Partial differential equations/Calculus of variations

(1)

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Partial differential equations/Calculus of variations

A maximum principle for the system u − ∇ ^W ( u ) = ⁰

Un principe du maximum pour le système u − ∇ ^W ( u ) = ⁰

Panagiotis Antonopoulos

^a

, Panayotis Smyrnelis

^b

aDepartmentofMathematics,UniversityofAthens,Panepistemiopolis,15784Athens,Greece bCentrodeModelamientoMatemático,UniversidaddeChile,Santiago,Chile

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received5August2015

Acceptedafterrevision1April2016 Availableonline14April2016 PresentedbyHaïmBrézis

Amaximumprincipleisestablishedforminimalsolutionstothesystemu− ∇^W(u)=^0, withapotential W vanishingattheboundaryofaclosed convexset C0⊂R^m^,^which^is eitherC²smoothorcoincideswithapoint{^a}^.

r é s um é

Nousétablissonsunprincipedumaximumpourlessolutionsminimalesdusystèmeu−

∇^W(u)=^0,^dont^le^potentiel^W^s’annule^à^la^frontière^d’unênsemble^fermé^convexe^C0⊂ R^m^,^de^classe^C²ôu^réduit^àûn^point{â}^.

1. Introductionandstatementofthemaximumprinciple

Wewillconsiderthesystem

u

− ∇

^W

(

u

) =

⁰

,

u

:

^A

→ R

^m

,

A

⊂ R

ⁿ

,

(1)

which isthe Euler–Lagrangeequation of thefree energy functional J

(

u

;

^A

) =

A

1

2

|∇

^u

|

²

+

^W

(

u

)

dx. Assumingthat the potential W satisﬁes:

(a) W

∈

^C¹

(R

^m

;

R)^,^W

(

a

) =

^{0 for}^some^a

∈

R^m^,^W

≥

^0,

(b) thereexistsr₀

>

0 suchthatforevery

ξ ∈

R^m^with

|ξ| =

^1:

(

0

,

r₀

)

^r

→

^W

(

a

+

^r

ξ )

isstrictlyincreasing, thefollowingmaximumprinciplewasprovedin[1].

E-mailaddresses:[email protected](P. Antonopoulos),[email protected](P. Smyrnelis).

http://dx.doi.org/10.1016/j.crma.2016.03.015

(2)

Theorem1.1.LetA

⊂

Rⁿ^be^an^open,^connected,^bounded^set,^with

∂

A Lipschitz,andsupposethatv

( · ) ∈

^W¹^,²

(

A

;

R^m

) ∩

^L^∞

(

A

;

R^m

)

minimizes J

(

u

;

^A

)

subjecttoitsboundaryconditionson

∂

A:

J

(

v

;

A

) =

min

{

J

(

u

;

A

) :

u

=

v on

∂

A

}.

Then,ifthereholds

|

^v

(

x

) −

^a

| ≤

^{r on}

∂

A,forsomer

>

0with0

<

2r

≤

^r0,therealsoholds

|

^v

(

x

) −

^a

| ≤

^{r on}^A.

The scopeofthispaperistoestablishageneralizationofTheorem 1.1,by consideringpotentials W thatvanishatthe boundary ofaclosedconvexsetC0

⊂

R^m^,^whichîsêither^C² ^smooth ôr^coincides^withâ^point

{

^a

}

^.^We^mention^that ^the convexityassumptionisessentialforsimilarproblemsinvolvingsystemsofPDEs(cf.[6,12,13]).Assumingthat

(i) W

∈

^C¹

(

R^m

;

R

)

,withW

∂C0

=

^0,^W

≥

^{0 on}R^m

\

^C0,

(ii) there existsr0

>

0 suchthat forevery outerunit normalvector

ξ

atthepoint p

∈ ∂

C0 (

| ξ | =

^1,^if^C0

= {

^a

}

^):

(

0

,

r0

)

r

→

^W

(

p

+

^r

ξ )

isincreasing,andmoreoverW

(

p

+

^r0

ξ ) >

0,

wecanstatethefollowingmaximumprinciple.

⊂

∂

A Lipschitz,andv

∈

^W¹^,²

(

A

;

R^m

) ∩

^L^∞

(

A

;

R^m

)

.Supposethat v minimizes J

(

u

;

^A

)

∂

A,

J

(

v

;

^A

) =

^min

{

^J

(

u

;

^A

) :

^u

=

^{v on}

∂

A

} .

Then,ifthereholds

d

(

x

) :=

^d

(

v

(

x

),

C₀

) ≤

^{r on}

∂

A

,

for some r

>

0with0

<

2r

≤

^r0

,

(2) whered istheEuclideandistance,therealsoholds

d

(

x

) ≤

r on A

.

(3)

Moreover,theattainmentofequalityin(3)ataninteriorpointofA,¹

d

(ˆ

x

) =

^r

,

for somex

ˆ ∈

^A

,

(4)

impliesthatd

(

x

) =

^r,

∀

^x

∈

Â,ândⁱⁿâdditionîf^C0isstrictlyconvex,v

(

x

) ≡

^constantⁱⁿ^A.

Thefollowingextensionisalsotrue:

⊂

∂

A Lipschitz,andlet

∂

DA

=

∅^be^a^Lipschitz^portion^of

∂

A.As- sumethatv

( · ) ∈

^W¹^,²

(

A

;

R^m

) ∩

^L^∞

(

A

;

R^m

)

minimizes J

(

u

;

^A

)

∂

DA:J

(

v

;

^A

) =

^min

{

^J

(

u

;

^A

) :

u

=

^{v on}

∂

DA

}

^.^Then,^the^condition^d

(

v

(

x

),

C0

) ≤

^{r on}

∂

DA

,

0

<

2r

≤

^r0,impliesthesameconclusionsasinTheorem 1.2above.

Notethat thestrictmonotonicityassumption(b)inTheorem 1.1hasbeenweakened.Theorem 1.2isacorollaryofthe replacementresultestablishedinLemma 3.2below.Foru

( · ) ∈

^W¹^,²

(

A

;

R^m

) ∩

^L^∞

(

A

;

R^m

)

,satisfyingtheboundarycondition (2),itisshownthat ifd

(

u

(

x

),

C₀

) >

ronanon-negligiblesubset,thenthereexists amap^u

˜

^coinciding^with^u^on

∂

A,and havinglessenergy.Toconstructthecompetitor^u,

˜

^we^utilize^the^followingdecompositionofu:

u

(

x

) =

^p

(

x

) + (

u

(

x

) −

^p

(

x

)),

(5)

where p

(

x

) :=

^p

(

u

(

x

))

istheprojectionofu

(

x

)

ontoC₀.Next,wedeﬁnethedistancefunctiond

(

x

) :=

^d

(

u

(

x

),

C₀

) = |

^u

(

x

) −

p

(

x

) |

^,^and^consider^adeformationofuoftheform:

˜

u

(

x

) =

^p

(

x

) +

^g

(

d

(

x

))(

u

(

x

) −

^p

(

x

)),

(6)

where g

:

R

→ [

⁰

, ∞)

îs â ^suitable ^locally^Lipschitz^function. ^We^point ôut ^that ^theâbove êxpression ôfû

˜

^by-passes^the diﬃcultyencounteredin[1],whichutilizesthenormalvector:n

(

x

) :=

_|û_u⁽₍^x_x⁾⁻₎₋â_a_| îfû

(

x

) =

^a,^andⁿ

(

x

) :=

^{0 if}^u

(

x

) =

â,ând^the polarrepresentationsofu(respectivelyû):

˜

^u

(

x

) =

^a

+ |

^u

(

x

) −

^a

|

ⁿ

(

x

)

(resp.^u

˜ (

x

) =

^a

+

^f

(|

^u

(

x

) −

^a

|)

ⁿ

(

x

)

),with f

:

R

→ [

⁰

, ∞)

^, locallyLipschitz.Indeed,from(6),thecomputationof

|∇ ˜

^u

|

^iselementary.In addition,thedecomposition(5)allowsusto dealinoneshotwiththepointcase(where p

(

x

) ≡

^a,^cf.Theorem 1.1),aswellaswiththecaseofsmoothconvexsets.

1 WeareindebtedtoA.Savas-Halilajforsuggestingtheuseoftheusualmaximumprincipleatthispoint.Thestrongmaximumprinciplepartofthe theoremfollowsfromtheusualmaximumprincipleasdevelopedin[12]and[13].

(3)

2. Comparisonwiththeusualmaximumprinciple

2Thetheoremsabovearedifferentfromtheusualmaximumprincipleinthefollowingrespects:

(a) W isnotconvex,hencetheusualmaximumprincipleisnotvalidevenforthescalarcasem

=

^1;

(b) the monotonicity condition (ii) on W is extremely mild, andallows applicability in situations where degeneracy is natural(cf.[3]);

(c) theusualmaximumprincipleisacalculusfactandappliestoallsolutions.Thisisnottruefortheresultabove.

ConsidertheO.D.E.

u

−

^W

(

u

) =

⁰

,

u

: R → R,

^with^W

: R → R,

^W

(

u

) =

¹

4

(

u²

−

¹

)

²

.

(7)

Notice that (7) hasperiodic solutions forr

>

0 as small asdesired such that min_Ru

= −

1

+

r andmax_Ru

=

1

−

r. By choosing A

= (

0

,

T

)

suchthatu

(

0

) =

^u

(

T

) =

¹

−

^r,^and^C0

= {

¹

}

^,^we^see^thatTheorem 1.1doesnotapplytothesesolutions.

Similarly, forthe potential H

(

u

) =

¹

+

^cos

(

π^u

)

,andfor C0

= [

¹

,

3

]

^,^the ^previous ^periodic ^solution ^u ^does^not ^satisfy ^the maximumprincipleofTheorem 1.2.

ThefollowingexampleshowsthatTheorem 1.1doesnotapplyeventolocalminimizers(stablesolutionsto(1),deﬁned intermsofthedeﬁnitenessofsignofthesecondvariation).ConsiderthescalarP.D.E.

²

u

−

^W

(

u

) =

^{0 in}

, ∂

u

∂

n

∂

=

⁰

,

(8)

whereW isasin(7)aboveand

isadumbbelldomain.Itiswellknown(cf.[5,9,10])thatfor

>

0 suﬃcientlysmall,(8) hasastablesolutionwhichon theleftandrightoftheneckisascloseto

−

^{1 and}

+

^1,respectively,asdesired(bytaking

>

0 suﬃciently small).Bychoosing the set A

= \

^B,^with ^B

⊂

a closedball located on therightof theneck, and

∂

DA

= ∂

B,wecansecurethat

|

^u

(

x

) −

¹

| ≤

^r^on

∂

DA,andthereforeweseethatTheorem 1.3doesnotapply.

Thesolutionsto(1)forwhichTheorems 1.2 and1.3apply areusuallycalledminimal.Theyhavethepropertythatthey minimizethefreeenergywithrespecttotheirDirichletvaluesontheboundaryofanyopenboundedsubsetoftheirdomain ofdeﬁnition.Thisisreminiscentofa familiarpropertyof minimalsurfaces.Thatitis sharedbythe minimalsolutions is notsurprising,sincethefunctional J islinkedtotheperimeterfunctionalifscaledappropriately.Moreprecisely,itiswell knownthattheGammalimit undera blow-downof J istheperimeterfunctional(cf.[11] forthescalarcase, and[2]for thevector),andthatthelevelsetsoftherescaledminimizersconvergetominimalsurfaces.

3. Proofofthemaximumprinciple

In what follows C₀

⊂

R^m îs â ^closed ^convex ^set, ^which îs êither ^C² ^smooth ôr^coincides ^with â ^point

{

^a

}

^. ^We ^ﬁrst compute

|∇ ˜

^u

|

² ^for^the^map^u

˜

^definedⁱⁿ^(6),ûtilizing^the^propertiesôf^the^projection ^p

(

x

)

.

Proposition3.1.Let A

⊂

Rⁿ ôpenând^bounded,^with^Lipschitz^boundary,^letû

(·) ∈

^W¹^,²

(

A

;

R^m

) ∩

^L^∞

(

A

;

R^m

)

,andlet^u

˜ (

x

) =

p

(

x

) +

^g

(

d

(

x

))(

u

(

x

) −

^p

(

x

))

,where p

(

x

) :=

^p

(

u

(

x

))

istheprojectionofu

(

x

)

ontoC₀,d

(

x

) :=

^d

(

u

(

x

),

C₀

) = |

^u

(

x

) −

^p

(

x

)|

^,^and g

:

R

→ [

⁰

, ∞ )

isalocallyLipschitzfunction.Then,u

˜ ( · ) ∈

^W¹^,²

(

A

;

R^m

) ∩

^L^∞

(

A

;

R^m

)

,and

|∇ ˜

^u

|

²

= |∇

^p

|

²

+

f

(

d

)

2

|∇

^d

|

²

+

g

(

d

)

2

|∇(

^u

−

^p

)|

²

− |∇

^d

|

²

+

^2g

(

d

)

n i=¹

^pxi

,

u_x_i

−

^pxi

,

wherewehaveset f

(

s

) :=

^sg

(

s

)

.Inaddition,if

|

^f

| ≤

¹^and⁰

≤

^g

≤

^1,^then

|∇ ˜

^u

|

²

≤ |∇

^u

|

²^. Theproofofthemaximumprincipleisbasedonthefollowingcut-offlemma:

Lemma3.2.LetA

⊂

Rⁿ^,^open,^bounded,^connected,^with

∂

A Lipschitz,andletW satisfyHypotheses(i),(ii)above.Supposethat u

(·) ∈

^W¹^,²

(

A

;

R^m

) ∩

^L^∞

(

A

;

R^m

)

.Ifthefollowingtwoconditionshold,

(I) d

(

x

) :=

^d

(

u

(

x

),

C0

) ≤

^{r on}

∂

A,0

<

2r

≤

^r0,

(II) Lⁿ

(

A

∩ {

^d

(

x

) >

r

}) >

0(Lⁿ

(

E

)

,then-dimensionalLebesguemeasure), then,thereisu

˜ ( · ) ∈

^W¹^,²

(

A

;

R^m

) ∩

^L^∞

(

A

;

R^m

)

suchthat

2 WewanttothankProf.N.D.Alikakosforprovidingtheexamplesofthissection.

(4)

˜

u

=

^{u on}

∂

A

,

(9a)

d

˜ (

x

) :=

^d

(

^u

˜ (

x

),

C₀

) ≤

^r

,

on A

,

(9b)

J

(

u

˜ ;

^A

) <

J

(

u

;

^A

).

(9c)

Proof. Case1.Wewillﬁrstestablishthelemmaundertheadditionalhypothesis:

d

(

x

) ≤

^r0

.

(10)

The argumenthereis easy sinceu stays inthe monotonicityregion of W about C0. Let f

(

s

) =

^min

{

^s

,

r

} =

^g

(

s

)

s. Clearly,

|

^f

| ≤

^{1 and}⁰

≤

^g

≤

^1.^ByProposition 3.1,u

˜ (

x

) =

^p

(

x

) +

^g

(

d

(

x

))(

u

(

x

) −

^p

(

x

))

satisﬁes

A

|∇ ˜

^u

(

x

)|

²^dx

≤

A

|∇

^u

(

x

)|

²^dx

.

(11)

Havingacloserlookweseethatincaseofequalityin(11),then 0

=

A

|∇ ˜

^u

|

²^dx

−

A

|∇

^u

|

²^dx

≤

A

|∇

^d

|

²

((

f

(

d

))

²

−

¹

)

dx

≤ −

A∩{d>r}

|∇

^d

|

²^dx

,

fromwhich itfollowsthat

∇

^d

=

^{0 a.e. on}^A

∩ {

^d

>

r

}

^,^and^therefore

∇(

d

˜ −

^d

) =

^{0 a.e. on}^A,^where d

˜ (

x

) =

^f

(

d

(

x

))

via(6).

Sinced

˜ −

^d

∈

^W¹^,²

(

A

)

,we haveby connectednessd

˜ (

x

) −

^d

(

x

) =

constant a.e. in A,andfromd

˜ −

^d

=

^{0 on}

∂

A inthesense of trace, we obtain _d

˜ (

x

) −

^d

(

x

) =

^{0 a.e. in}^A, ⁱⁿ contradictionto assumption (II) inLemma 3.2. Thereforewe havestrict inequalityin(11).

Ontheotherhand,since f

(

d

) =

^g

(

d

)

d

≤

^d

≤

^r0,wehaveby(ii):

A

W

p

(

x

) +

^g

(

d

(

x

))(

u

(

x

) −

^p

(

x

))

dx

≤

A

W

p

(

x

) + (

u

(

x

) −

^p

(

x

))

dx

,

A

W

(

u

˜ (

x

))

dx

≤

A

W

(

u

(

x

))

dx

,

henceCase1issettled.Noticethatinthiscasethestrictnessin(9c)wasobtainedviathegradientterm.

Case2.Assume

Lⁿ

(

A

∩ {

^d

>

r₀

}) >

⁰

.

(12)

Considerthefollowingcut-offfunctions:

α (

s

) :=

⎧ ⎪

⎨

⎪ ⎩

1 fors

≤

^r

2r−^s

r forr

≤

^s

≤

^2r 0 fors

≥

^2r

,

f

(

s

) =

^min

{

^s

,

r

} α (

s

) =

^g

(

s

)

s

.

Again,itisclearthat

|

^f

| ≤

^{1 and}⁰

≤

^g

(

s

) ≤

^1,^thus^u

˜ (

x

) =

^p

(

x

) +

^g

(

d

(

x

))(

u

(

x

) −

^p

(

x

))

satisﬁesthankstoProposition 3.1:

|∇ ˜

u

(

x

)|

²

≤ |∇

u

(

x

)|

²

.

(13)

Wenote inpassingthat u

˜

isareﬂectionofualongd

(

u

,

C0

) =

^r,ând^thus⁽¹³⁾îsêxpected.Ûnlikeⁱⁿ^Case^1,^the^strictness oftheinequalityin(9c)willfollowfromthepotentialterm.Wewillneedthefollowingproposition.

Proposition3.3.(‘Continuity’ofSobolevfunctions,cf.[4].)LetA

⊂

Rⁿ^,ôpen,^boundedând^connected,^with^Lipschitz^boundary,ând assumethatf

∈

^W¹^,²

(

A

;

R)^satisﬁes

f

≤ ˆ

^{r on}

∂

A (in the sense of trace) andLⁿ

(

A

∩ {ˆ

^s

<

f

} ) >

0for somer

ˆ < ˆ

s

.

(14) Then,Lⁿ

(

A

∩ {ˆ

^r

<

f

≤ ˆ

^s

} ) >

0.

Proof. Let

σ , τ :

^A

→

R^be^deﬁned^by

σ (

x

) =

^min

{

^f

(

x

), ˆ

s

} =

f

(

x

)

forx

∈

^E1

:=

^A

∩ {

^f

≤ ˆ

^r

} ˆ

s forx

∈

^E3

:=

^A

∩ {ˆ

^s

<

f

}, τ (

x

) =

^max

{ σ (

x

),

r

ˆ } =

r

ˆ

forx

∈

^E1

ˆ

s forx

∈

^E3

.

(5)

Suppose,for thesake ofcontradiction, that Lⁿ

(

A

∩ {ˆ

^r

<

f

≤ ˆ

^s

} ) =

^0.^Therefore,

τ

is astep function.Thus

∇ τ ₌

0 a.e.

in A.Ontheotherhand,

σ

,

τ

areinW¹^,²

(

A

;

R)^(cf.^[8,^p.^130]).^This^and^theconnectednessof Aimplythat

τ ≡

^constant (cf.[7, p.307]). Hence

τ ≡ ˆ

^s ^since Lⁿ

(

E3

) >

0. Itfollows thatLⁿ

(

E1

) =

^{0 and} ^f

>

s

ˆ

a.e. in A. Thus, f

≥ ˆ

^s ^on

∂

A inthe senseoftrace,whichiscontradicting(14).Theproofiscomplete. 2

Conclusion:Let

>

0 suchthatW

(

u

) >

0 onr0

≤

^d

(

u

,

C0

) ≤

^r0

+

.Wedeﬁnethesets E₁

:=

^A

∩ {

^d

≤

^r0

},

^E2

:=

^A

∩ {

^r0

<

d

≤

^r0

+ },

^E3

:=

^A

∩ {

^d

>

r₀

+ }.

From (12), we obtain that in the event that Lⁿ

(

E₂

) =

^0, ^then necessarily Lⁿ

(

E₃

) >

0. But d

≤

^r

<

r₀ on

∂

A, hence by Proposition 3.3:

Lⁿ

(

E₂

) >

0

.

(15)

Therefore,(15)holdsunderanycircumstances.On A

∩ {

⁰

≤

^d

≤

^2r

}

^we^have:

W

(

u

˜ (

x

)) =

^W

p

(

x

) +

^g

(

d

(

x

)(

u

(

x

) −

^p

(

x

))

≤

^W

p

(

x

) + (

u

(

x

) −

^p

(

x

))

=

^W

(

u

(

x

)),

since g

(

d

)

d

=

^f

(

d

) ≤

^d

≤

^2r

≤

^r0. On the other hand, on A

∩ {

^d

>

2r

}

^we ^have ⁰

=

^W

(

^u

˜ (

x

)) ≤

^W

(

u

(

x

))

, while on E2: 0

=

^W

(

u

˜ (

x

)) <

W

(

u

(

x

))

.Therefore,wehave J

(˜

u

;

^A

) <

J

(

u

;

^A

)

andtheproofofLemma 3.2iscomplete. 2

NowtheproofofTheorem 1.2isstraightforward.

ProofofTheorem 1.2. Weproceedbycontradiction.Sosupposethat(3)doesnothold,henceLⁿ

(

A

∩

^d

(

v

(

x

),

C0

) >

r

} ) >

0.

ButthiscontradictstheminimalityofvbyLemma 3.2.Thus(3)holds.Next,suppose(4)holdsandnoticethatd²

=

^d²

(

v

,

C₀

)

satisﬁes

(

d²

)(

x

) =

^2d

(

x

) ∇

^d

(

x

), ∇

^W

(

v

(

x

)) +

n i=¹

D²

(

d²

)(

v

(

x

))(

v_x_i

(

x

),

v_x_i

(

x

)) ≥

⁰

,

foreveryx

∈

^A^such^that⁰

<

d

(

v

(

x

),

C0

) ≤

^r0,thankstoHypothesis(ii)onW,totheconvexityofthefunctionu

→

^d²

(

u

,

C0

)

inthecomplementofC₀,andtothefactthattheminimizervsolvessystem(1).Bythestrongmaximumprinciple,itfollows thatd

(

v

,

C0

)

isconstantin A.Inaddition,ifC0 isstrictly convex,thenthefunction u

→

^d²

(

u

,

C0

)

isstrictlyconvexinthe complementofC0.Thisimpliesthat

(

d²

)(

x

) ≥ _|∇

v

(

x

) |

²

, ∀

^x

∈

^A

:

⁰

<

d

(

v

(

x

),

C₀

) ≤

^r0

,

forsome

>

0.Thus,

∇

^v

≡

^{0 since}^d

(

v

,

C0

)

isconstant,andasaconsequencev isconstantin A. 2

ProofofTheorem 1.3. First we note that if in Lemma 3.2, speciﬁcallyin condition (I), one replaces

∂

A with

∂

DA, then the sameconclusion (9)holds,where

∂

A isnow replacedwith

∂

DA. Theargumentis completelyunaltered. Similarly,in Proposition 3.3,

∂

A isreplacedwith

∂

DA,withoutchangeintheproof.TheproofofTheorem 1.3iscomplete. 2

Acknowledgements

PanayotisSmyrneliswaspartiallysupportedbyFondoBasalCMM-Chile,FONDECYTpostdoctoralgrantNo.3160055,and through theproject PDEGE PartialDifferential Equations Motivatedby Geometric Evolution, co-ﬁnancedby the European Union European Social Fund (ESF) and national resources, in the framework of the program Aristeia of the Operational ProgramEducationandLifelongLearningoftheNationalStrategicReferenceFramework(NSRF).

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Partial differential equations/Calculus of variations

C. R. Acad. Sci. Paris, Ser. I