Flat flow solution to the mean curvature flow with volume constraint

This is a sel -a chi ed e sion o an o iginal a icle. This e sion may di e om he o iginal in pagina ion and ypog aphic de ails. Au ho (s): Ti le: Yea : Ve sion: Copy igh : Righ s: Righ s u l: Please ci e he o iginal e sion: CC BY 4.0 h ps://c ea i ecommons.o g/licenses/by/4.0/ 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 .