iv) from Assumption
(A3)
−A(t)∇ψt
X+∇ψ
t→ − A0(0)∇ψ− ∇ ˙
ψXstrongly in L2(Ω,Rd),
−ξ(t)ψt
X+ψ
t→ − ξ0(0)ψ−˙
ψXstrongly in L2(Ω,Rd).
We are now well-prepared for the derivation of the material derivative.
Theorem 4.8. Let (A1)-(A3) be satisfied. The weak material derivative ˙yof t7→ yt
exists in all directions X∈C1
c(D, Rd)and is characterised as the unique solution of the
following variational inequality
(˙y∈˜
Sy(K)and ∀ϕ∈˜
Sy(K) :
h∂˜
A(y) ˙y , ϕ −˙yiH1≥ −h ˜
A0(y), ϕ −˙yiH1,(52)
where ˜
Sy(K)denotes the closed and convex cone
˜
Sy(K) = Ty(K)∩kern(˜
A(y)).(53)
The functional derivatives ∂˜
Aand ˜
A0are given by
h∂˜
A(y) ˙y , ϕi=a( ˙y+˙
ψX, ϕ) + ZΩ
∂yw(x, y +ψ) ˙yϕ dx, (54)
h˜
A0(y), ϕi=ZΩ
A0(0)∇y· ∇ϕ+ξ0(0)λy +w(x, y +ψ)ϕdx
+ZΩ
˙wX(x, y +ψ)ϕ+∂yw(x, y +ψ)˙
ψXϕdx
+ZΩ
A0(0)∇ψ· ∇ϕ+ξ0(0)λψϕ dx.
(55)
Proof.
Existence of ˙y:We want to apply Theorem 3.5. For this we need to check Assumption
(O2). To this end we notice that by Corollary 4.6 ytn→ustrongly and (ytn−y)/tn z
weakly in H1(Ω) for a suitable subsequence tn&0.
•We check (O2) (ii): Let vn v be a given weakly convergence sequence in H1(Ω).
Then
*˜
Atn(ytn)−˜
A(ytn)
tn
, vn+
=ZΩ
A(tn)−I
tn
∇ytn· ∇vndx
| {z }
→RΩA0(0)∇y·∇vdx
+ZΩ
ξ(tn)−1
tnλytn+wtn
X(x, ytn+ψtn
X)vndx
| {z }
→RΩξ0(0)(λy+w(x,y+ψ))vdxby Remark 4.7 (ii)
26
+ZΩ
wtn
X(x, ytn+ψtn
X)−w(x, ytn+ψtn
X)
tn
vndx
| {z }
→RΩ˙wX(x,y+ψ)vdxby Assumption (A3) (ii) and (iv)
+ZΩ
w(x, ytn+ψtn
X)−w(x, y +ψ)
tn
vndx−ZΩ
w(x, ytn+ψ)−w(x, y +ψ)
tn
vndx
| {z }
→RΩ∂yw(x,y+ψ)(z+˙
ψX)vdx−RΩ∂yw(x,y+ψ)zv dx=RΩ∂yw(x,y+ψ)˙
ψXvdxby (A3) (iii)-(iv)
+ZΩ
A(tn)−I
tn
∇ψtn
X· ∇vn+ξ(tn)−1
tn
ψtn
Xvndx
| {z }
→− RΩA0(0)∇ψ·∇v+ξ0(0)ψv dx
.
•We check (O2) (iii):
*˜
A(ytn)−˜
A(y)
tn
,ytn−y
tn+
=ZΩ∇ytn−y
tn2+λytn−y
tn2dx
| {z }
lim inf ≥RΩ|∇z|2+λ|z|2dx
+ZΩ
w(x, ytn+ψ)−w(x, y +ψ)
tn
ytn−y
tn
dx
| {z }
→RΩ∂yw(x,y+ψ)|z|2by Assumption (A3) (iii)
and for all ϕn→ϕstrongly in H1(Ω):
*˜
A(ytn)−˜
A(y)
tn
, ϕn+
=ZΩ
∇ytn−y
tn
· ∇ϕn+λytn−y
tn
ϕndx
| {z }
→RΩ∇z·∇ϕ+λzϕ dx
+ZΩ
w(x, ytn+ψ)−w(x, y +ψ)
tn
ϕndx
| {z }
→RΩ∂yw(x,y+ψ)zϕ by Assumption (A3) (iii)
.
•Property (O2) (i) follows from the above calculations.
Uniqueness of ˙y:Assume two solutions ˙yand ˙zfor (52). Testing their variational
inequalities with ˙zand ˙y, respectively, and adding the result yields
h∂˜
A(y) ˙y−∂˜
A(y) ˙z, ˙y−˙zi ≤ 0.
The left-hand side calculates as
h∂˜
A(y) ˙y−∂˜
A(y) ˙z, ˙y−˙zi
=a( ˙y−˙z, ˙y−˙z) + ZΩ
∂yw(x, y +ψ)|˙y−˙z|2dx.
Due to the convexity assumption in (A1) (i) we find ∂yw≥0 and see that
a( ˙y−˙z, ˙y−˙z)≤0.
We obtain ˙y−˙z= 0.
27
By exploiting the specific structure of ˜
Atand Assumption (A3) we can even show
that the strong material derivative exists.
Corollary 4.9. We have for all X∈C1
c(D, Rd)
yt
X−y
t→˙yXstrongly in H1(Ω).(56)
Proof. We test the variational inequality (48) with ϕ=ytand for t= 0 with ϕ=y.
Adding both inequalities yields
h˜
At(yt)−˜
A(y), yt−yi ≤ 0.
Dividing by t2and rearranging the terms we obtain by setting zt:= (yt−y)/t
a(zt, zt)
≤ − ZΩ
A(t)−I
t∇yt· ∇ztdx−ZΩ
λξ(t)−1
tytztdx
−ZΩξ(t)−1
twt
X(x, yt+ψt
X) + wt
X(x, yt+ψt
X)−w(x, yt+ψt
X)
tztdx
−ZΩ
w(x, yt+ψt
X)−w(x, y +ψ)
tztdx
−ZΩ
A(t)∇ψt
X− ∇ψ
t· ∇ztdx−ZΩ
λξ(t)ψt
X−ψ
tztdx
=: B(t).
(57)
The known convergence properties shows as t&0 for a subsequence
B(t)→ −h ˜
A0(y),˙yi − ZΩ
∂yw(x, y +ψ)|˙y|2dx−ZΩ
∇˙
ψX· ∇ ˙ydx−ZΩ
λ˙
ψX˙ydx
| {z }
=:B(0)
.
However testing (52) with ϕ= 2 ˙y∈˜
Sy(K) we also obtain h∂˜
A(y) ˙y, ˙yiH1≥ −h ˜
A0(y),˙yiH1
which is precisely
a( ˙y, ˙y)≥B(0).
All in all we get
lim sup
t&0
a(zt, zt)≤lim sup
t&0
B(t) = B(0) ≤a( ˙y, ˙y).(58)
The weak convergence zt˙yin H1(Ω) implies lim inft&0a(zt, zt)≥a( ˙y, ˙y). Together
with (58) this gives a(zt, zt)→a( ˙y, ˙y) as t&0. This finishes the proof.
Remark 4.10. If we assume that
˙wX(x, y) := T0(x, y)·X(x) + T1(x, y) : ∂X(x) (59)
28
for functions T0(·,·):Ω×R→Rdand T1(·,·):Ω×R→Rd×dwe may rewrite the
variational inequality in (52) by using Lemma 4.4 as
a( ˙y , ϕ −˙y) + ZΩ
∂yw(x, y +ψ) ˙y(ϕ−˙y) dx
≥ZΩ
L1(x, y +ψ;ϕ−˙y) : ∂X +L0(x, y +ψ;ϕ−˙y)·Xdx
−a(˙
ψ, ϕ −˙y) + ZΩ
∂yw(x, y +ψ)˙
ψ(ϕ−˙y) dx,
where we use the abbreviations
L1(x, y +ψ;ϕ) := −∇(y+ψ)· ∇ϕ+λ(y+ψ) + w(x, y +ψ)ϕI
+∇ϕ⊗ ∇(y+ψ) + ∇(ψ+y)⊗ ∇ϕ−T1(x, y +ψ)ϕ,
L0(x, y +ψ;ϕ) := −T0(x, y +ψ)ϕ.
4.4 Limiting system for the material derivative
So far we have derived an equation for ˙y. Since we are interested in the original problem
(40), we may now use Theorem 4.8 and the transformation y=u−ψto obtain the
material derivative equation for (40). It is clear that ˙y= ˙u−˙
ψXand we conclude with
the following result:
Corollary 4.11. Under the assumptions (A1)-(A3) the material deriative ˙u= ˙u(X)of
solutions of the perturbed problem to (40) in direction X∈C1
c(D, Rd)exists and is given
as the solution of the following variational inequality:
˙u∈SX
u(Kψ)and ∀ϕ∈SX
u(Kψ) :
a( ˙u, ϕ −˙u) + ZΩ
∂yw(x, u) ˙u(ϕ−˙u) dx
≥ − ZΩ
A0(0)∇u· ∇(ϕ−˙u) + ξ0(0)λu +w(x, u)(ϕ−˙u) dx
−ZΩ
˙wX(x, u)(ϕ−˙u) dx,
(60)
where
SX
u(Kψ) := Tu(Kψ)∩kern(A(u)) + ˙
ψX.
In particular under the additional assumption in Remark 4.10
a( ˙u, ϕ −˙u) + ZΩ
∂yw(x, u) ˙u(ϕ−˙u) dx
≥ZΩ
L1(x, u;ϕ−˙u) : ∂X +L0(x, u;ϕ−˙u)·Xdx.
29
Proof. We obtain from Theorem 4.8 that ˙u∈˜
Sy(K) + ˙
ψXand for all ϕ∈˜
Sy(K) + ˙
ψX:
h∂˜
A(u−ψ)( ˙u−˙
ψX), ϕ −˙uiH1≥ −h ˜
A0(u−ψ), ϕ −˙uiH1,
which is precisely the inequality in (60).
It remains to show SX
u(Kψ) = ˜
Sy(K)+ ˙
ψXwhich is equivalent to Tu(Kψ)∩kern(A(u)) =
Ty(K)∩kern(˜
A(y)). Indeed, by definition (47) we find
kern(A(u)) = kern(˜
A(y))
as well as by (1)-(3)
Tu(Kψ) = Tu−ψ(K) = Ty(K)
Note that we get the following characterisation of SX
uby using Lemma 2.5 and the
definition in (53):
ϕ∈SX
u(Kψ)⇔ϕ−˙
ψX∈Tu(Kψ)∩kern(A(u))
⇔(ϕ∈H1(Ω) with ϕ≤˙
ψXq.e. on {u=ψΩ},
hA(u), ϕ −˙
ψXi= 0.
Moreover under an additional assumptions we obtain the subsequent translation prop-
erty:
Lemma 4.12. Suppose that u, ψ ∈H2(Ω) and let ζ∈H1(Ω) be with
˜
ζ= 0 q.e. on the coincidence set {x∈Ω : ˜u(x) = ˜
ψ(x)},
where ˜
ζ,˜uand ˜
ψdenote quasi-continuous representatives for ζ,uand ψ. Then we have
±ζ∈Tu(Kψ)∩kern(A(u)).
In particular
ζ+SX
u(Kψ) = SX
u(Kψ).(61)
Proof. It is clear from the assumption that ±˜
ζ= 0 q.e. on the coincidence set {u=ψ}.
Thus ±ζ∈Tu(Kψ). Furthermore y=u−ψsatisfies the variational inequality (see (48)
with t= 0)
h˜
A(y), ϕ −yi ≥ 0 for all ϕ∈H1(Ω) and ϕ≤0 a.e. in Ω.
From the H2(Ω)-regularity of uand ψwe deduce that (in a pointwise formulation)
˜
A(y) = 0 a.e. in {x∈Ω : u(x)< ψ(x)}. In particular we see that
h˜
A(y), ϕi= 0 for all ϕ∈H1(Ω) with {x∈Ω : ϕ(x)=0}⊇{x∈Ω : u(x) = ψ(x)}a.e.
Testing with ϕ=±ζyields ±ζ∈kern(˜
A(y)) = kern(A(u)).
Finally, ζ∈Tu(Kψ)∩kern(A(u)) implies ζ+SX
u(Kψ)⊆SX
u(Kψ), and −ζ∈Tu(Kψ)∩
kern(A(u)) implies ζ+SX
u(Kψ)⊇SX
u(Kψ).
30
In the following ψΩis referred to as a static obstacle if there exists a fixed function
ψ∈H2(D) such that ψ˜
Ω=ψ|˜
Ωfor all Lipschitz domains ˜
Ω⊆D.
Remark 4.13. Let X∈C1
c(D, Rd). Suppose that ψΩis a static obstacle, u∈H2(Ω)
and {X=0} ⊇ {˜u=˜
ψΩ}q.e. in Ω. Then ˙
ψX=∇ψΩ·Xand the assumptions from
Lemma 4.12 are satisfied for ζ=˙
ψXand we obtain
±˙
ψX∈Tu(Kψ)∩kern(A(u)).
In particular
SX
u(Kψ) = Tu(Kψ)∩kern(A(u))
and
ϕ∈SX
u(Kψ)⇔(ϕ∈H1(Ω) with ϕ≤0q.e. on {u=ψΩ},
hA(u), ϕi= 0.
4.5 Limiting system for the state-shape derivative
The state shape derivative of uat Ω in direction X∈C1
c(D, Rd) is defined by
u0=u0(X) := ˙u−∂Xuon Ω (62)
where usolves (40), ˙usolves (60) and ∂Xu:= ∇u·X. It is clear that u0∈L2(Ω).
Thus in general the state shape derivative is less regular than the material derivative.
Another important observation is that the boundary conditions imposed on ˙uon ∂Ω are
not carried over to u0.
Lemma 4.14. Let X∈C1
c(D, Rd)be a vector field satisfying X·n= 0 on ∂Ω. Then
the state shape derivative vanishes identically, that is, u0(X)=0a.e. on Ω.
Proof. The X-flow Φtleaves the domain Ω unchanged, i.e., Φt(Ω) = Ω for all t∈[0, τ ].
Consequently, ut=u(Ωt) = u(Ω) = uand thus ut=ut◦Φt=u◦Φtfor all t∈[0, τ ].
Hence by Lemma 2.9 (ii) we may calculate the material derivative ˙uas
ut−u
t=u◦Φt−u
t→∂Xustrongly in L2(Ω).
Thus ˙u=∂Xuand consequently u0= 0.
Now we are prepared to prove the main result of this section which gives a simplified
variational inequality for the state-shape derivative u0under certain conditions. To
derive this result we will assume the enhanced regularity u∈H2(Ω). Preliminarily we
observe from Corollary 4.11 and by using the relation (62) that
u0∈ˆ
SX
u(Kψ) and ∀ϕ∈ˆ
SX
u(Kψ) :
a(u0, ϕ −u0) + ZΩ
∂yw(x, u)u0(ϕ−u0) dx
≥ − ZΩ
A0(0)∇u· ∇(ϕ−u0) + ξ0(0)λu +w(x, u)(ϕ−u0) dx
−a(∂Xu, ϕ −u0)−ZΩ
∂yw(x, u)∂Xu(ϕ−u0) dx,
(63)
31
where ˆ
SX
u(Kψ) := Tu(Kψ)∩kern(A(u)) + ˙
ψX−∂Xu.
We notice that in general the cone b
SX(K) depend on the vector field X. In the case of
a static obstacle problem (see Remark 4.1 (i)) we derive the following result:
Theorem 4.15. Suppose that (A1)-(A3), (59) and u∈H2(Ω) hold. Furthermore let
ψΩbe a static obstacle function.
Then ˆ
SX
uis independent of X∈C1
c(D, Rd)with
ˆ
SX
u(Kψ) = Tu(Kψ)∩kern(A(u)) =: Su(Kψ) (64)
and the state shape derivative is the unique solution of
u0∈Su(Kψ)and ∀ϕ∈Su(Kψ) :
a(u0, ϕ −u0) + ZΩ
∂yw(x, u)u0(ϕ−u0) dx
≥ZΓ
S1(x, u;ϕ−u0)n·n(X·n) ds,
(65)
where
S1(x, u;ϕ) := L1(x, u;ϕ)− ∇u⊗ ∇ϕ.
with L1from Remark 4.10.
Proof. By using the assumption ˙
ψX=∇ψΩ·Xwe find on the coincidence set {u=ψΩ}
(here we resort to quasi-continuous representants):
˙
ψX−∂Xu=˙
ψX−∂XψΩ= 0.
Lemma 4.12 applied to ζ=˙
ψX−∂Xuyields ±(˙
ψX−∂Xu)∈Su(Kψ) and therefore (64).
Furthermore by using the notation in Remark 4.10 and the identity (note that u∈
H2(Ω) by assumption)
∇(∂Xu)=(∂X )T(∇u)+(∂2u)X,
the variational inequality in (63) rewrites to u0∈Su(Kψ) and for all ϕ∈Su(Kψ):
a(u0, ϕ −u0)+ ZΩ
∂yw(x, u)u0(ϕ−u0) dx
≥ZΩ
S1(x, u;ϕ−u0) : ∂X +S0(x, u;ϕ−u0)·Xdx,
(66)
where
S0(x, u, ϕ) := L0(x, u, ϕ)−∂yw(x, u)ϕ∇u−(∂2u)∇ϕ,
S1(x, u, ϕ) := L1(x, u, ϕ)− ∇u⊗ ∇ϕ.
32
Picking any vector field X∈C1
c(D, Rd) with X·n= 0 on Γ we know from Lemma 4.14
that u0(±X) = 0 and it follows from (66)
ZΩ
S1(x, u; ˜ϕ) : ∂X +S0(x, u; ˜ϕ)·Xdx= 0 (67)
for all ˜ϕ∈Su(Kψ). Then integrating by parts in (67) shows the pointwise identity
−div(S1(x, u(x); ˜ϕ(x))) + S0(x, u(x); ˜ϕ(x)) = 0 a.e. on Ω.(68)
Now for an arbitrary X∈C1
c(D, Rd) and ˜ϕ∈Su(Kψ) we consider the additive splitting
X=Xn+XTfor Xn, XT∈C1
c(D, Rd) such that Xn=n(X·n) and XT=X−n(X·n)
on Γ. Then XT·n= 0 on Γ and we get
ZΩ
S1(x, u; ˜ϕ) : ∂X +S0(x, u; ˜ϕ)·Xdx
=ZΩ
S1(x, u; ˜ϕ) : ∂XT+S0(x, u; ˜ϕ)·XTdx
| {z }
=0 by (67)
+ZΩ
S1(x, u; ˜ϕ) : ∂Xn+S0(x, u; ˜ϕ)·Xndx
| {z }
partial integration and (68)
=ZΓ
S1(x, u; ˜ϕ)n·Xnds.
(69)
We may test (69) with ˜ϕ=u0since u0∈Su(Kψ). Then multiplying the resulting identity
with −1 and exploiting linearity of S0and S1with respect to ϕyields
ZΩ
S1(x, u;−u0) : ∂X +S0(x, u;−u0)·Xdx=ZΓ
S1(x, u;−u0)n·Xnds. (70)
Now we find by letting ˜ϕ=ϕ∈Su(Kψ) be arbitrary, adding (69) and (70), and again
exploiting linearity
ZΩ
S1(x, u;ϕ−u0) : ∂X +S0(x, u;ϕ−u0)·Xdx=ZΓ
S1(x, u;ϕ−u0)n·Xnds.
In combination with (66) we obtain (65). Uniqueness of u0is implied by uniqueness of
˙y(see Theorem 4.8).
It is readily checked that
S1(x, u;ϕ)n·n=−∇Γu· ∇Γϕ−λu +w(x, u)ϕfor all ϕ∈Su(Kψ).
Thus we conclude this section with an explicit formula for the shape derivative in the
case of a static obstacle.
33
Corollary 4.16. Under the assumption of Theorem 4.15 the shape derivative u0is the
unique solution of the following variational inequality:
u0∈Su(Kψ),a(u0, ϕ −u0) + ZΩ
∂yw(x, u)u0(ϕ−u0) dx
≥ − ZΓ∇Γu· ∇Γ(ϕ−u0) + λu +w(x, u)(X·n)(ϕ−u0) ds
for all ϕ∈Su(Kψ).
4.6 Eulerian semi-derivative of certain shape functions
We adopt the notation from Subsection 2.3 and denote by J: Ξ →Ra shape function.
Application of Corollary 4.11, Lemma 2.8 and the chain rule yield the following result:
Corollary 4.17. Let (A1)-(A3) be satisfied and let Ω∈Ξbe a Lipschitz domain, X∈
C1
c(D, Rd)and Φt: Ω →Ωtbe the associated flow. Suppose that for all small t > 0
J(Ωt) = J(Φt, ut),
where
J=J(Φ, u) : C0,1(Ω; Rd)×H1(Ω) →R
is assumed to be a Fr´echet differentiable functional and ut∈H1(Ω) the transported state
ut=ut◦Φtwith the unique solution utof (40) on Ωt.
Then the Eulerian semi-derivative exists and is given as
dJ(Ω)(X) = hdΦJ(I d, u0), XiC0,1(Ω;Rd)+hduJ(Id, u0),˙uXiH1(Ω),
where ˙uXdenotes the unique solution of (60).
In particular dJ(Ω)(·)is positively 1-homogeneous.
5 Applications to damage phase field models
In this section we investigate shape optimization problems for a coupled inclusion/pde
system describing damage processes in linear elastic materials. Our aim is to apply
the abstract results from Section 4 designed for semilinear variational inequalities with
dynamic obstacles to such concrete application scenarios. In this way we demonstrate
how necessary optimality conditions for shape problems can be derived for relevant
engineering tasks.
5.1 Physical model
The physical model under consideration was derived in [9] and is described in the time-
continuous setting by the following relations:
utt −divC(χ)ε(u)=`,(71a)
34
0∈∂I(−∞,0] (χt) + χt−∆χ+1
2C0(χ)ε(u) : ε(u) + f0(χ),(71b)
with the damage-dependent stiffness tensor Cand the damage potential function f.
The variable udenotes the displacement field, ε(u) := 1
2(∂u+ (∂u)T) the linearised
strain tensor and χis an internal variable (a so-called phase field variable) indicating
the degree of damage. In terms of damage mechanics χis interpreted as the density of
micro-defects and is therefore valued in the unit interval (cf. [17]). In this spirit we may
use the following interpretation:
χ(x) =
1↔no damage in x,
∈(0,1) ↔partial damage in x,
0↔maximal damage in x.
The system is supplemented with initial-time values for χ,uand ut, Dirichlet boundary
condition for uand homogeneous Neumann boundary condition for χ. The governing
state system (71) can be derived by balance equations and suitable constitutive relations
such that the laws of thermodynamics from continuum physics are fulfilled. We refer to
[9] for more details on the derivation of the model.
A main feature of the evolution system (71) is the uni-directionality constraint χt≤0
enforced by the subdifferential ∂I(−∞,0](χt). This leads to non-smooth/switching be-
haviour of the evolution law by noticing that (71b) rewrites as
χt=(d, if d≤0,
0,if d > 0with the driving force d= ∆χ−1
2C0(χ)ε(u) : ε(u)−g0(χ).
A weak formulation of (71) and existence of weak solution can be found in [11] with
minor adaption. Existence and uniqueness results for strong solutions for the above
system with higher-order viscous terms are established in [7]. For the analysis of quasi-
linear variants of (71) and for rate-independent as well as rate-dependent cases, we refer
to [16] and the references therein.
The following remark justifies that the phase field variable χtakes only admissible
values provided H1(0, T ;H1(Ω))-regularity and mild growth assumptions on Cand g. In
that case it is not