Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift

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xPreface

Herbert Amann has been a steady source of new ideas, and he has inﬂuenced

many researchers. His unwavering dedication to research and teaching has been

an example to all of his colleagues and students, in particular to his 24 doctoral

students. In 2001 he became foreign corresponding member of the Real Academia

de Ciencias Exactas, F´ısicas y Naturales, Madrid, and, one year later, received a

Doctor Honoris Causa from the Universidad Complutense, Madrid. As of 2004,

Herbert Amann is Professor Emeritus of the Universit¨at Z¨urich.

During his long and ongoing career he has enjoyed the invaluable support of

his wife, Gisela Amann.

The present volume contains original research papers and reﬂects the wide-

ranging scientiﬁc interests of Herbert Amann. It is inspired by the conference

“Nonlinear Parabolic Problems: In honor of Herbert Amann” held May 10–16,

2009, at the Banach Center in Bedlewo, Poland.

We are grateful to all the participants of the conference and all the contrib-

utors of this volume.

References

[1] H. Amann. Monte-Carlo-Methoden und lineare Randwertprobleme. Z. Angew. Math.

Mech. 48 (1968), 109–116.

[2] H. Amann.

Uber die n¨aherungsweise L¨osung nichtlinearer Integralgleichungen. Nu-

mer. Math. 19 (1972), 29–45.

[3] H. Amann. Lusternik-Schnirelman theory and non-linear eigenvalue problems. Math.

Ann. 199 (1972), 55–72.

[4] H. Amann, S.A. Weiss. On the uniqueness of the topological degree. Math. Z. 130

(1973), 39–54.

[5] H. Amann. Saddle points and multiple solutions of diﬀerential equations. Math. Z.

169 (1979), no. 2, 127–166.

[6] H. Amann. On the number of solutions of nonlinear equations in ordered Banach

spaces. J. Functional Analysis 11 (1972), 346–384.

[7] H. Amann. Existence and multiplicity theorems for semi-linear elliptic boundary

value problems. Math. Z. 150 (1976), no. 3, 281–295.

[8] H. Amann. Periodic solutions of semilinear parabolic equations. Nonlinear analysis

(collection of papers in honor of Erich H. Rothe), pp. 1–29, Academic Press, New

York, 1978.

[9] H. Amann. Multiple positive ﬁxed points of asymptotically linear maps. J. Functional

Analysis 17 (1974), 174–213.

[10] H. Amann, M.G. Crandall. On some existence theorems for semi-linear elliptic equa-

tions. Indiana Univ. Math. J. 27 (1978), no. 5, 779–790.

[11] H. Amann. Fixed point equations and nonlinear eigenvalue problems in ordered Ba-

nach spaces. SIAM Rev. 18 (1976), no. 4, 620–709.

[12] H. Amann, P. Hess. A multiplicity result for a class of elliptic boundary value prob-

lems. Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), no. 1-2, 145–151.

Preface xi

[13] H. Amann, E. Zehnder. Nontrivial solutions for a class of nonresonance problems

and applications to nonlinear diﬀerential equations. Ann. Scuola Norm. Sup. Pisa

Cl.Sci.(4)7 (1980), no. 4, 539–603.

[14] H. Amann. Parabolic evolution equations and nonlinear boundary conditions. J. Dif-

ferential Equations 72 (1988), no. 2, 201–269.

[15] H. Amann. Existence and regularity for semilinear parabolic evolution equations.

Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 4, 593–676.

[16] H. Amann. Parabolic evolution equations in interpolation and extrapolation spaces.

J. Funct. Anal. 78 (1988), no. 2, 233–270.

[17] H. Amann. Dynamic theory of quasilinear parabolic equations. I. Abstract evolution

equations. Nonlinear Anal. 12 (1988), no. 9, 895–919.

[18] H. Amann. Dynamic theory of quasilinear parabolic equations. II. Reaction-diﬀusion

systems. Diﬀerential Integral Equations 3 (1990), no. 1, 13–75.

[19] H. Amann. Dynamic theory of quasilinear parabolic systems. III. Global existence.

Math. Z. 202 (1989), no. 2, 219–250.

[20] H. Amann. Nonhomogeneous linear and quasilinear elliptic and parabolic bound-

ary value problems. Function spaces, diﬀerential operators and nonlinear analysis

(Friedrichroda, 1992), 9–126, Teubner-Texte Math., 133, Teubner, Stuttgart, 1993.

[21] H. Amann. Coagulation-fragmentation processes. Arch. Ration. Mech. Anal. 151

(2000), no. 4, 339–366.

[22] H. Amann. Operator-valued Fourier multipliers, vector-valued Besov spaces, and ap-

plications. Math. Nachr. 186 (1997), 5–56.

[23] H. Amann. Compact embeddings of vector-valued Sobolev and Besov spaces. Dedicated

to the memory of Branko Najman. Glas. Mat. Ser. III 35 (55) (2000), no. 1, 161–177.

[24] H. Amann. Elliptic operators with inﬁnite-dimensional state spaces. J. Evol. Equ. 1

(2001), no. 2, 143–188.

[25] H. Amann. Multiplication in Sobolev and Besov spaces. Nonlinear analysis, 27–50,

Sc. Norm. Super. di Pisa Quaderni, Scuola Norm. Sup., Pisa, 1991.

[26] H. Amann. On the strong solvability of the Navier-Stokes equations. J. Math. Fluid

Mech. 2 (2000), no. 1, 16–98.

[27] H. Amann. Stability of the rest state of a viscous incompressible ﬂuid. Arch. Rational

Mech. Anal. 126 (1994), no. 3, 231–242.

[28] H. Amann. Time-delayed Perona-Malik type problems. Acta Math. Univ. Comenian.

(N.S.) 76 (2007), no. 1, 15–38.

[29] H. Amann. Maximal regularity and quasilinear parabolic boundary value problems.

Recent advances in elliptic and parabolic problems, 1–17, World Sci. Publ., Hacken-

sack, NJ, 2005.

[30] H. Amann. Ordinary diﬀerential equations. An introduction to nonlinear analysis.

Translated from the German by Gerhard Metzen. de Gruyter Studies in Mathemat-

ics, 13. Walter de Gruyter & Co., Berlin, 1990. xii+438 pp.

[31] H. Amann. Linear and quasilinear parabolic problems. Vol. I. Abstract linear the-

ory. Monographs in Mathematics, 89. Birkh¨auser Boston, Inc., Boston, MA, 1995.

xxxvi+335 pp.

xii Preface

[32] H. Amann, J. Escher. Analysis. I. Translated from the 1998 German original by

Gary Brookﬁeld. Birkh¨auser Verlag, Basel, 2005. xiv+426 pp.

[33] H. Amann, J. Escher. Analysis. II. Translated from the 1999 German original by

Silvio Levy and Matthew Cargo. Birkh¨auser Verlag, Basel, 2008. xii+400 pp.

[34] H. Amann, J. Escher. Analysis. III. Translated from the 2001 German original by

Silvio Levy and Matthew Cargo. Birkh¨auser Verlag, Basel, 2009. xii+468 pp.

[35] H. Amann. Anisotropic function spaces and maximal regularity for parabolic prob-

lems. Part 1:Functionspaces.Jindrich Necas Center for Mathematical Modeling

Lecture Notes, Prague, Volume 6, 2009.

Joachim Escher

Leibniz Universit¨at Hannover

Patrick Guidotti

University of California, Irvine

Matthias Hieber

Technische Universit¨at Darmstadt

Piotr Mucha

Warsaw University

Jan Pr¨uß

Martin-Luther-Universit¨at

Halle-Wittenberg

Yoshihiro Shibata

Waseda University

Gieri Simonett

Vanderbilt University

Christoph Walker

Leibniz Universit¨at Hannover

Wojciech Zajaczkowski

Polish Academy of Sciences

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 80, 1–20

 2011 Springer Basel AG

Double Obstacle Limit for a

Navier-Stokes/Cahn-Hilliard System

Helmut Abels

Dedicated to Herbert Amann on the oc casion of his 70th birthday

Abstract. We consider the double obstacle limit for a Navier-Stokes/Cahn-

Hilliard type system. The system describes a so-called diﬀuse interface model

for the two-phase ﬂow of two macroscopically immiscible incompressible vis-

cous ﬂuids in the case of matched densities, also known as Model H. Starting

with a suitable class of singular free energies, which keep the concentration

strictly inside the physically reasonable interval [a, b], we analyze a certain

singular limit, where the equation for the chemical potential converges to

a diﬀerential inclusion related to the subgradient of the indicator function

of [a, b].

Mathematics Subject Classiﬁcation (2000). Primary 76T99; Secondary 76D27,

76D03, 76D05, 76D45, 35B40, 35B65, 35Q30, 35Q35,

Keywords. Two-phase ﬂow, diﬀuse interface model, mixtures of viscous ﬂuids,

Cahn-Hilliard equation, Navier-Stokes equation, double obstacle problem.

1. Introduction and main result

In the present contribution we study a system describing the ﬂow of viscous incom-

pressible Newtonian ﬂuids of the same density, but diﬀerent viscosity. Although

it is assumed that the ﬂuids are macroscopically immiscible, the model takes a

partial mixing on a small length scale measured by a parameter ε>0intoac-

count. Therefore the classical sharp interface between both ﬂuids is replaced by an

interfacial region and an order parameter related to the concentration diﬀerence

of both ﬂuids is introduced. This makes it possible to describe the ﬂow beyond the

occurrence of topological singularities of the separating interface (e.g., coalescence

or formation of droplets), cf. Anderson and McFadden [5] for a review on that

topic.

2H.Abels

This model, also known as “model H”, cf. Hohenberg and Halperin [10] and

Gurtin et al. [9], leads to a coupled Navier-Stokes/Cahn-Hilliard system:

∂

v + v ·∇v − div(ν(c)Dv)+∇p = −ε div(∇c ⊗∇c)inΩ×(0, ∞), (1.1)

div v =0 inΩ× (0, ∞), (1.2)

∂

c + v ·∇c = mΔμ in Ω ×(0, ∞), (1.3)

μ = ε

−1

φ(c) − εΔc in Ω ×(0, ∞). (1.4)

Here v is the mean velocity, Dv =

(∇v + ∇v

), p is the pressure, c is an or-

der parameter related to the concentration of the ﬂuids (e.g., the concentration

diﬀerence or the concentration of one component), and Ω is a suitable bounded

domain. Moreover, ν(c) > 0 is the viscosity of the mixture, ε>0 is a (small)

parameter, which will be related to the “thickness” of the interfacial region, and

φ =Φ



, where Φ is the homogeneous free energy density speciﬁed below.

It is assumed that the densities of both components as well as the density of

the mixture are constant and for simplicity equal to one. We note that capillary

forces due to surface tension are modeled by an extra contribution ε∇c ⊗∇c in

the stress tensor leading to the term on the right-hand side of (1.1). Moreover, we

note that in the modeling diﬀusion of the ﬂuid components is taken into account.

Therefore mΔμ is appearing in (1.3), where m>0 is the mobility coeﬃcient,

which is assumed to be constant.

We close the system by adding the boundary and initial conditions

∂Ω

=0 on∂Ω × (0, ∞), (1.5)

∂

∂Ω

= ∂

μ|

∂Ω

=0 on∂Ω × (0, ∞), (1.6)

(v, c)|

t=0

=(v

)inΩ. (1.7)

Here (1.5) is the usual no-slip boundary condition for viscous ﬂuids, n is the

exterior normal on ∂Ω, ∂

μ|

∂Ω

= 0 means that there is no ﬂux of the components

through the boundary, and ∂

∂Ω

= 0 describes a “contact angle” of π/2ofthe

diﬀused interface and the boundary of the domain.

We note that (1.1) can be replaced by

∂

v + v ·∇v − div(ν(c)Dv)+∇g = μ∇c (1.8)

with g = p +

|∇c|

+ ε

−1

Φ(c)since

μ∇c = ∇



|∇c|

+ ε

−1

Φ(c)



− ε div(∇c ⊗∇c). (1.9)

The total energy of the system above is given by E(c, v)=E

free

(c)+E

kin

(v),

where

free

(c)=



ε|∇c(x)|

dx +



−1

Φ(c(x)) dx, (1.10)

kin

(v)=



|v(x)|

dx.

Double Obstacle Limit 3

Here the free energy E

free

region where c is not close to the minima of Φ(c)andE

kin

(v) is the kinetic energy of

the ﬂuid. The system is dissipative. More precisely, for suﬃciently smooth solutions

E(c(t),v(t)) = −



ν(c(t))|Dv(t)|

dx − m



|∇μ(t)|

dx.

Since we will consider (1.1)–(1.7) only for ﬁxed ε>0, we will assume for simplicity

that ε = 1 in the following. But all statements remain true for arbitrary ε>0

(with constants depending on ε).

The assumptions on the homogeneous free energy density Φ are motivated by

the so-called regular solution model free energy suggested by Cahn and Hilliard [7]:

Φ(c)=

((1 + c)ln(1+c)+(1− c)ln(1− c)) −

(1.11)

with θ, θ

> 0. We note that Φ(c) is not convex if and only if 0 <θ<θ

.Butwe

have the decomposition

Φ(s)=θΦ

(s) −

,φ(s)=θφ

(s) − θ

where Φ

∈ C([−1, 1]) ∩C

∞

((−1, 1)) is convex and θ, θ

> 0. Finally, we note that

(s) →

s→±1

±∞.

−1

−0.8

−0.6

−0.4

−0.2

0.2

0.4

0.6

0.8

−0.5

−0.4

−0.3

−0.2

−0.1

0.1

0.2

Figure 1. Logarithmic free energy density for θ =0.8, 0.7, 0.6,

0.5, 0.4, 0.2, 0.0 (from top to bottom) and θ

4H.Abels

In Figure 1 the free energy density is plotted for some choices of θ, θ

.Note

that, if θ>0, then the minima are never ±1. (Although Figure 1 for θ =0.2

suggests that the minima are ±1.) But the minima are close to ±1ifθ is small in

comparison with θ

. Since for two macroscopically immiscible ﬂuids mixing costs

a lot of energy, we think that a small θ in comparison with θ

is physically the

most meaningful choice. Since qualitatively the free energy density for θ =0.2in

Figure 1 looks already the same as for θ = 0, it is very reasonable to choose θ =0

directly, i.e., to choose the free energy density

Φ(c)=



−

if c ∈ [−1, 1]

+∞ else.

The free energy density is called to be of double obstacle type because of the con-

straint c ∈ [−1, 1] for all c with Φ(c) < ∞. The Cahn-Hilliard equation with the

latter free energy was ﬁrst studied by Blowey and Elliott [6]. Elliott and Luck-

haus [8] have shown that as θ → 0 the solutions of the Cahn-Hilliard equation

with the logarithmic free energy density converge to solutions of the latter free

energy of double obstacle type. Physically this limit describes the dynamics of

phase separating binary mixture, where the absolute temperature θ is far from the

critical temperature θ

below which phase separation occurs.

The main result of this contribution is that weak solutions of the Model H

(1.1)–(1.7) converge as θ → 0 (for a suitable subsequence) to weak solutions of the

corresponding Navier-Stokes/Cahn-Hilliard system, where (1.4) is replaced by a

diﬀerential inclusion related to the subgradient of the double obstacle free energy,

cf. Section 4 for details. In the following we will assume a slightly more general

form of the free energy than (1.11). More precisely, we assume that

Φ(s)=θΦ

(s) −

,φ(s)=θφ

(s) − θ

s (1.12)

where θ, θ

> 0, Φ

∈ C([a, b]) ∩ C

((a, b)) is convex, φ(s)=Φ



(s), a<b,and

(s) →

s→a

−∞ φ

(s) →

s→b

∞. (1.13)

We note that this assumption implies the (θ-independent) assumption made in [3]

for the free energy density.

The structure of the article is as follows: First we ﬁx some notation and recall

some basic lemmas in Section 2. Then we study the double obstacle limit for the

convex part of the free energy E

free

and the convective Cahn-Hilliard equation

(1.3)–(1.4) in Section 3. In Section 4 we state and prove our main result on con-

vergence of weak solutions of (1.1)–(1.7) as θ → 0. We conclude with two results

on uniqueness and regularity of weak solutions for the limit system, which are the

same in [3] for the case θ>0. These results are part of the author’s Habilitation

thesis [1].

Double Obstacle Limit 5

2. Notation and preliminaries

Throughout the article Ω ⊂ R

, d =2, 3, will denote a bounded domain with

-boundary and Q

:= Ω × (0,T),Q:= Q

∞

We use the same notation as in [3] and refer to the latter article for precise

deﬁnitions and references. Let us just recall some notation. The inner product of

a Hilbert space H is denoted by (., .)

and we use the abbreviation (., .)

for

(., .)

(M)

. The duality product of a Banach space X and its dual X



is denoted

by ., .



or just ., ..

Moreover,

(Ω) =



f ∈ L

(Ω)

:divf =0,n· f|

∂Ω



(Ω) =



(Ω)

∩ H

(Ω)

∩L

(Ω) if s ≥ 1,

(Ω)

∩ L

(Ω) if 0 ≤ s<

and V

(Ω) := V

(Ω).

For m ∈ R we set

(m)

(Ω) :=



f ∈ L

(Ω) : m(f):=

|Ω|



f(x) dx = m



, 1 ≤ q ≤∞,

and P

f := f − m(f) is the orthogonal projection onto L

(0)

(Ω). Furthermore, we

deﬁne

(0)

≡ H

(0)

(Ω) = H

(Ω) ∩L

(0)

(Ω),H

−1

(0)

≡ H

−1

(0)

(Ω) = H

(0)

(Ω)



We equip H

(0)

(Ω) with the inner product (c, d)

(0)

(Ω)

:= (∇c, ∇d)

(Ω)

. Then the

Riesz isomorphism R: H

(0)

(Ω) → H

−1

(0)

(Ω) is given by

Rc, d

−1

(0)

=(c, d)

(0)

=(∇c, ∇d)

,c,d∈ H

(0)

(Ω),

i.e., R = −Δ

is the negative (weak) Laplace operator with Neumann boundary

conditions. We note that, if u ∈ H

(0)

(Ω) solves Δ

u = f for some f ∈ L

(0)

(Ω),

1 <q<∞,and∂ΩisofclassC

, then it follows from standard elliptic theory

that u ∈ W

(Ω) and Δu = f a.e. in Ω and ∂

∂Ω

= 0 in the sense of traces. If

additionally f ∈ W

(Ω) and ∂Ω ∈ C

,thenu ∈ W

(Ω). Moreover,

u

k+2

(Ω)

≤ C

f

(Ω)

for all f ∈ W

(Ω) ∩ L

(0)

(Ω),k =0, 1, (2.1)

with a constant C

depending only on 1 <q<∞, d, k, and Ω. Finally we denote

p,N

(Ω) =



u ∈ W

(Ω) : ∂

∂Ω



where 1 <p<∞.

Concerning vector-valued spaces, we recall that BC(0,T; X) is the Banach

space of all bounded and continuous f :[0,T) → X equipped with the supremum

norm and BUC(0,T; X) is the subspace of all bounded and uniformly contin-

uous functions, where X is a Banach space. Moreover, we deﬁne BC

(0,T; X)

as the topological vector space of all bounded and weakly continuous functions

6H.Abels

f :[0,T) → X.Furthermore,f ∈ L

loc

([0, ∞); X) if and only if f ∈ L

(0,T; X)

for every T>0. Moreover, L

uloc

([0, ∞); X) denotes the uniformly local variant of

(0, ∞; X) consisting of all strongly measurable f :[0, ∞) → X such that

f

uloc

([0,∞);X)

=sup

t≥0

f

(t,t+1;X)

< ∞.

In order to derive some regularity estimates we will use vector-valued Besov spaces

q∞

(I; X), where s ∈ (0, 1), 1 ≤ q ≤∞, I is an interval, and X is a Banach space.

They are deﬁned as

q∞

(I; X)=



f ∈ L

(I; X):f 

q∞

(I;X)

< ∞



f

q∞

(I;X)

= f

(I;X)

+sup

0<h≤1

Δ

f(t)

;X)

where Δ

f(t)=f (t + h) − f(t)andI

= {t ∈ I : t + h ∈ I}.Moreover,weset

(I; X)=B

∞∞

(I; X), s ∈ (0, 1). Finally, B

q∞,uloc

([0, ∞); X) is deﬁned in the

obvious way replacing L

(0, ∞; X)-norms by L

uloc

([0, ∞); X)-norms.

Let us conclude with two useful lemmas for the following.

Lemma 2.1. Let X, Y be two Banach spaces such that Y→ X and X



→ Y



densely and let 0 <T≤∞.ThenL

∞

(0,T; Y )∩BUC([0,T]; X) → BC

([0,T]; Y ).

We refer to [2, Lemma 4.1] for a proof.

Lemma 2.2. Let E :[0,T) → [0, ∞), 0 <T ≤∞, be a lower semi-continuous

function and let D :(0,T) → [0, ∞) be an integrable function. Then

E(0)ϕ(0) +



E(t)ϕ



(t) dt ≥



D(t)ϕ(t) dt (2.2)

holds for all ϕ ∈ W

(0,T) with ϕ(T )=0if and only if

E(t)+



D(τ) dτ ≤ E(s) (2.3)

holds for all s ≤ t<T and almost all 0 ≤ s<T including s =0.

See [2, Lemma 4.3] for a proof.

3. Double obstacle limit for the Cahn-Hilliard equation

3.1. Limit of the energy

In this section we study the “convex part” of E

free

as in (1.10), namely

(c)=



|∇c|

dx +



θΦ

(c(x)) dx, θ > 0, (3.1)

as θ → 0, where Φ

is the same as in (1.12), (1.13).

Firstly, E

is deﬁned on L

(m)

(Ω), m ∈ (a, b), with

dom E



c ∈ H

(Ω) ∩ L

(m)

(Ω) : c(x) ∈ [a, b]a.e.



Double Obstacle Limit 7

But we will assume in the following m = 0 without loss of generality. By a simple

shift of c and Φ by m we can always reduce to this case.

We denote by ∂E

(c): L

(0)

(Ω) →P(L

(0)

(Ω)) the subgradient of E

at c ∈

dom E

, i.e., w ∈ ∂E

(w, c



− c)

≤ E



) − E



∈ L

(0)

(Ω).

From [3, Corollary 1], see also [4, Corollary 4.4], we recall:

Lemma 3.1. Let E

be as above and extend E

toafunctional



: H

−1

(0)

(Ω) →

R ∪{+∞} by setting



(c)=E

and



(c)=+∞ else. Then



is a proper, convex, and lower semi-continuous functional, ∂



is a maximal

monotone operator with ∂



(c)=−Δ

∂E

(c),and

D(∂





c ∈D(∂E

):∂E

(c)=−Δc + θP

(0)

(Ω)



, (3.2)

where

D(∂E



c ∈ H

(Ω) ∩ L

(0)

(Ω) : φ

,φ



(c)|∇c|

∈ L

,∂

∂Ω



and ∂E

(c)=−Δc + θP

(c). Moreover, for every c ∈D(∂



)

c

+ φ

(c)

≤ C

r,θ



∂E

(c)

(0)

+ c



, (3.3)

where r =6if d =3and 2 ≤ r<∞ is arbitrary if d =2.

Remark 3.2. We note that in [3, Corollary 1] the case θ = 1 is considered. This

implies the lemma for every θ>0whereC

r,θ

in (3.3) depends on θ>0. But

one crucial observation for the following is that the estimate (3.3) is valid with a

constant C

independent of 0 <θ≤ 1.

Proposition 3.3. Let E

, 0 <θ≤ 1, be as above, let



be the extension to H

−1

(0)

(Ω),

let R>0 and let r =6if d =3and 2 ≤ r<∞ arbitrary if d =2. Then there are

constants C(R),C



(r, R) > 0 independent of 0 <θ≤ 1 such that

c

(Ω)

+ θφ

(c)

(Ω)

≤ C(R)



∂E

(c)

(Ω)



(3.4)

for all c ∈D(∂E

) with c

(Ω)

≤ R and

c

(Ω)

+ θφ

(c)

(Ω)

≤ C



(r, R)



∂



(c)

−1

(0)

(Ω)



(3.5)

for all c ∈D(∂



) with c

(Ω)

≤ R.

Proof. Let c ∈D(∂E

). First we show a suitable estimate for Δc and θφ

(c)in

(Ω) which is independent of θ ∈ (0, 1]. Taking the L

-inner product of ∂E

(c)=

−Δc + P

θφ

(c)and−Δc, we conclude that



|Δc|

dx + θ





(c)|∇c|

dx = −(∂E

(c), Δc)