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Fla low solu ion o he mean cu a u e low wi h olume cons ain
© 2024 he au ho (s), published by De G uy e
Published e sion
Julin, Vesa
Julin, V. (2024). Fla low solu ion o he mean cu a u e low wi h olume cons ain . Ad ances
in Calculus o Va ia ions, Ea ly online. h ps://doi.o g/10.1515/ac -2023-0047
2024
Ad . Calc. Va . 2024; aop
Resea ch A icle
Vesa Julin*
Fla low solu ion o he mean cu a u e low
wi h olume cons ain
h ps://doi.o g/10.1515/ac -2023-0047
Recei ed Ap il 26, 2023; accep ed Ma ch 10, 2024
Abs ac : In his pape I will e isi he cons uc ion o a global weak solu ion o he olume p ese ing mean
cu a u e low ia disc e e minimizing mo emen scheme by Mugnai, Seis and Spada o [L. Mugnai, C. Seis
and E. Spada o, Global solu ions o he olume-p ese ing mean-cu a u e low, Calc. Va . Pa ial Di e en ial
Equa ions 55 (2016), no. 1, A icle ID 18]. This me hod is based on he g adien low app oach due o Almg en,
Taylo and Wang [F. Almg en, J. E. Taylo and L. Wang, Cu a u e-d i en lows: a a ia ional app oach, SIAM
J. Con ol Op im. 31 (1993), no. 2, 387–438] and Luckhaus and S u zenhecke [S. Luckhaus and T. S u zenhecke ,
Implici ime disc e iza ion o he mean cu a u e low equa ion, Calc. Va . Pa ial Di e en ial Equa ions 3
(1995), no. 2, 253–271] and my aim is o eplace he olume penaliza ion wi h he olume cons ain di ec ly in
he disc e e scheme, which om p ac ical poin o iew is pe haps mo e na u al. A echnical no el y is he p oo
o he densi y es ima e which is based on second a ia ion a gumen .
Keywo ds: Mean-cu a u e low, olume cons ain , g adien low, ime disc e iza ion
MSC 2020: 35K93, 53C42

Communica ed by: F ank Duzaa
1In oduc ion
A smoo h amily o se (E ) ≥0is said o e ol e acco ding o olume p ese ing mean cu a u e low i he
no mal eloci y V is p opo ional o he mean cu a u e HE as
V =−(HE −
HE )on ∂E ,(1.1)
whe e 
HE =−
∫∂E HE dHn. Such a geome ic equa ion has been p oposed in he physical li e a u e o model
coa sening phenomena, whe e he sys em consis ing o se e al subdomains e ol es such ha i dec eases he
in e acial a ea while keeping he o al olume unchanged [7, 18]. F om pu ely ma hema ical poin o iew he
equa ion (1.1) can be seen as he L2-g adien low o he su ace a ea unde he olume cons ain [18]. One has
o be ca e ul in his in e p e a ion as he Riemannian dis ance be ween wo se s is in gene al degene a e [16]. In
o de o o e come his one may use he idea due o Almg en, Taylo and Wang [2] and Luckhaus and S u zen-
hecke [14] and o iew (1.1) as he g adien low o he su ace a ea wi h espec o a di e en , non-degene a e,
dis ance. Using he g adien low s uc u e, one may hen cons uc a disc e e-in- ime app oxima ion o he
solu ion o (1.1) ia he Eule implici me hod, also known as he minimizing mo emen s scheme. By le ing he
ime s ep o ze o, one hen ob ains a candida e o a weak solu ion o (1.1) called la low, as he con e gence is
measu ed in e ms o he “ la no m”. This me hod is implemen ed o he olume p ese ing se ing in [19].
In [19] he au ho s obse e ha om echnical poin o iew i is easie o eplace he olume cons ain
o he p oblem wi h olume penaliza ion, as his simpli ies ce ain egula i y issues a he le el o he disc e e
app oxima ion. My aim he e is o show ha one may cons uc he la low solu ion o (1.1) by implemen ing
he olume cons ain in he minimizing mo emen s scheme di ec ly and hus a oid he olume penaliza ion.
*Co esponding au ho : Vesa Julin, Depa men o Ma hema ics and S a is ics, Uni e si y o Jy äskylä, P. O. Box 35, 40014 Jy äskylä,
Finland, e-mail: [email p o ec ed]. h ps://o cid.o g/0000-0002-1310-4904
Open Access. ©2024 he au ho (s), published by De G uy e . This wo k is licensed unde he C ea i e Commons A ibu ion 4.0 In e -
na ional License.
2V. Julin, Fla low solu ion o he mean cu a u e low wi h olume cons ain
Le me quickly ecall he disc e e minimizing mo emen s scheme o (1.1). One de ines a sequence o
se s (Eh
k)k, wi h ixed ime s ep h>0, i e a i ely such ha Eh
0=E0, whe e E0is he gi en ini ial se , and Eh
k+1is
a minimize o he unc ional
P(E)+1
h∫
E

dEh
kdxunde he cons ain |E|=|Eh
k|.
He e P(E)deno es he pe ime e (gene alized su ace a ea) o he se Eand 
dFis he signed dis ance unc ion
o he se F(see nex sec ion). One hen de ines an app oxima i e la low solu ion o (1.1) (Eh
) ≥0 om he
p e ious sequence by Eh
=Eh
k o ∈[kh,(k+1)h). Any clus e poin o (Eh
) ≥0is hen de ined as la low
solu ion o (1.1). The ad an age is ha such a solu ion is de ined o all imes and o ough ini ial da a. The
main esul in he pape is he exis ence o a la low solu ion.
Theo em 1. Assume ha E0⊂ℝn+1is an open and bounded se wi h ini e pe ime e and le (Eh
) ≥0be an app ox-
ima i e la low solu ion o (1.1) s a ing om E0(see De ini ion 2.1). Then he e exis s a amily o bounded se s o
ini e pe ime e (E ) ≥0and a subsequence hk→0such ha
lim
hk→0|Ehk
ΔE |=0 o a.e. ≥0
and o e e y 0< <si holds |E |=|E0|,P(E )≤P(E0)and
|E ΔEs|≤C√s− ,
whe e Cdepends on he dimension and on E0. Mo eo e , i he ini ial se E0is C1,1- egula , hen any such limi low
(E ) ≥0ag ees wi h he unique classical solu ion o (1.1) as long as he la e exis s.
The abo e heo em hus p o ides he exis ence o a la low solu ion and gua an ees ha his no ion is consis-
en wi h he classical solu ion when he ini ial se is egula enough. The disad an age o he la low is ha i
is no clea i i p o ides a solu ion o he o iginal equa ion (1.1) in any weak sense a e he i s singula ime.
Howe e , he condi ional esul in he spi i o Luckhaus and S u zenhecke [14] holds also in his case.
Theo em 2. Le (Eh
) ≥0be an app oxima i e la low solu ion o (1.1) and le (Ehk
) ≥0be he con e ging subse-
quence in Theo em 1. Assume u he ha i holds
lim
hk→0P(Ehk
)=P(E ) o a.e. ≥0.
Then o n≤6 he la low (E ) ≥0is a dis ibu ional solu ion o (1.1) (see De ini ion 4.5).
One may also y o iew equa ion (1.1) as a mean cu a u e low wi h o cing, whe e he o cing e m depends
on he low i sel . In his way one may y o use di e en me hods o cons uc a solu ion o he equa ion see,
e.g., [5, 6]. I also e e he ecen wo k [13] o a weak-s ong uniqueness esul ela ed o (1.1).
As I al eady men ioned, he la low is de ined o all imes and one may s udy i s asymp o ical beha io .
Indeed, by using he me hods om [10, 11] one may deduce he con e gence o he low in low dimensions. I will
s a e his me ely as a ema k as i ollows om he abo e me hods wi hou any modi ica ions.
Rema k 1.1. Assume ha E0⊂ℝn+1, wi h n≤2, is as in Theo em 1 and le (E ) ≥0be a limi la low. When
n=1, he low E con e ges o a union o disjoin balls exponen ially as and when n=2 he low con e ges o
a union o disjoin balls up o a possible ansla ion o he componen s.
The main echnical challenge in p o ing Theo em 1 is o ob ain he sha p densi y es ima e o he disc e e
low. This is also he main echnical no el y o his pape . The e a e se e al echniques o deal wi h he olume
cons ain in a ia ional p oblems, e.g., by using he a gumen om [3] (see also [15, Lemma 17.21]) o om [8]
(see also [4, 9]). Howe e , due o he p esence o he dissipa ion e m in he ene gy i is no ob ious how o apply
hese a gumen s in o de o ob ain sha p densi y es ima es in e ms o he ime s ep h. I will use an a gumen
which is based on he second a ia ion condi ion o he ene gy o p o e he densi y es ima e in P oposi ion 3.1.
A e his he p oo o Theo em 1 ollows exac ly as in [14, 19] and he consis ency ollows almos di ec ly using
V. Julin, Fla low solu ion o he mean cu a u e low wi h olume cons ain 3
he a gumen in [12]. The p oo also p o ides he dissipa ion inequali y and he e o e he esul s in [10, 11]
hold and one ob ains he esul s a ed in Rema k 1.1. Finally, I would like o poin ou ha his a icle is no
sel -consis en as I will ake se e al well-known a gumen s o g an ed, in pa icula , in Sec ion 4.
2P elimina ies
In his sec ion I will b ie ly in oduce he no a ion, he de ini ion o he la low solu ion and ecall some o i s
basic p ope ies.
Gi en a se E⊂ℝn+1, he dis ance unc ion dis (⋅,E):ℝn+1→[0,∞)is de ined, as usual, as
dis (x,E):=in
y∈E|x−y|
and deno e he signed dis ance unc ion by 
dE:ℝn+1→ℝ,

dE(x):={
{
{−dis (x,∂E) o x∈E,
dis (x,∂E) o x∈ℝn+1 E.
Then clea ly i holds dis (⋅,∂E)=|
dE|. I deno e he ball wi h adius cen e ed a xby B (x)and by B i i is
cen e ed a he o igin.
Fo a measu able se E⊂ℝn+1 he pe ime e in an open se U⊂ℝn+1is de ined as
P(E,U):=sup{∫
E
di Xdx:X∈C1
0(U,ℝn+1),‖X‖L∞≤1}
and w i e P(E)=P(E,ℝn+1). I P(E)<∞, hen Eis called a se o ini e pe ime e . Fo an in oduc ion o he
opic I e e o [15]. The educed bounda y o a se o ini e pe ime e Eis deno ed by ∂∗Eand he gene alized
uni ou e no mal by νE. No e ha i holds P(E,U)=Hn(∂∗E∩U) o open se s U. Recall also ha i Eis egula
enough, say wi h Lipschi z bounda y, hen P(E)=Hn(∂E). Fo a gi en ec o ield X∈C1(ℝn+1,ℝn+1)and a se
o ini e pe ime e Edeno e he angen ial di e gence on ∂E∗as di τX=di X−⟨DXνE,νE⟩. The dis ibu ional
mean cu a u e HE∈L1(∂∗E,ℝ)is de ined ia he di e gence heo em such ha o e e y es ec o ield
X∈C1
0(ℝn+1,ℝn+1)i holds ∫
∂∗E
di τXdHn=∫
∂∗E
HE⟨X,νE⟩dHn.
I will conside a la low solu ion o (1.1) in he spi i o Almg en, Taylo and Wang [2] and Luckhaus and
S u zenhecke [14]. To his end, o a ixed h∈(0,1)and a gi en (open) se F⊂ℝn+1, I de ine he unc ional
Fh(E,F)=P(E)+1
h∫
E

dFdx.(2.1)
The la low solu ion is de ined analogously as in [19].
De ini ion 2.1. Le E0⊂ℝn+1be an open and bounded se o ini e pe ime e and ix h∈(0,1). De ine he
sequence o se s (Eh
k)∞
k=0i e a i ely as Eh
0=E0and Eh
k+1is a minimize o he p oblem
min{Fh(E,Eh
k):|E|=|E0|}.
Mo eo e , de ine an app oxima i e la low (Eh
) ≥0 o (1.1) s a ing om E0as
Eh
=Eh
k o ∈[kh,(k+1)h).
One has o be ca e ul in he de ini ion o he unc ional (2.1) i he se Fis me ely a se o ini e pe ime e
as i s alue depends on he choice o he ep esen a i e o F. One may o e come his by choosing a p ope
ep esen a i e o he se F. Howe e , his is no necessa y as he egula i y heo em below implies ha one
4V. Julin, Fla low solu ion o he mean cu a u e low wi h olume cons ain
may in ac assume he se s Eh
k o be open. The di e ence in De ini ion 2.1 o he scheme in [19] is ha he e
he minimizing p oblem is unde olume cons ain . On one hand his makes he minimiza ion p oblem mo e
na u al, bu on he o he hand, i makes he quan i a i e densi y es ima es mo e di icul o p o e.
Fo a gi en open and bounded se F⊂ℝn+1conside he minimiza ion p oblem
min{Fh(E,F):|E|=|F|},(2.2)
whe e Fh(⋅,F)is de ined in (2.1). One may use an a gumen simila o [9] o [15, Lemma 17.21] o emo e he
olume cons ain in (2.2) and deduce ha a minimize o (2.2) is a minimize also o
min{Fh(E,F)+
Λ||E|−|F||},(2.3)
when 
Λis chosen la ge. No e ha he cons an 
Λmay ha e nonop imal dependence on Eand on h. Howe e , he
p ope y (2.3) is enough o deduce quali a i e egula i y p ope ies since i implies ha he minimize inhe i s
he egula i y om he heo y o he pe ime e minimize s [15]. One may also w i e he Eule –Lag ange equa-
ion and by s anda d calcula ions (see, e.g., [1]) we ha e he second a ia ion condi ion. We s a e his in he
ollowing p oposi ion.
P oposi ion 2.2. Le F⊂ℝn+1be an open and bounded se , ix h∈(0,1)and le Ebe a minimize o (2.2). Then E
can be chosen o be open, which opological bounda y is C2,α- egula up o a ela i ely closed singula se which
Hausdo dimension is a mos n−7. The egula pa is exac ly he educed bounda y ∂∗E.
The Eule –Lag ange equa ion
dF
h=−HE+λ,(2.4)
whe e λ∈ℝis he Lag ange-mul iplie , holds poin -wise on ∂∗Eand in a dis ibu ional sense on ∂E. The
quad a ic o m associa ed wi h he second a ia ion o he ene gy is non-nega i e, i.e., o all φ∈H1(∂∗E)
wi h ∫∂∗EφdHn=0i holds
∫
∂∗E|∇τφ|2−|BE|2φ2dHn+1
h∫
∂∗E⟨∇ 
dF,νE⟩φ2dHn≥0,(2.5)
whe e BE(x)deno es he second undamen al o m o Ea x∈∂∗E.
P oo . Since he a gumen is s anda d, I will only gi e he ou line. As I al eady men ioned, he minimize Eis
also a minimize o p oblem (2.3) o some la ge cons an 
Λ, which depends on hand on Ei sel . This implies ha
he se Eis a Λ-minimize o he pe ime e and hus he educed bounda y ∂∗Eis ela i ely open, C1,α- egula
hype su ace and he singula se ∂E ∂∗Ehas dimension a mos n−7(see [15]). The C2,α- egula i y hen
ollows om he Eule –Lag ange equa ion and om s anda d Schaude -es ima es o ellip ic PDEs.
One may ob ain he second a ia ion condi ion (2.5) by using he a gumen om [1]. Indeed, gi en a unc-
ion φ∈C2
0(∂∗E)wi h ∫∂∗EφdHn=0, we may cons uc a amily o di eomo phisms Φ such ha Φ0=id,
|Φ (E)|=|E|and ∂
∂ 󵄨󵄨󵄨󵄨 =0Φ (x)⋅νE=φ. Then he inequali y ollows om he minimali y o Eas
∂2
∂ 2󵄨󵄨󵄨󵄨󵄨󵄨󵄨 =0
Fh(Φ (E),F)≥0
and ollowing he s anda d calcula ion o he second a ia ion (see, e.g., [1]). Finally, one ob ains (2.5) o all
φ∈H1(∂∗E)by app oxima ion a gumen and by he ac ha he singula se has ze o capaci y.
3Densi y es ima es
This sec ion is he heo e ical co e o he pape . The aim is o p o e he ollowing densi y es ima e.
P oposi ion 3.1. Le F⊂ℝn+1be an open and bounded se o ini e pe ime e , ix h∈(0,1)and le Ebe a minimize
o (2.2). Then he e is a cons an c>0, which depends on he dimension n,|F|and on P(F)such ha o all ≤√h
and all x∈∂E i holds
min{|E∩B (x)|,|B (x) E|}≥c n+1

V. Julin, Fla low solu ion o he mean cu a u e low wi h olume cons ain 5
and o all ≤C0√h, whe e C0≥1, i holds
c n≤P(E,B (x))≤C1 n,
whe e C1depends also on C0. Mo eo e , he ollowing es ima es hold
‖HE‖L∞(∂∗E)≤1
c√hand ‖
dF‖L∞(∂E)≤c−1√h.
I is in e es ing ha in [19, Co olla y 3.3] he au ho s ob ain simila esul o hei scheme o a cons an which
is independen o P(F).
I need se e al lemmas in o de o p o e P oposi ion 3.1 and he e o e I pos pone i s p oo o he end o he
sec ion. Be o e p oceeding o echnical de ails, I s a e a use ul consequence o P oposi ion 3.1.
P oposi ion 3.2. Le F,E⊂ℝn+1be as in P oposi ion 3.1. Then he e a e cons an s C≥1,c>0and h0>0,
depending on he dimension, |F|and P(F)such ha Eis he (Λ, )-minimize o he pe ime e o Λ=C
√h, =c√h
and o h<h0. To be mo e p ecise, o se s G⊂ℝn+1wi h EΔG⊂Bc√h(x0)i holds
P(E)≤P(G)+C
√h|EΔG|.
P oo . The a gumen is s anda d bu I ecall i o he eade ’s con enience. Le me i s show ha he e is
x∈Eand 
c>0such ha o ρ=
c√hi holds Bρ(x)⊂E. Fix ρand apply he Besico i ch co e ing heo em o
ind disjoin balls {Bρ(xi)}N
i=1such ha xi∈Eand
N
∑
i=1|Bρ(xi)|=N|B1|ρn+1≥c|E|.(3.1)
I claim ha o some i=1,2,...,Ni holds Bρ
2(xi)⊂E. Indeed, i his is no he case, hen P oposi ion 3.1 implies
P(E,Bρ(xi))≥cρn
o all i. Since he balls a e disjoin , one has by he abo e and by (3.1) ha
P(E)≥N
∑
i=1
P(E,Bρ(xi))≥cN ρn≥c|E|
ρ≥c
√h.
This is a con adic ion when his small enough.
Fix x0and Gas in he claim. No e ha in gene al he se Gdoes no ha e he same measu e as Eand one
needs o modi y i o 
Gwi h |
G|=|E|, e.g., by using he a gumen om [9] as ollows. Assume ha |G|<|E|
( he case |G|>|E| ollows om simila a gumen ). Since Bρ(x)⊂E, by dec easing ρand i needed, i holds
Bρ(x)⊂G. By con inui y he e is z∈ℝn+1such ha |z−x0|≥2ρand |G∪Bρ(z)|=|E|. De ine 
G=G∪Bρ(z).
Then by he minimali y o Eand P oposi ion 3.1 i holds
P(E)≤P(
G)+C
√h|
GΔE|.
A guing as in [9], one hen deduces
P(
G)−P(G)≤Hn(∂Bρ(z) G)−Hn(∂G ∩Bρ(z))
≤C
ρ|Bρ(z) G|≤C
√h|
GΔE|
and he claim ollows as |
GΔE|≤2|GΔE|.
The i s echnical esul which I need is he classical densi y es ima e which can be ound, e.g., in [20].
Lemma 3.3. Assume E⊂ℝn+1is a se o ini e pe ime e wi h dis ibu ional mean cu a u e HEwhich sa is ies
‖HE‖L∞(B2R(x0)) ≤Λ. Then o all x∈BR(x0)which a e on he bounda y o Eand ≤min{R, Λ−1}i holds
P(E,B (x))≥cn n
o a dimensional cons an cn>0.
6V. Julin, Fla low solu ion o he mean cu a u e low wi h olume cons ain
Fo a minimize o (2.2) i holds he in e se o he isope ime ic inequali y.
Lemma 3.4. Le F⊂ℝn+1be an open and bounded se o ini e pe ime e , ix h∈(0,1)and le Ebe a minimize
o (2.2). Then o all x∈∂E and ≤C0√hi holds
P(E,B (x))≤C
min{|E∩B2 (x)|,|B2 (x) E|}
o a cons an which depends on he dimension and on C0>0. In, pa icula i holds P(E,B (x))≤C n.
P oo . Fix h∈(0,1),xand >0as in he claim and wi hou loss o gene ali y assume ha x=0. One may also
conside only he case |E∩B2 |≤|B2 E|as he o he case is simila . In pa icula , i holds |E∩B2 |≤1
2|B2 |.
Since 2
∫
0
Hn(∂Bρ∩E)=|E∩B2 |,
he e is ρ∈( ,2 )such ha
Hn(∂Bρ∩E)≤Cn|E∩B2 |
and |Bρ|≥2
3|B2 |.(3.2)
Conside i s he se E1=E 
Bρ. In o de o ha e a compe ing se wi h he olume o E, de ine 
ρ≤ρ o be
a adius such ha |B
ρ|=|E∩Bρ|and de ine E2=E1∪B
ρ. Then i holds by cons uc ion ha |E2|=|E|,
ρ<ρ
and
Hn(∂B 
ρ)=cn|B
ρ|n
n+1=cn|E∩Bρ|n
n+1≤cn|E∩B2 |n
n+1≤Cn|E∩B2 |
.(3.3)
By he minimali y o Ewe ha e
P(E)+1
h∫
E

dFdx≤P(E2)+1
h∫
E2

dFdx.
Es ima e he pe ime e o E2using (3.2) and (3.3) as
P(E2)≤P(E,ℝn+1 Bρ)+Hn(∂Bρ∩E)+Hn(∂B 
ρ)
≤P(E,ℝn+1 Bρ)+Cn|E∩B2 |
.
Use hen EΔE2⊂B2 ,|E|=|E2|and he ac ha he signed dis ance unc ion is 1-Lipschi z o es ima e
󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨∫
E

dFdx−∫
E2

dF󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨≤4 |E∩Bρ|≤4 |E∩B2 |.
The e o e one ob ains by combining he h ee abo e inequali ies and ≤C0√h
P(E,B )≤P(E,Bρ)≤Cn|E∩B2 |
+4
h|E∩B2 |≤C|E∩B2 |
.
By Lemma 3.3 and Lemma 3.4 i is clea ha o P oposi ion 3.1 i is c ucial o p o e he cu a u e es ima e
‖HE‖L∞≤C
√h. The nex lemma is a s ep owa ds his.
Lemma 3.5. Le F,Eand hbe as in P oposi ion 3.1. Then i holds
‖HE‖L2(∂∗E)≤C1
√h,
whe e he cons an C1depends on he dimension and on |F|and P(F).
P oo . The p oo elies on he second a ia ion inequali y in P oposi ion 2.2. I would like o poin ou ha in
he case o he mean cu a u e low, when he e is no olume cons ain , he p oo is conside able easie as
one could choose cons an unc ion in (2.5). In he olume p ese ing case I will choose a cu -o unc ion o
a es unc ion.
V. Julin, Fla low solu ion o he mean cu a u e low wi h olume cons ain 7
To his end, use i s [11, P oposi ion 2.3] (see also [17, Lemma 2.1]) o ind a poin x0∈ℝn+1and a adius
∈(c,1), whe e c=c(n,|F|,P(F)), such ha
|E∩B (x0)|=1
2|B |.
No e ha he minimali y o Eyields P(E)≤P(F)≤C. Mo eo e , by he isope ime ic inequali y i holds
P(E)≥cn|E|n
n+1=cn|F|n
n+1≥c. These es ima es a e used epea edly om now on wi hou men ioning. Wi hou
loss o gene ali y assume ha x0=0. Choose ρ< such ha |B Bρ|=1
4|B |. No e ha hen −ρ≥cn>0and
3
4|Bρ|≥|E∩Bρ|≥1
4|Bρ|.(3.4)
De ine i s a cu -o unc ion ζ∈C1
0(ℝn+1)such ha 0≤ζ≤1,ζ=1in Bρ,ζ=0ou side B and |∇ζ|≤Cn. Choose
hen φ=ζ−
ζ, whe e 
ζ=−
∫∂∗EζdHn, as a es unc ion in (2.5), use |⟨∇ 
dF,νE⟩|≤1and |φ|≤1, and ob ain
∫
∂∗E|BE|2(ζ−
ζ)2dHn≤∫
∂∗E|∇τζ|2dHn+P(E)
h.(3.5)
Since HE=T ace(BE), i holds poin -wise on ∂∗E
|BE|2≥H2
E
n.(3.6)
Recall ha 0≤ζ≤1. Mo eo e , by he isope ime ic inequali y and by (3.4) i holds P(E,Bρ)≥cn|E∩Bρ|n
n+1≥c.
The e o e 
ζ≥c o c=c(n,|F|,P(F)). In pa icula , i holds |ζ(x)−
ζ|≥c o x∈∂∗E B . Hence, we ha e
by (3.5), ∫
∂∗E B
|HE|2dHn≤C
hP(F).
We epea he same a gumen by de ining a cu -o unc ion ζ∈C1(ℝn+1)as ζ=0in B ,ζ=1ou side BR
and |∇ζ|≤Cn, whe e R> is such ha |BR B |=1
4|B |. Using φ=ζ−
ζin (2.5) and a guing as abo e yields
∫
∂∗E∩B
|HE|2dHn≤C
hP(F)
and he claim ollows.
The las lemma I need is a bound on he Lag ange mul iplie in he Eule –Lag ange equa ion (2.4).
Lemma 3.6. Le F,Eand hbe as in P oposi ion 3.1. Then o he Lag ange mul iplie in (2.4), i.e.,

dF
h=−HE+λon ∂∗E
i holds |λ|≤C2
√h,
whe e he cons an C2depends on he dimension, on |F|and on P(F).
P oo . Le Λ≥0be such ha |λ|=Λ
√h. Below all he cons an s depend on n,|F|and P(F). I only ea he case
when λis posi i e as in he nega i e case he p oo is simila . De ine he se
Σ={x∈∂∗E:|HE(x)|<
C
√h}.
I claim ha we may choose 
C>2such ha i depends on n,|F|,P(F)and on C1 om Lemma 3.5 and i holds
Hn(Σ)≥P(E)
2.(3.7)
Indeed, by Lemma 3.5 and by 
C>2i holds

C2
hHn(∂∗E Σ)≤∫
∂∗E Σ
H2
EdHn≤∫
∂∗E
H2
EdHn≤C2
1
h.
By choosing 
Cla ge enough one hen ob ains Hn(∂∗E Σ)<P(E)
2and (3.7) ollows.
8V. Julin, Fla low solu ion o he mean cu a u e low wi h olume cons ain
By he Besico i ch co e ing heo em one inds disjoin balls o adius √h, deno e hem by {B√h(xi)}N
i=1, wi h
xi∈Σsuch ha N
∑
i=1
P(E,B√h(xi))≥cnHn(Σ)≥cP(E).(3.8)
By he Eule –Lag ange equa ion (2.4) and by he de ini ion o he se Σi holds o all x∈∂∗E∩B√h(xi), wi h
xi∈Σ, ha
|HE(x)|≤󵄨󵄨󵄨󵄨󵄨󵄨󵄨
dF(x)
h−λ󵄨󵄨󵄨󵄨󵄨󵄨󵄨≤|
dF(x)−
dF(xi)|
h+󵄨󵄨󵄨󵄨󵄨󵄨󵄨
dF(xi)
h−λ󵄨󵄨󵄨󵄨󵄨󵄨󵄨≤1
√h+|HE(xi)|≤2
C
√h.(3.9)
The e o e by Lemma 3.3 i holds
P(E,B√h
2(xi))≥P(E,B√h
2
C(xi))≥ch n
2.
On he o he hand, applying Lemma 3.4 i s wi h =√h
2yields
|E∩B√h(xi)|≥c√hP(E,B√h
2(xi))
and hen wi h =√hyields hn
2≥cP(E,B√h(xi)). In conclusion, i holds
|E∩B√h(xi)|≥c√hP(E,B√h(xi)) (3.10)
o all balls in he co e .
The minimali y o Eimplies
1
h∫
E F

dFdx≤P(E)+1
h∫
EΔF|
dF|dx≤P(F).(3.11)
No e ha by (3.9), by λ=Λ
√hand by he Eule –Lag ange equa ion (2.4) we ha e o all x∈B√h(xi) ha

dF(x)≥λh −|H(x)|h≥(Λ−2
C)√h.
The e o e ei he Λ≤4
C, in which case he claim ollows i ially, o

dF(x)≥Λ
2√h.
I assume he la e and show ha also in his case Λis bounded. Indeed, by he abo e discussion he balls B√h(xi)
a e in he ex e io o F. The e o e we es ima e by (3.10) and by (3.8) ha
1
h∫
E F

dFdx≥1
h
N
∑
i=1∫
E∩B√h(xi)
dFdx
≥1
h
N
∑
i=1(Λ
2√h|E∩B√h(xi)|)
≥cΛ
N
∑
i=1
P(E,B√h(xi))
≥cΛHn(Σ)≥cΛP(E).
Since P(E)≥cn|E|n
n+1=cn|F|n
n+1, he abo e and (3.11) gi es a bound o Λand he claim ollows.
He e is he p oo o he densi y es ima e.
P oo o P oposi ion 3.1. By Lemma 3.3, Lemma 3.4, Lemma 3.6 and by he Eule –Lag ange equa ion (2.4) i is
enough o p o e ‖
dF‖L∞(∂∗E)≤C√h.(3.12)
A gue by con adic ion and assume ha he e is x0∈∂∗Esuch ha
|
dF(x0)|=Λ√h
o la ge Λ≫1. Wi hou loss o gene ali y assume ha x0=0and conside only he case 
dF(x0)>0as he case

dF(x0)<0is simila .