A two-phase problem with Robin conditions on the free boundary
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A wo-phase p oblem wi h Robin condi ions on he ee bounda y
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Gua ino Lo Bianco, Se ena; La Manna, Domenico Angelo; Velichko , Bozhida
Gua ino Lo Bianco, S., La Manna, D. A., & Velichko , B. (2021). A wo-phase p oblem wi h Robin
condi ions on he ee bounda y. Jou nal de l'École poly echnique : Ma héma iques, 8, 1-25.
h ps://doi.o g/10.5802/jep.139
2021
Se ena Gua ino Lo Bianco, Domenico Angelo La Manna,
&Bozhida Velichko
A wo-phase p oblem wi h Robin condi ions on he ee bounda y
Tome 8 (2021), p. 1-25.
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Tome 8, 2021, p.1–25 DOI: 10.5802/jep.139
A TWO-PHASE PROBLEM WITH ROBIN CONDITIONS
ON THE FREE BOUNDARY
by Se ena Gua ino Lo Bianco, Domenico Angelo La Manna
& Bozhida Velichko
Abs ac . — We s udy o he i s ime a wo-phase ee bounda y p oblem in which he
solu ion sa is ies a Robin bounda y condi ion. We conside he case in which he solu ion is
con inuous ac oss he ee bounda y and we p o e an exis ence and a egula i y esul o
minimize s o he associa ed a ia ional p oblem. Finally, in he appendix, we gi e an example
o a class o S eine symme ic minimize s.
Résumé (Un p oblème à on iè e lib e à deux phases a ec condi ions au bo d de Robin)
Nous é udions pou la p emiè e ois un p oblème à on iè e lib e à deux phases pou lequel
la solu ion sa is ai à une condi ion de Robin au bo d. Nous considé ons le cas où la solu ion es
con inue au bo d e nous mon ons un ésul a d’exis ence e de égula i é pou les minimiseu s
du p oblème a ia ionnel associé. En in, nous donnons dans l’appendice un exemple d’une classe
de minimiseu s a ec une symé ie de S eine .
Con en s
1. In oduc ion.................................................................. 2
2. P elimina ies.................................................................. 7
3. A amily o app oxima ing p oblems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4. Exis ence o an op imal se . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 13
5. Regula i y o he ee bounda y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Appendix. Examples o minimize s.............................................. 24
Re e ences....................................................................... 25
2020 Ma hema ics Subjec Classi ica ion. — 35R35, 49Q10.
Keywo ds. — F ee bounda y p oblems, wo-phase, Robin bounda y condi ions, egula i y.
The i s au ho was pa ially suppo ed by PRIN 2017 Nonlinea Di e en ial P oblems ia Va i-
a ional, Topological and Se - alued Me hods (G an 2017AYM8XW) and he INdAM-GNAMPA
p ojec 2020 “P oblemi di o imizzazione con incoli ia aspo o o imo e ince ezza”. The second
au ho was pa ially suppo ed by he Academy o Finland g an 314227. The hi d au ho has been
pa ially suppo ed by he Eu opean Resea ch Council (ERC) unde he Eu opean Union’s Ho izon
2020 esea ch and inno a ion p og amme (g an ag eemen VAREG, No. 853404).
e-ISSN: 2270-518X h p://jep.cen e-me senne.o g/
2 S. Gua ino Lo Bianco, D. A. La Manna & B. Velichko
1. In oduc ion
Fo a ixed a cons an β > 0and a smoo h bounded open se D⊂Rd,d⩾2,
we conside he unc ional
Jβ(u, Ω) = ZD
|∇u|2dx +βZ∂∗Ω
u2dHd−1,
de ined on he pai s (u, Ω), whe e u∈H1(D),Ω⊂Rdis a se o ini e pe ime e in
he sense o De Gio gi (see Sec ion 2) and ∂∗Ωdeno es he educed bounda y o Ω
(see Sec ion 2); when Ωis smoo h, ∂∗Ωis he opological bounda y o Ω.
In his pape we s udy he exis ence and he egula i y o minimize s o he unc-
ional Jβamong all pai s (u, Ω), which a e ixed ou side he domain D. P ecisely,
h oughou he pape , we ix a se E⊂Rdo ini e pe ime e , a cons an s m > 0and
a unc ion
∈H1
loc(Rd)such ha ⩾min Rdand Z∂∗E
2dHd−1<+∞;
we de ine he admissible se s
V=u∈H1
loc(Rd) : u− ∈H1
0(D),
E=Ω⊂Rd: Pe (Ω) <+∞and Ω = Ein Rd D,
and we conside he a ia ional minimiza ion p oblem
(1.1) min Jβ(u, Ω) : u∈ V,Ω∈ E.
Ou main esul is he ollowing.
Theo em 1.1 (Exis ence and egula i y o minimize s). — Le β >0,D⊂Rd, ,E,V
and Ebe as abo e. Then he ollowing holds.
(i) The e exis s a solu ion (u, Ω) ∈ V × E o he a ia ional p oblem (1.1).
(ii) Fo e e y solu ion (u, Ω) o (1.1),uis Hölde con inuous and bounded om
below by a s ic ly posi i e cons an in D.
(iii) I (u, Ω) is a solu ion o (1.1), hen he ee bounda y ∂Ω∩Dcan be decomposed
as he disjoin union o a egula pa Reg(∂Ω) and a singula pa Sing(∂Ω), whe e:
–Reg(∂Ω) is a C∞hype su ace and a ela i ely open subse o ∂Ω, and he
unc ion uis C∞smoo h on Reg(∂Ω);
–Sing(∂Ω) is a closed se , which is emp y i d⩽7, disc e e i d= 8, and o
Hausdo dimension d−8, i d > 8.
Rema k 1.2. — We no ice ha i (u, Ω) is a solu ion o (1.1), hen uis ha monic in he
in e io o Ωand D Ω. Thus, as a consequence o Theo em 1.1(iii), in a neighbo hood
o a egula poin x0∈Reg(∂Ω), he unc ions u: Ω →Rand u:D Ω→Ra e C∞
up o he ee bounda y ∂Ω.
J.É.P.—M., 2021, ome8
A wo-phase p oblem wi h Robin condi ions on he ee bounda y 3
1.1. Ou line o he p oo and o ganiza ion o he pape . — The main di icul y
in he p oo o Theo em 1.1 is o p o e he exis ence o a minimizing pai s (u, Ω)
and o show ha he unc ion uis Hölde con inuous and bounded om below by a
s ic ly posi i e cons an in D. The almos -minimali y o he solu ions is p o ed in
Theo em 5.1. Finally, in he Appendix, we gi e examples o minimize s in domains D
symme ic wi h espec o he hype plane {xd= 0}.
1.1.1. Exis ence. — The exis ence o a solu ion (u, Ω) and he egula i y o u(Hölde
egula i y and non-degene acy) a e ea ed simul aneously. The eason is ha i
(un,Ωn)is a minimizing sequence o (1.1), hen in o de o ge he compac ness
o Ωn, we need a uni o m bound ( om abo e) on he pe ime e Pe (Ωn), o which
we need he unc ions un o be bounded om below by a s ic ly posi i e cons an .
Now, no ice ha we canno simply eplace unby un∨ε, o some ε > 0; his is due
o he ac ha he second e m in Jβis inc easing in u:
Z∂∗Ωn
u2
ndHd−1⩽Z∂∗Ωn
(ε∨un)2dHd−1.
Thus, we selec a minimizing sequence which is in some sense op imal. P ecisely, we
ake (un,Ωn) o be solu ion o he auxilia y p oblem
(1.2) min Jβ(u, Ω) : u∈ V,Ω∈ E, u ⩾1/n in D,
o which he exis ence o an op imal se is much easie (see Sec ion 3, P oposi ion 3.1).
S ill, we do no ha e a uni o m (independen om n) bound om below o he
unc ions un, so we s ill miss he uni o m bound on he pe ime e o Ωn.
On he o he hand, we a e able o p o e ha he sequence unis uni o mly Hölde
con inuous in D(see Sec ion 3, Lemma 3.5). This enables us o ex ac a subse-
quence un ha con e ges locally uni o mly in D o a non-nega i e Hölde con inuous
unc ion u∞:D→R(see Sec ion 4). Now, on each o he se s {u∞> }, > 0,
he sequence Ωnhas uni o mly bounded pe ime e . This enables us o ex ac a sub-
sequence Ωn ha con e ges poin wise almos -e e ywhe e on {u∞>0} o some Ω∞.
Thus, we ha e cons uc ed ou candida e o a solu ion: (u∞,Ω∞).
In o de o p o e ha (u∞,Ω∞)is an admissible compe i o in (1.1), we need o
show ha Ω∞has ini e pe ime e . We do his in Sec ion 4. We i s use he op imali y
o (un,Ωn) o p o e ha (u∞,Ω∞)is op imal when compa ed o a special class o
compe i o s. This op imali y condi ion can be w i en as (we e e o Lemma 4.1 o
he p ecise s a emen ):
(1.3) Jβ(u∞,Ω∞)⩽Jβ(u ,Ω ),whe e u =u∞∨ and Ω = Ω∞∪ {u∞⩽ },
o any > 0. Nex , om his special op imali y condi ion we deduce ha he unc-
ion u∞is bounded om below by a s ic ly posi i e cons an (see P oposi ion 4.2).
F om his, in Sec ion 4, we deduce ha Ω∞has ini e pe ime e in Rdand ha he
pai s (u∞,Ω∞)is a solu ion o (1.1).
J.É.P.—M., 2021, ome8
4 S. Gua ino Lo Bianco, D. A. La Manna & B. Velichko
1.1.2. Hölde con inui y and non-degene acy o u. — Le now (u, Ω) be any solu ion
o (1.1). In o de o p o e he Hölde con inui y and he non-degene acy o ui is
su icien o exploi some o he es ima es ha we al eady used o p o e he exis ence.
Indeed, we can es he op imali y o (u, Ω) wi h he compe i o s om (1.3). Thus,
o > 0small enough, we ha e
(1.4) Jβ(u, Ω) ⩽Jβ(u ,Ω )whe e u =u∨ and Ω = Ω ∪ {u⩽ }.
In pa icula ,
ZD
|∇u|2dx +βZ∂∗Ω
u2⩽ZD
|∇(u∨ )|2dx +βZ∂∗(Ω∪{u< })
u2
⩽ZD
|∇(u∨ )|2dx +β 2Pe ({u < }) + βZ{u> }∩∂∗Ω
u2,
which p o es ha usa is ies he op imali y condi ion (4.1) om Lemma 4.1:
(1.5) Z{u< }
|∇u|2dx ⩽β 2Pe {u< }.
Now, applying P oposi ion 4.2, we ge ha uis bounded om below by a s ic ly
posi i e cons an in D. Finally, P oposi ion 3.5 gi es ha uis Hölde con inuous
in D. This p o es Theo em 1.1(iii).
1.1.3. Regula i y o he ee bounda y. — In o de o p o e he egula i y o he ee
bounda y (Theo em 1.1(iii)), we use he Hölde con inui y and he non-degene acy
o u o show ha a solu ion Ωis an almos -minimize o he pe ime e . We do his in
Theo em 5.1. Now, om he classical egula i y heo y o almos -minimize s o he
pe ime e (see [8]), we ob ain ha (inside D) he ee bounda y ∂Ωcan be decomposed
in o a C1,α- egula pa Reg(∂Ω) and a (possibly emp y) singula pa o Hausdo
dimension smalle han d−8.
Finally, in Theo em 5.2, we p o e he C∞ egula i y o Reg(∂Ω). In o de o do so,
we i s show (see Lemma 5.3) ha in a neighbo hood o a egula poin x0, he
es ic ions u+and u−o uon Ωand D Ωa e solu ions o he ollowing ansmission
p oblem:
∆u+= 0 in Ω,
∆u−= 0 in D Ω,
u+=u−=uon ∂Ω,
∂u+
∂νΩ
−∂u−
∂νΩ
+ 2βu = 0 on ∂Ω,
whe e νΩis he no mal de i a i e o ∂Ω. Now, using he ecen esul s [4] and [5],
we ge ha u+and u−a e as egula as he ee bounda y ∂Ω(see Lemma 5.4).
On he o he hand, using a ia ions o ualong smoo h ec o ields, we ob ain ha
Reg(∂Ω) sol es an equa ion o he o m
“Mean cu a u e o ∂Ω"=F(∇u+,∇u−, u±)on ∂Ω,
J.É.P.—M., 2021, ome8
A wo-phase p oblem wi h Robin condi ions on he ee bounda y 5
whe e Fis an explici ( a ional) unc ion o ∇u±and u. In pa icula , his implies ha
∂Ωgains one mo e de i a i e wi h espec o u, ha is, u∈Ck,α ⇒∂Ω∈Ck+1,α.
Thus, by a boo s ap a gumen , he egula pa o he ee bounda y is C∞.
1.2. On he non-degene acy o he solu ions. — We no ice ha he compe i o s
(u ,Ω )in (1.3) a e he wo-phase analogue o he ones used by Ca a elli and
K i en so in [3], whe e he au ho s s udy a one-phase e sion o (1.1). Ne e heless,
he unc ional in [3] in ol es he measu e o Ω, which means ha he op imali y
condi ion he e co esponds o
Jβ(u, Ω) + C|Ω∩ {u⩽ }| ⩽Jβ(u ,Ω ),whe e u =u∨ and Ω = Ω {u⩽ },
whe e C > 0. The p esence o he cons an Cenables us o p o e he bound om
below by using a di e en ial inequali y o a sui ably chosen unc ion ( ), which is
gi en in e ms o uand {u< }(see P oposi ion 4.2 and [3, Th. 3.2]). In P oposi ion
4.2, we exploi he same idea, bu since we do no ha e he cons an C, we can only
conclude ha ( )⩾ε (which is no in con adic ion wi h he ac ha ( )is de ined
o e e y > 0). So, we con inue, and we use his lowe bound o ob ain a bound o
he o m
(1.6) c⩽β1/2Pe ({u< })1/2|{u < }|1/2 o e e y > 0,
whe e u:= u∞and cis a cons an depending on βand d. Then, we no ice ha his
en ails
c⩽β3/4Pe ({u< })1/4|{u < }|3/4 o e e y > 0.
and we use an i e a ion p ocedu e o ge ha
c⩽β1−1/2nPe ({u < })1/2n|{u < }|1−1/2n o e e y > 0.
Passing o he limi as n→ ∞, we ge ha i uis no bounded away om ze o, hen
(1.7) c⩽β|{u < }| ⩽β|D| o e e y > 0.
Now, his means ha he measu e o he ze o-se |{u= 0}| is bounded om below.
Thus, using again he op imali y o u, we ge ha (1.6) holds wi h an a bi a y small
ε > 0in place o β, we ge ha
c⩽ε|{u < }| o e e y > 0,
which is impossible.
A simila non-degene acy esul was p o ed by Bucu and Giacomini in [1] by
a De Gio gi i e a ion scheme(1). P ecisely, one can p o e ha any solu ion o (1.1)
sa is ies he op imali y condi ion om [1, Rem. 3.7]. Thus, [1, Th. 3.5] also applies
o he solu ions o (1.1). Con e sely, he a gumen om 4.2 can be applied o he
minimize s o [1] o ob ain he bound om below o [1, Th. 3.5].
(1)We a e g a e ul o he anonymous e e ee o b inging o ou a en ion he e e ence [1].
J.É.P.—M., 2021, ome8
6 S. Gua ino Lo Bianco, D. A. La Manna & B. Velichko
1.3. One-phase and wo-phase p oblems wi h Robin bounda y condi ions
The p oblem (1.1) is he i s ins ance o a wo-phase ee bounda y p oblem wi h
Robin bounda y condi ions. P ecisely, we no ice ha i Ωis a ixed se wi h smoo h
bounda y and i uminimizes he unc ional Jβ(·,Ω) in H1(D), hen he unc ions
u+:= uon Ωand u−:= uon D Ω,
a e ha monic in Ωand D Ω, and sa is y he ollowing condi ions:
(1.8) u+=u−and ∂u+
∂ν+
+β
2u++∂u−
∂ν−
+β
2u−= 0 on ∂Ω∩D,
whe e ν+and ν−a e he ex e io and he in e io no mals o ∂Ω. No ice ha (1.8) is
a wo-phase coun e pa o he one-phase p oblem
(1.9) ∆u= 0 in Ω,∂u
∂ν +βu = 0 on ∂Ω∩D,
which was s udied by Bucu -Luckhaus in [2] and Ca a elli-K i en so in [3].
As explained in [3], he Robin condi ion in (1.9) na u ally a ises in he physical
si ua ion in which he hea di uses eely in Ω, he empe a u e is se o be ze o on
he su ace ∂Ω, which is sepa a ed om he in e io o Ωby an in ini esimal insula o .
The wo-phase p oblem (1.8) also may be in e p e ed in his way, in his case he
hea di uses eely bo h inside Ωand ou side, in D Ω; he empe a u e is se o
be ze o on he su ace ∂Ω, which is insula ed om bo h sides; he con inui y o he
empe a u e means ha he hea ans e is allowed also ac oss ∂Ω, which happens
o ins ance i he su ace ∂Ωis eplaced by a e y hin (in ini esimal) ne .
E en i he p oblems in [2, 3] and in he p esen pape lead o he ee bounda y
condi ions o he same ype, he echniques a e comple ely di e en . Fo ins ance,
he p oblem s udied in [2, 3] is a ee discon inui y p oblem as he unc ion ujumps
om posi i e in Ω o ze o in D Ω. Thus, he co esponding a ia ional minimiza ion
p oblem can be na u ally s a ed in he class o SBV unc ions, which clea ly in luences
bo h he exis ence and he egula i y echniques; oughly speaking, he exis ence is
ob ained h ough a compac ness heo em in he SBV class, while he egula i y elies
on echniques ela ed o he Mum o d-Shah unc ional.
In ou case, he p oblem can be s a ed o he unc ions (u1, u2)wi h disjoin
suppo s (u1u2= 0 almos -e e ywhe e in D) which sa is y he ollowing cons ain s:
he sum u1+u2should be a Sobole unc ion ( his co esponds o he con inui y
condi ion in (1.8)); u2
1and u2
2a e SBV unc ions whose jump se s a e con ained in
he bounda y o he posi i i y se s {u1>0}and {u2>0}. Now, i is easonable o
expec ha an exis ence esul can be p o ed also in his class, bu hen, in o de o
p o e ha a solu ion o (1.1) exis s, one should show ha u1and u2a e o he o m
u1=u
1
Ωand u2=u
1
D Ω o a se o ini e pe ime e Ω⊂Rd,ubeing he sum
u1+u2. Summa izing, wo king in he class o SBV unc ions would allow o s a e
(1.1) in a weake o m, bu i doesn’ seem o be a sho cu o he exis ence o a
solu ion (o (1.1)) as i will equi e he analysis o he jump se s o he op imal pai s
J.É.P.—M., 2021, ome8
A wo-phase p oblem wi h Robin condi ions on he ee bounda y 7
in he SBV class. Thus, we p e e no o ely on he ad anced compac ness esul s o
SBV unc ions, bu o p o e he exis ence o a solu ion om sc a ch.
Finally, as explained in Sec ion 1.1, once we know ha an op imal pai s (u, Ω)
exis s, and ha uis non-degene a e and Hölde con inuous, he egula i y o he ee
bounda y ∂Ω ollows immedia ely since he se Ωbecomes an almos -minimize o he
pe ime e .
2. P elimina ies
2.1. Se s o ini e pe ime e . — Le A⊂Rdbe a an open se in Rd. We ecall ha
he se E⊂Rdis said o ha e a ini e pe ime e in Ai
(2.1) Pe (E, A) = sup nZA
di ξ(x)dx :ξ∈C1
c(A;Rd),sup
x∈Rd
|ξ(x)|⩽1o
is ini e. We say ha Ehas a locally ini e pe ime e in A, i o e e y open se B⊂Rd
such ha B⊂A, we ha e ha Pe (E, B)<∞. We say ha Eis o ini e pe ime e i
Pe (E) := Pe (E, Rd)<+∞.
By he De Gio gi s uc u e heo em (see o ins ance [7, Th. II.4.9]), i he se E⊂Rd
has locally ini e pe ime e in A, hen he e is a se ∂∗E⊂A∩∂E called educed
bounda y such ha
Pe (E, B) = Hd−1(B∩∂∗E) o e e y se BbA,
whe e Hd−1is he (d−1)-dimensional Hausdo measu e in Rd. Mo eo e , he e is
aHd−1-measu able unc ion νE:∂∗E→Rd, called gene alized no mal such ha
|νE|= 1 and
ZE
di ξ(x)dx =Z∂∗E
νE·ξ dHd−1 o e e y ξ∈C1
c(A;Rd).
2.2. Capaci y and aces o Sobole unc ions. — We de ine he capaci y (o he
2-capaci y) o a se E⊂Rdas
cap(E) = in kuk2
H1(Rd):u∈H1(Rd), u ⩾1in a neighbo hood o E.
Suppose now ha d⩾3. I is well-known ha he se s o ze o capaci y ha e ze o
d−1dimensional Hausdo measu e (see o ins ance [6, §4.7.2, Th. 4]):
I cap(E)=0, hen Hd−1(E)=0.
The Sobole unc ions a e de ined up o a se o ze o capaci y (i.e., quasi-e e ywhe e),
ha is, i A⊂Rdis an open se and u∈H1(A), hen he e is a se Nu⊂Rdsuch
ha cap (Nu)=0and
u(x0) = lim
→0
1
|B |ZB (x0)
u(x)dx o e e y x0∈A Nu.
Mo eo e , o e e y unc ion u∈H1(A) he e is a sequence un∈C∞(A)∩H1(A)
and a se N ⊂ Ao ze o capaci y such ha :
–uncon e ges o us ongly in H1(A);
–u(x) = limn→∞ un(x) o e e y x∈A (N ∪ Nu).
J.É.P.—M., 2021, ome8
14 S. Gua ino Lo Bianco, D. A. La Manna & B. Velichko
The cons uc ion o Ω0is mo e delica e. Fi s , we ix > 0and δ > 0and we
no ice ha he pe ime e o Ωεnis bounded on he open se {u0> } ∩ Dδ. Indeed,
he uni o m con e gence o uεn o u0implies ha , o nla ge enough (n⩾N ,δ, o
some ixed N ,δ ∈N),
uεn⩾
2on Dδ∩ {u0> }.
Thus, we ha e
Jβ( , E)⩾βZDδ∩{u0> }∩∂∗Ωεn
u2
εndHd−1⩾β 2
2Pe Ωεn;Dδ∩ {u0> }.
Now, i we choose such ha Pe ({u0> })<∞(which, by he co-a ea o mula,
is ue o almos -e e y > 0), hen we ha e ha
Pe Ωεn∩ {u0> } ∩ Dδ⩽C ,δ o e e y n⩾N ,δ,
o some cons an C ,δ >0. Now, since all he se s Ωεn∩ {u0> } ∩ Dδa e con ained
in Dand ha e uni o mly bounded pe ime e , we can ind a se Ω0and a subsequence
o which
1
Ωεn∩{u0> }∩Dδ(x)−→
1
Ω0∩{u0> }∩Dδ(x) o almos -e e y x∈D.
Thus, by a diagonal sequence a gumen , we can ex ac a subsequence o εn(s ill
deno ed by εn) and we can de ine he se Ω0⊂Rdas he poin wise limi
1
Ω0(x) = lim
n→∞
1
Ωεn∩{u0>0}(x) o almos -e e y x∈ {u0>0},
and we no ice ha , by cons uc ion, Ω0⊂ {u0>0}. No ice ha , we do no know a
p io i ha Ω0has ini e pe ime e . We only know ha
Pe (Ω0∩ {u0> } ∩ Dδ)<∞ o e e y δ > 0and almos -e e y > 0.
which means ha Ω0∩ {u0> }has locally ini e pe ime e in D o a.e. > 0.
4.2. An op imali y condi ion. — As poin ed ou abo e, we do no know i he pai s
(u0,Ω0)is e en an admissible compe i o o (1.1) (we need o show ha Ω0∈ E).
Ne e heless, we can s ill p o e ha i sa is ies a sui able op imali y condi ion.
Lemma 4.1 (The op imali y condi ion a he limi ). — Le u0and Ω0be as in Sec-
ion 4.1. Then, o almos -e e y > 0, we ha e
(4.1) Z{u0< }
|∇u0|2dx ⩽β 2Pe {u0< }.
P oo . — Le now > 0be ixed and such ha he se {u0< }has ini e pe ime e .
Then, o nla ge enough, we can use he pai s (u0∨ , Ω0∪ {u0< }) o es he
op imali y o (uεn,Ωεn). No ice ha he se Ω0∪ {u0< }has ini e pe ime e o
J.É.P.—M., 2021, ome8
A wo-phase p oblem wi h Robin condi ions on he ee bounda y 15
a.e. ∈(0, m), as obse ed in he p e ious sec ion. Fo he sake o simplici y, we w i e
uεn=un,Ωεn= Ωn,u0=uand Ω0= Ω. Thus, we ha e
ZD
|∇un|2dx+βZ{u> }∩∂∗Ωn
u2
ndHd−1
⩽ZD
|∇un|2dx +βZ∂∗Ωn
u2
ndHd−1
⩽ZD
|∇(u∨ )|2dx +βZ∂∗(Ω∪{u< })
u2dHd−1
(4.2)
⩽ZD
|∇(u∨ )|2dx +β 2Pe ({u < }) + βZ{u> }∩∂∗Ω
u2dHd−1.
Now, by he weak con e gence o un o u, we ge ha
ZD
|∇u|2dx ⩽lim in
n→∞ ZD
|∇un|2dx.
On he o he hand, se ing U ,δ o be he open se
U ,δ =Rd Dδ∩ {u⩽ },
o some ixed δ > 0, and applying Lemma 2.4, we ha e ha
ZU ,δ∩∂∗Ω
u2dHd−1⩽lim in
n→∞ ZU ,δ∩∂∗Ωn
u2
ndHd−1⩽lim in
n→∞ Z{u> }∩∂∗Ωn
u2
ndHd−1.
Taking he limi as δ→0, by he mono one con e gence heo em, we ge ha
lim
δ→0ZU ,δ∩∂∗Ω
u2dHd−1=ZRd (D∩{u⩽ })∩∂∗Ω
u2dHd−1
Now, since
u(x) = h(x) o quasi-e e y x∈Rd Dand o Hd−1-almos -e e y x∈Rd D,
and since h⩾m > on ∂D, we ha e ha
(4.3) ZRd (D∩{u⩽ })∩∂∗Ω
u2dHd−1=Z{u> }∩∂∗Ω
u2dHd−1.
Thus, we ge ha
(4.4) Z{u> }∩∂∗Ω
u2dHd−1⩽lim in
n→∞ ZD∩{u> }∩∂∗Ωn
u2
ndHd−1.
Now, using (4.4) and (4.2), we ob ain
ZD
|∇u|2dx+βZ{u> }∩∂∗Ω
u2dHd−1
⩽lim in
n→∞ ZD
|∇un|2dx +βZ{u> }∩∂∗Ωn
u2
ndHd−1
⩽ZD
|∇(u∨ )|2dx +β 2Pe ({u < }) + βZ{u> }∩∂∗Ω
u2dHd−1,
which gi es (4.1).
J.É.P.—M., 2021, ome8
16 S. Gua ino Lo Bianco, D. A. La Manna & B. Velichko
4.3. Non-degene acy. — The c ucial obse a ion in his sec ion is ha he unc-
ions usa is ying he op imali y condi ion (4.1) a e non-degene a e in he sense o
he ollowing p oposi ion.
P oposi ion 4.2 (Non-degene acy). — Le β > 0,m > 0,Dbe a bounded open se
o Rdand u∈H1(D)be a non-nega i e unc ion in Dsuch ha u⩾mon ∂D. Le
Ω⊂Dbe a se o ini e pe ime e in D. Suppose ha uand Ωsa is y he op imali y
condi ion
(4.5) ZΩ
|∇u|2dx ⩽β 2Pe (Ω )whe e Ω ={u⩽ },
o almos -e e y ∈(0, m). Then, |Ω |= 0 o some > 0.
P oo . — By con adic ion, suppose ha
|Ω |>0 o e e y > 0.
Le ∈(0, m)be ixed. By he co-a ea o mula, he Cauchy-Schwa z inequali y and
he op imali y condi ion (4.5), we ge
(4.6) ZΩ
|∇u|=Z
0
Pe (Ωs)ds ⩽ZΩ
|∇u|21/2
|Ω |1/2⩽ β1/2Pe (Ω )1/2|Ω |1/2.
We now se
( ) := Z
0
Pe (Ωs)ds =ZΩ
|∇u|dx.
Using (4.6), we will es ima e ( ) om below.
S ep 1. Non-degene acy o . — By he isope ime ic inequali y and he es ima e (4.6),
he e is a dimensional cons an Cdsuch ha
Z
0
Pe (Ωs)ds ⩽ β1/2CdPe (Ω )(2d−1)/(2d−2).
Using he de ini ion o , we can e-w i e his inequali y as
( )(2d−2)/(2d−1) ⩽ (2d−2)/(2d−1)β1/2Cd(2d−2)/(2d−1) 0( ).
A e ea anging he e ms and in eg a ing om 0 o , we ob ain
( )1/(2d−1) − (0)1/(2d−1) ⩾ 1/(2d−1)
β1/2Cd(2d−2)/(2d−1) .
Now, since uis non-nega i e in D, we ha e ha (0) = 0. Thus
( )⩾
β1/2Cd2d−2.
Se ing
(4.7) C=βCd1−d,
we ob ain he lowe bound
( )⩾C .
J.É.P.—M., 2021, ome8
A wo-phase p oblem wi h Robin condi ions on he ee bounda y 17
In pa icula , as a consequence o (4.6), we ge ha
(4.8) C⩽β1/2Pe (Ω )1/2|Ω |1/2.
S ep 2. Non-degene acy o |Ω |. — Le α∈(0,1) be ixed. Then, we ha e ha
Z
0
Pe (Ωs)α|Ωs|1−αds ⩽Z
0
Pe (Ωs)dsαZ
0
|Ωs|ds1−α
⩽ β1/2Pe (Ω )1/2|Ω |1/2α |Ω |1−α
= βα/2Pe (Ω )α/2|Ω |1−α/2.
Thus, we ob ain ha o ixed T∈(0, m)and C > 0, he ollowing implica ion holds:
(4.9) (I C⩽Pe (Ω )α|Ω |1−α o e e y ∈(0, T ),
hen C⩽βα/2Pe (Ω )α/2|Ω |1−α/2 o e e y ∈(0, T ).
We claim ha , o e e y n⩾1and e e y ∈(0, m), we ha e he inequali y
(4.10) C⩽β1−1/2nPe (Ω )1/2n|Ω |1−1/2n.
In o de o p o e (4.10), we a gue by induc ion on n. When n= 1, (4.10) is p e-
cisely (4.8). In o de o p o e ha he claim (4.10) o n∈Nimplies he same claim
o n+ 1, we apply (4.9) o α= 2−n,n∈N, which gi es p ecisely (4.10) wi h n+ 1.
This concludes he p oo o (4.10). Nex , passing o he limi as n→ ∞, we ob ain
ha
C⩽β|Ω | o e e y ∈(0, T),
whe e Cis gi en by (4.7). Thus, he e is a dimensional cons an Cd>0such ha
(4.11) β−dCd⩽|Ω | o e e y ∈[0, m).
S ep 3. Conclusion. — We now no ice ha
lim
→0|Ω |=|Ω0|>0.
Thus, o e e y ε > 0, he e is Tεsuch ha o all ∈(0, Tε)we ha e
ZΩ
|∇u|=Z
0
Pe (Ωs)ds ⩽ZΩ
|∇u|21/2
|Ω Ω0|1/2
⩽ ε1/2Pe (Ω )1/2|Ω |1/2.
(4.12)
Now, epea ing he a gumen o S ep 1 and S ep 2, we ge ha (4.11) should hold
wi h εin place o β. Since ε > 0is a bi a y, his is a con adic ion.
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18 S. Gua ino Lo Bianco, D. A. La Manna & B. Velichko
4.4. Exis ence o a solu ion. — We a e now in posi ion o p o e ha he pai s
(u0,Ω0), cons uc ed in Sec ion 4.1, is a solu ion o (1.1).
P oposi ion 4.3 (Exis ence o a solu ion). — The e is a dimensional cons an Cd>0
such ha i Dis a bounded open se o Rdand β > 0is a gi en posi i e cons an , hen
he ollowing holds. Fo e e y se E⊂Rdo ini e pe ime e and e e y ∈H1(Rd)
sa is ying
⩾mon D o some cons an m > 0,
he e is a solu ion (u, Ω) o he p oblem (1.1).
P oo . — Le (u0,Ω0)be as in Sec ion 4.1. Then, by Lemma 4.1, (u0,Ω0)sa is ies
he op imali y condi ion (4.5). Now, by P oposi ion 4.2 we ge ha u0⩾ in D, o
some > 0. In pa icula , Ω0has ini e pe ime e in D. P ecisely, o e e y δ > 0, we
ha e
Pe (Ω0;Dδ)⩽lim in
n→∞ Pe (Ωεn;Dδ)⩽4
2lim in
n→∞ ZDδ∩∂∗Ωεn
u2
εndHd−1
⩽4
β 2lim in
n→∞ Jβuεn,Ωεn⩽4
β 2Jβ( , E).
Passing o he limi as δ→0, we ge
Pe (Ω0;D)⩽4
β 2Jβ( , E).
In pa icula , his implies ha Ω0is a se o ini e pe ime e in Rd. Indeed,
Pe (Ω0)⩽Pe (Ω0;D) + 2Pe (D) + Pe (Ω0;Rd D)
⩽4
β 2Jβ( , E) + 2Pe (D) + Pe (E;Rd D).
Thus, he pai s (u0,Ω0)is admissible in (1.1); i now emains o p o e ha i is
op imal. Le eu∈H1(D)be non-nega i e on Dand such ha u− ∈H1
0(D). Le
e
Ω⊂Rdbe a se o ini e pe ime e such ha e
Ω = Eon Rd D. I is su icien o
p o e ha
Jβ(u0,Ω0)⩽Jβ(eu, e
Ω).
Le ε > 0be ixed. We now use he pai s (eu∨ε, e
Ω) o es he op imali y o uεn,Ωεn:
Jβuεn,Ωεn⩽Jβ(eu∨ε, e
Ω).
Passing o he limi as ε→0, we ge
Jβuεn,Ωεn⩽Jβ(eu, e
Ω).
Now, Lemma 2.4 and he semicon inui y o he H1no m gi es ha Jβ(u0,Ω0)⩽
Jβ(eu, e
Ω), which concludes he p oo .
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A wo-phase p oblem wi h Robin condi ions on he ee bounda y 19
5. Regula i y o he ee bounda y
In his sec ion, we p o e he egula i y o he ee bounda y. In Theo em 5.1, we
p o e ha he solu ions o (1.1) a e almos -minimize s o he pe ime e in D. As a
consequence, ∂Ωcan be decomposed in o a egula and a singula pa and ha he
egula pa is C1,α mani old. Then, in Theo em 5.2, we p o e ha he egula pa
o he ee bounda y is C∞smoo h.
Theo em 5.1. — Le (u, Ω) be a solu ion o (1.1). he e is a cons an C > 0such
ha Ωis an almos -minimize o he pe ime e in he ollowing sense:
Pe Ω ; B (x0)⩽1 + C 1/3Pe Ω0;B (x0),
o e e y ball B (x0)⊂Dand e e y se Ω0⊂Rdsuch ha Ω=Ω0ou side B (x0).
In pa icula , he ee bounda y ∂Ω∩Dcan be decomposed as he disjoin union o
a egula pa Reg(∂Ω) and a singula pa Sing(∂Ω), whe e
(i) Reg(∂Ω) is a ela i ely open subse o ∂Ωand is a C1,α smoo h mani old;
(ii) Sing(∂Ω) is a closed se , which is emp y i d⩽7, disc e e i d= 8, and o
Hausdo dimension d−8, i d > 8.
P oo . — We i s no ice ha by Lemma 3.4, u∈C0,1/3(D). Le δ > 0,x0∈Dδand
< δ/2. We conside a se Ω0⊂Rdsuch ha Ω0∆Ω bB (x0). Tes ing he op imali y
o (u, Ω) agains (u, Ω0)we ge ha
ZB (x0)∩∂∗Ω
u2dHn−1⩽ZB (x0)∩∂∗Ω0
u2dHn−1,
which implies ha
min
B (x0)u2Pe Ω ; B (x0)⩽max
B (x0)u2Pe Ω0;B (x0).
By egula i y o u, we ha e ha
max
B (x0)u2⩽min
B (x0)u2+C 1/3⩽min
B (x0)u21 + C
1/3,
whe e in he second inequali y, we used ha u⩾ > 0. Thus, we ob ain
Pe Ω ; B (x0)⩽1 + C
1/3Pe Ω0;B (x0),
which p o es ha Ωis an almos -minimize o he pe ime e in D.
We nex p o e ha egula pa he ee bounda y Reg(∂Ω) is C∞.
Theo em 5.2. — Le (u, Ω) be a solu ion o (1.1). Le
D∩∂Ω = Reg(∂Ω) ∪Sing(∂Ω)
be he decomposi ion o he ee bounda y om Theo em 5.1. Then, in a neighbo hood
o any poin x0∈Reg(∂Ω),∂Ωis C∞- egula and he unc ion uis C∞on ∂Ω.
P oo . — We ix a poin x0∈Reg(∂Ω). Wi hou loss o gene ali y, we assume x0= 0.
J.É.P.—M., 2021, ome8
20 S. Gua ino Lo Bianco, D. A. La Manna & B. Velichko
S ep 1. No a ion. — Fo any x∈Rd, we use he no a ion x= (x0, xd), whe e x0∈Rd−1
and xd∈R. By he C1,α egula i y o Reg(∂Ω), in B0×(−ε, ε)⊂Rd−1×R,∂Ωis
he g aph o a C1,α egula unc ion η:B0→R, whe e B0is a ball in Rd−1; he se Ω
coincides wi h he subg aph o ηin a neighbo hood o he o igin:
B0×(−ε, ε)∩Ω = (x0, xd)∈B0×(−ε;ε) : xd< η(x0).
and he ex e io no mal νΩis gi en by
(5.1) νΩ=(−∇x0η, 1)
p1 + |∇x0η|2,
whe e ∇x0ηis he g adien o ηin he i s d−1 a iables. Le u+and u−be he
es ic ions o uon he se s Ωand D Ω; since uis con inuous ac oss ∂Ω, we ha e
u+=u−on ∂Ω. Mo eo e , we w i e he g adien s o u+and u−as
∇u±=∇x0u±, ∂xdu±∈Rd−1×R.
S ep 2. T ansmission condi ion and C1,α egula i y o u. — In Lemma 5.3, we keep ixed
he ee bounda y ∂Ωand we use e ical pe u ba ions o he unc ion u o ob ain
a Robin- ype ansmission condi ion on ∂Ω. We no ice ha he ecen esul s [4, 5]
imply he C1,α- egula i y o u+and u−, up o he bounda y ∂Ω. Thus, he g adien
is well-de ined and he ansmission condi ions (5.2) hold in he classical sense.
S ep 3. Op imali y condi ion and C2,α egula i y o Reg(∂Ω). — In Lemma 5.5 we pe -
o m a ia ions o he op imal se o ind he geome ic equa ion sol ed by ∂Ω. P e-
cisely, we ind ha he cu a u e o he op imal se sol es an equa ion o he o m
“Mean cu a u e o ∂Ω"=F(∇u+,∇u−, u±)on ∂Ω.
In pa icula , his implies ha i uis Ck,α, o some k⩾1, hen ∂Ωis Ck+1,α.
S ep 4. Boo s ap. — In Lemma 5.4 we use he ecen esul s o [5] o show ha i
he bounda y ∂Ωis Ck,α o some k⩾2, hen he solu ions u+and u−a e also Ck,α
egula up o he bounda y ∂Ω. Finally, applying his esul (Lemma 5.4) and he
esul om he p e ious s ep (Lemma 5.5), we ge ha ∂Ωis C∞.
Lemma 5.3 (Robin and con inui y condi ions on ∂Ω). — Suppose ha ∂Ωis C1,α
egula in he neighbo hood o he o igin. Le η:B0→R,u+and u−be as abo e.
Then, o e e y x0∈B0we ha e
(5.2) (∇x0η· ∇x0u+− ∇x0η· ∇x0u−=−∂xdu+−∂xdu−|∇x0η|2
p1 + |∇x0η|2∂xdu+−∂xdu−+βu = 0,
whe e u+,u−and hei pa ial de i a i es a e calcula ed in (x0, η(x0)) ∈∂Ω.
J.É.P.—M., 2021, ome8
A wo-phase p oblem wi h Robin condi ions on he ee bounda y 21
P oo . — Le φ∈C∞
c(D)be a smoo h unc ion suppo ed in B0×(−ε, ε). Then, he
op imali y o ugi es ha
0 = ∂
∂ =0Jβ(u+ φ, Ω) = ZD ∂Ω
2∇u· ∇φ dx +βZ∂Ω
2uφ dHd−1
=Z∂Ω
2νΩ· ∇u+−νΩ· ∇u−+βuφ dHd−1,
whe e in he las inequali y we in eg a ed by pa s u+in Ωand u−in D Ω. Since φ
is a bi a y we ge ha usa is ies he Robin- ype condi ion on ∂Ω
(5.3) νΩ· ∇u+−νΩ· ∇u−+βu on ∂Ω.
Now, using (5.1), we can e-w i e his as
(5.4) − ∇x0η· ∇x0u++∂xdu+−− ∇x0η· ∇x0u−+∂xdu−+βup1 + |∇x0η|2= 0.
On he o he hand uis con inuous ac oss ∂Ω. This means ha
∇x0u+(x0, η(x0)) + ∂xdu+(x0, η(x0))∇x0η=∇x0u−(x0, η(x0)) + ∂xdu−(x0, η(x0))∇x0η.
Mul iplying by ∇x0η, we ge
(5.5) ∇x0η· ∇x0u++∂xdu+|∇x0η|2=∇x0η· ∇x0u−+∂xdu−|∇x0η|2,
whe e u+,u−and hei pa ial de i a i es a e calcula ed in (x0, η(x0)). Pu ing o-
ge he (5.4) and (5.5), we ge (5.2).
Lemma 5.4 (Smoo h bounda y ⇒smoo h unc ion). — Le (u, Ω) be a solu ion
o (1.1). Suppose ha , in a neighbo hood o ze o, ∂Ωis Ck,α- egula o some k⩾1.
Then, in a neighbo hood o he o igin, he unc ions u+and u−a e Ck,α up o he
bounda y ∂Ω.
P oo . — We a gue by induc ion. The case k= 1 ollows by [5]. We suppose ha
k⩾2and ha he claim holds o k−1. Suppose ha ∂Ωis he g aph o η:B0→R,
η∈Ck,α(B0), and conside he unc ions
+(x0, xd) := u+(x0, xd+η(x0)) and −(x0, xd) := u−(x0, xd+η(x0)),
de ined on he hal -space {xd⩾0}. We se
Aη=Nd−1−(∇x0η)
−∇x0η|∇x0η|2,
whe e Nd−1is he null (d−1) ×(d−1) ma ix and we no ice ha Aηhas Ck−1,α
egula coe icien s. Now, since u+and u−a e ha monic in Ωand D Ω, we ha e
ha +and −a e solu ions o he ansmission p oblem
−di ((Id + Aη)∇ +)=0 in {xd>0}
−di ((Id + Aη)∇ −)=0 in {xd<0}
+= −on {xd= 0}
∂xd +−∂xd −+β
2p1 + |∇x0η|2( ++ −)=0 on {xd= 0}.
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22 S. Gua ino Lo Bianco, D. A. La Manna & B. Velichko
We now ix k−1di ec ions i1, . . . , ik−1,ij6=d o e e y j, and we conside he
unc ions
w+:= ∂i1∂i2. . . ∂ik−1 +and w−:= ∂i1∂i2. . . ∂ik−1 −.
We no ice ha , in {xd>0}and {xd<0} he unc ions w+and w−a e solu ions o
−di (Id + Aη)∇w±+X
I,J
di ∂IAη∂J∇u±= 0,
whe e he sum is o e all mul iindices Iand Jsuch ha he se s Iand Ja e disjoin
subse s o {i1, i2, . . . , ik−1},I∪J={i1, i2, . . . , ik−1}and Iis non-emp y. In pa icula ,
using ha Aη∈Ck−1,α and ∇u∈Ck−2,α (since by hypo hesis u±∈Ck−1,α), we ge
ha w±sol e
−di (Id + Aη∇w±) + di (F±)=0 in {±xd>0},
whe e F+and F−a e C0,α con inuous unc ions (depending on i1, . . . , ik). On he
o he hand, on he bounda y {xd= 0}we ha e ha w+=w−and
∂xdw+−∂xdw−+∂i1∂i2. . . ∂ik−1β(u++u−)
2p1 + |∇x0η|2= 0 on {xd= 0}.
Reasoning as abo e, we no ice ha his condi ion can be w i en as
∂xdw+−∂xdw−=gon {xd= 0},
whe e gis a C0,α unc ion. Now, applying [5, Th. 1.2], we ge ha w+and w−a e
C1,α egula up o he bounda y {xd= 0}. Thus, he ace u+=u−is Ck,α smoo h
on {xd= 0}. Finally, he classical Schaude es ima es gi e ha u+and u−a e Ck,α
on {xd⩾0}and {xd⩽0}, espec i ely.
Lemma 5.5 (Smoo h unc ion ⇒smoo h bounda y). — Le (u, Ω) be a solu ion
o (1.1). Suppose ha , in a neighbo hood o ze o, ∂Ωis C1,α- egula and ha he
unc ions u+and u−a e Ck,α up o he bounda y ∂Ω, o some k⩾1. Then, ∂Ωis
Ck+1,α- egula in a neighbo hood o ze o.
P oo . — Le ξ∈C∞
c(D;Rd)be a gi en ec o ield wi h compac suppo in Dand
le Ψ be he unc ion
Ψ (x) = x+ ξ(x) o e e y x∈D.
Then, o small enough, Ψ :D→Dis a di eomo phism and se ing Φ := Ψ−1
,
he unc ion u := u◦Φ is well-de ined and belongs o H1(D); he unc ion
7−→ ZD
|∇u |2dx
is di e en iable a = 0 and
∂
∂ =0 ZD
|∇u |2dx =ZD−2∇u Dξ · ∇u+|∇u|2di ξdx.
I is immedia e o check ha
−2∇u Dξ · ∇u+|∇u|2di ξ=di |∇u|2ξ−2(ξ· ∇u)∇uin D ∂Ω.
J.É.P.—M., 2021, ome8
A wo-phase p oblem wi h Robin condi ions on he ee bounda y 23
We now ake ξ o be smoo h ou side ∂Ωand such ha
ξ=φνΩon ∂Ω,
whe e νΩis he ex e io no mal o ∂Ωand φ:∂Ω→Ris con inuous and wi h compac
suppo . In eg a ing by pa s, we ge
∂
∂ =0 ZD
|∇u |2dx =Z∂Ω|∇u+|2(ξ·νΩ)−2(ξ· ∇u+)(νΩ· ∇u+)dHd−1
−Z∂Ω|∇u−|2(ξ·νΩ)−2(ξ· ∇u−)(νΩ· ∇u−)dHd−1,
whe e u+:= uon Ω, and u−:= uon D Ω. Now, i
ξ=φedand νΩ=(−∇x0η, 1)
p1 + |∇x0η|2,
hen
p1 + |∇x0η|2|∇u+|2(ξ·νΩ)−2(ξ· ∇u+)(νΩ· ∇u+)
−p1 + |∇x0η|2|∇u−|2(ξ·νΩ)−2(ξ· ∇u−)(νΩ· ∇u−)
=φ|∇u+|2− |∇u−|2
−2φ∂xdu+− ∇x0η· ∇x0u++∂xdu+−∂xdu−− ∇x0η· ∇x0u−+∂xdu−.
We now suppose ha x0∈Reg(∂Ω) and ha ∂Ωis he g aph o he (C1,α) unc ion
η:B0→R, whe e B0is a ball in Rd−1. Taking ξ=edφand Ω = Φ (Ω), we ha e
∂
∂ =0 Z∂Ω
u2
dHd−1=∂
∂ =0 ZB0
u2x0, η(x0)p1 + |∇x0η+ ∇x0φ|2dx0
=ZB0
u2x0, η(x0)
p1 + |∇x0η|2∇x0η· ∇x0φ dx0
=ZB0
u2x0, η(x0)Hx0, η(x0)φ(x0)dx0
−2ZB0
φ(x0)ux0, η(x0)∇x0u+∂xdu∇x0η· ∇x0η
p1 + |∇x0η|2dx0.
In pa icula , combining hese wo compu a ions and using he op imali y o (u, Ω),
we ge
0 = ∂
∂ =0Jβ(u ,Ω ) = ZB0
βu2H(x0)φ(x0)dx0+ZB0|∇u+|2− |∇u−|2φ(x0)dx0
−ZB0
21 + |∇x0η|2(∂xdu+)2−(∂xdu−)2φ(x0)dx0
Since φis a bi a y, we ob ain ha ηis a solu ion o he p oblem
−di x0∇x0η
p1 + |∇x0η|2= (x0)in B0,
J.É.P.—M., 2021, ome8