Well-posedness of general boundary-value problems for scalar conservation laws

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scalar conservation laws

Boris Andreianov, Karima Sbihi

To cite this version:

Boris Andreianov, Karima Sbihi. Well-posedness of general boundary-value problems for scalar conservation laws. Transactions AMS, 2015, 367 (6), pp. 3763-3806. �10.1090/S0002-9947-2015-05988-1�.

�hal-00708973v3�

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BORISANDREIANOVANDKARIMASBIHI

Abstrat. Inthispaper we investigate well-posednessfor the problemut+ divϕ(u) = f

on(0, T)×Ω^,Ω⊂R^N,withinitialonditionu(0,·) =u0 ônΩând^with^generaldissipative boundary onditions ϕ(u)·ν ∈ β_(t,x)(u) ôn(0, T)×∂Ω^. ^Here^for â.e. (t, x) ∈ (0, T)×∂Ω^, β_(t,x)(·)îs â ^maximal^monotone ^graphôn R. This inludes, as partiular ases, Dirihlet, Neumann,Robin,obstaleboundaryonditionsandtheirpieewiseombinations.

Asforthe well-studied aseofthe Dirihletondition, onehas to interprete the formal

boundaryonditiongivenbyβ^by^replaingît^with^theâdequateêetive^boundaryôndition.

Suh eetiveondition an beobtained through a studyof the boundary layer appearing

inapproximationproessessuhasthevanishingvisosityapproximation. Welaimthatthe

formalboundaryonditiongivenbyβ^should^beinterpretedastheeetiveboundaryondition givenbyanothermonotonegraph β˜^,^whih^is^dened^fromβ ^by^the^projetion^proedure^we

desribe. Wegiveseveral equivalentdenitionsofentropysolutionsassoiatedwithβ˜^(and

thusalsowithβ^).

Forthenotionofsolutiondenedinthisway,weproveexistene,uniquenessandL¹^on-

tration,monotoneandontinuousdependeneonthegraphβ^.^Convergene^ofapproximation proeduresandstabilityofthenotionofentropysolutionareillustratedbyseveralresults.

Keywords: salar onservation law, boundary-value problem, entropy solution,

vanishing visosity limit, formal boundary ondition, eetive boundary ondition,

maximalmonotonegraph,strongboundarytrae,L¹ ^ontration,well-posedness

Contents

1. Introdution 2

1.1. Dissipativeboundaryonditionsforonservationlaws 2

1.2. ClassialresultsontheDirihletase 3

1.3. Strong traesofentropysolutionsontheboundary 4

1.4. Interpretationofageneralboundaryondition 6

1.5. Formerresultsandasummaryofthepaper 7

2. TheeetiveBCgraph 8

2.1. Preliminaries: undershoot andovershootsets,inreasingenvelopes 8

2.2. Denition andequivalentharaterizationsofβ˜ ¹⁰

2.3. Orderand metristrutureonB_x 12

3. Notionofsolution 13

3.1. Equivalentdenitionsofentropysolutions,sub-solutions,super-solutions 14

3.2. Proofoftheequivaleneofdierentdenitions 16

4. Uniqueness,omparison,ontinuousdependene 18

5. Existene: aformalproof 19

6. Justiationoftheeetiveboundaryondition 21

2010MathematisSubjetClassiation. Primary35L65,35L04;Seondary35A01,35A02.

TheworkoftherstauthorwaspartiallysupportedbytheFrenhANRprojetCoToCoLa.

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6.1. Convergeneofthevanishingvisosityapproximation 21

6.2. Stabilityofthenotionofentropysolution 25

7. Furtherexisteneandonvergeneresults 30

7.1. Theone-dimensionalase: existeneviaBVloc ^estimate ³¹

7.2. Entropy-proesssolutions 32

Conlusion 34

Appendix: existeneforthevisosityregularizedproblem 34

Referenes 37

1. Introdution

Whilethere existsan extensiveliteratureonthe Cauhyand Cauhy-Dirihletproblems for

salaronservation lawut+ divϕ(u) = 0^, ^otherinitial-boundaryvalue problemshavereeived very few attention. This is the purpose of this paper to dene a notion of entropy solution

forawidelass ofboundaryonditionsthat wealldissipativeboundary onditions;to justify

thisdenition throughonvergeneof naturalapproximationproedures;andto establishwell-

posedness resultsforthesodenedentropysolutions.

1.1. Dissipative boundary onditions for onservation laws. LetΩ ^be^an ^open ^domain

inR^N withLipshitzboundary, N ≥1^,^and T >0^. ^We^onsider ^the^followinginitial-boundary valueproblemforasalaronservationlaw:

(Hϕ,β(u0, f)⁾





ut+ divϕ(u) =f ⁱⁿ QT := (0, T)×Ω

u|^t=0=u0 ⁱⁿ Ω

ϕν(x)(u) :=ϕ(u)·ν(x)∈β(t,x)(u) ^on Σ := (0, T)×∂Ω.

Here ϕ:R−→R^N is aontinuousfuntion (forthesakeof simpliity, thereadermayassume that ϕ îs ^Lipshitz ôntinuous, âlthough ^most ôf ôur ^results ^hold ^without ^this assumption)

1

;

u0∈L^∞(Ω)^; ândf îsâ^measurable^funtion ônQT ^withRT

0kf(t,·)k^L^∞^(Ω)<∞^.

Further, in (Hϕ,β(u0, f)^{) ,} ^the ûnit ôutward^normal ^vetor ôn ∂Ω îs ^denoted ^by ν^, ând ^the

boundaryonditionis presribed(formally)in termsof β ^that îsâ^map^from Σ^to ^the^set Bof allmaximal monotone graphs on R. Clearly, some measurability assumption is neededon the mapβ : (t, x)∈Σ7→β(t,x)∈B. Inthesequel,wealwaysextendβ(t,x)^to â^maximal^monotone

graphfrom Rto Randrequirethefollowing:

(1.1)

forallk∈R, (t, x)7→infβ(t,x)(k)^and(t, x)7→supβ(t,x)(k)

aremeasurable R-valuedfuntions w.r.t. theHausdormeasureonΣ^.

This enompasses dierent lassial boundary onditions. For instane, the graph β(t,x) = {u^D(t, x)} ×Rpresribesthe Dirihletboundary onditionu=u^D ^onΣ^; ^the^graphβ(t,x):=

R× {−g(t, x)}^presribes^theôndition−ϕ(u)·ν(x) =g ^that ^we^willâll^Neumannôndition,

byanalogywiththeNeumannboundaryonditionsforthegeneralonvetion-diusionproblems

ofthe kindut−diva(u,∇u) = f^. Îtîs âlso êasy^to înlude ^the^more^generalônditionsôf ^the

kindλu+ (1−λ)(−ϕ(u)·ν) =g^,λ∈(0,1)^,^onditions^thatinterpolatebetweentheDirihletand the Neumannones (these are known asRobin onditions in the onvetion-diusion ontext).

1

Note thatthe the resultsof the presentpaper an beeasily extended to the aseof x^-dependent ϕ^(and x^-dependentβ⁾ⁱⁿône^spae^dimension,ûsing^the^nonlinear^semigroup^theory. ^We^refer^to^[5℄^for^thisêxtension

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Togive onemoreexample, the (bilateral)obstale boundary onditions u^m ≤u ≤u^M ^onΣ

orrespondto thegraph

β_(t,x)=

{u^m(t, x)}×R⁻

∪

[u^m(t, x), u^M(t, x)]×{0}

∪

{u^M(t, x)}×R⁺ .

Forthesakeofsimpliity,thereadermayonsider

(1.2)

β(t,x)(r) =β_(t,x)⁰ (r−u^D(t, x))−g(t, x) ^withu^D∈L^∞(Σ)^,g∈L^∞(Σ)

andwithamaximalmonotonegraphβ_(t,x)⁰ ^suh^thatβ⁰_(t,x)(0)∋0;

thisontainstheaforementionedasesand,e.g.,theaseofmixedDirihlet-Neumannboundary

onditions.

Intheontextofparaboliproblemsut−diva(u,∇u) =f^,^it^is^well^known^that^the^boundary

onditions of the kind β(t,x)(u) +a(u,∇u)·ν(x) ∋ 0 ^lead ^to ^the L¹ ^ontration ^property ^(see

e.g. [38℄ for a study of the assoiated stationary ellipti problem; see also [3℄); that's why we

alltheseonditionsdissipativeboundaryonditions. Itisustomarytointerpretethephysially

admissible weak solutions(alled entropy solutions sine thefounding work [21℄ of Kruzhkov)

ofasalaronservationlawaslimitsof thevanishing visosity approximation that,in ourase,

wouldtaketheform

(1.3)





u^ε_t−div (−ϕ(u^ε)+ε∇u^ε) =f, u^ε|^t=0=u0, β_(t,x)(u^ε) + (−ϕ(u^ε) +ε∇u^ε)·ν(x)

|(t,x)∈Σ∋0.

Thenitislearthattheboundaryonditionin(Hϕ,β(u0, f)⁾îs^the^formal^limitôf^thedissipative boundaryonditionβ(t,x)(u^ε) + (−ϕ(u^ε) +ε∇u^ε)·ν(x)∋0ⁱⁿ^(1.3)^(here^we^shouldâssume^some

regularityof β(t,x) ⁱⁿ (t, x) ⁱⁿ ôrder ^that â^solutionu^ε êxist; ^forînstane, ^for ^the^Dirihlet ^BC

ase we need u^D ∈ L²(0, T;H^−1/2(∂Ω)) ^). ^Moreover^let u^ε^, uˆ^ε ^be ^solutions ^of ^problem ^(1.3)

with thesamedissipativeboundaryondition andwith datau0, f ^anduˆ0,fˆ^, respetively. The

L¹ ôntration^property^holdsûnder^rather^weakrestritionsonΩândϕ^(see,ê.g.,^[26,^6℄):

ku^ε(t,·)−uˆ^ε(t,·)kL¹(Ω)≤ ku0−uˆ0kL¹(Ω)+kf −fˆkL¹(Ω)

Provided the L¹(QT) ômpatness ôf ^the ^sequenes (u^ε)ε^, (û^ε)ε ^with ε → 0 îs ^known, ît îs

inherited at thelimitε→0^. ^Therefore ^weêxpet ^that ^the^boundaryôndition^satisedât^the

limitisalsoadissipativeone.

But what isthis limitboundaryondition asε→0 ⁱⁿ ^(1.3)^? ^The^ompatness^of (u^ε)ε ⁱⁿ

L¹(QT)^gives^noinformationononvergeneofu^ε^on^the^boundary,^the^termε∇u^ε·ν(x)^on^the

boundarybeomessingularasε→0^,^therefore^passage^to^the^limitⁱⁿ^boundary^onditions^is^by

nomeansstraightforward. Asamatteroffat,ingeneral

theboundaryondition ϕ(u)·ν(x)∈β(t,x)(u) ^is^not^the^orret^limit

obtainedfromtheboundaryonditions β(t,x)(u^ε) + (−ϕ(u^ε) +ε∇u^ε)·ν(x)∋0^.

TheDirihletonditionasedisussedbelowisawell-knownillustrationofthisfat.

1.2. Classialresultsonthe Dirihlet ase. Withinthewholevarietyofdissipativebound-

aryonditions,onlytheDirihletasereeivedmuhattentionintheframeworkofonservation

laws. The elebratedresultof Bardos, LeRouxand Nédéle [10℄ states that theDirihlet on-

dition u=u^D ônΣ ^should ^be^seenâsâ^formal ôndition; ând^that ît^must ^be interpreted by stating that the trae (γu)(t, x) ôf u ât â ^point (t, x) ∈ Σ ^belongs ^to ^the ^subset I(t, x) ôf R

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denedin termsofu^D(t, x)ândôf^the^funtionr7→ϕ_ν(x)(r) =ϕ(r)·ν(x)âs^follows:

(1.4)

I(t, x) =

z∈Rsign (z−u^D(t, x))(ϕν(x)(z)−ϕν(x)(k))≥0

∀k∈[u^D(t, x)∧z, u^D(t, x)∨z]

.

Here and in the sequel, ∧ (respetively, ∨⁾ ^denotes ^the min ^(resp., ^the max⁾ ^operation. ^We

denotebyH^N ^theN−dimensionalHausdormeasureonΣ^.

Theeetive boundaryondition

(1.5) (γu)(t, x)∈I(t, x) H^N^-a.e. ^onΣ

isknownastheBLNondition;inthispaper,wewillusethereformulationoftheBLNondition

intermsofamaximalmonotone (sub)graph. Suhgraphinterpretationwasrstmadeexpliit,

forthe Dirihlet ase, byDubois andLeFloh in [18℄(see in partiular [18, Fig.1.1℄). Another

usefulinterpretationoftheBLNonditiongoingbakto[18℄isthefollowing:

I(t, x) =n

z∈Rϕν(x)(z) =^God[ϕν(x)](z, u^D(t, x))o ,

whereGod[ψ] :R²→RistheGodunovnumerialuxassoiatedtoagivensalaruxψ:R→R. ReallthattheGodunovux isgivenbytheexpression

(1.6) God[ψ](a, b) =

minz∈[a,b]ψ(z), ^ifa≤b maxz∈[b,a]ψ(z), ^ifb≤a^.

Thefuntional framework of thepaper[10℄ is the spae L^∞(0, T;BV(Ω)) ^(atually, ^the^so-

lutions belong to the spae BV(QT)^). ^There ^are ^two ^good ^reasons ^for ^that. ^Firstly^, ^the BV

in spaeregularityofu^guarantees^theêxisteneôf â^traeγuôf uônΣ^,^neessaryⁱⁿ ôrder ^to

givesense to theBLN ondition. Seondly, uniform in ε BV êstimatesôn ^the^solutionsôf ^the

approximatingproblems(1.3)areavailable,forBV ^datau0ându^Dând^for^Lipshitzôntinuous

ux funtion ϕ^. ^Bardos, ^LeRoux ând ^Nédéle ^show ^that ^for ^the âbove ^data ând ûx, ^there

existsauniqueL^∞(0, T;BV(Ω))êntropy ^solution ôf ^theonservationlawsatisfying(pointwise onΣ⁾^the^BLN^boundaryôndition;ând^that ^this^solutionîs^the^limitôf^the^vanishing^visosity

approximation.

Morereently,Otto in[27, 28℄(see also[25℄)providedaformulationsuitableformerelyL^∞

data u0 ând u^D^; ^Porretta, ^Vovelle ^[35℄ ând Âmmar, ^Carrillo ând ^Wittbold ^[2℄ êxtended ^the

denition and resultsto theframeworkof L¹ ^data^(see ^the^papers^for ^the^preise assumptions on u^D⁾ ând ^merely ôntinuous ûx ^funtion ϕ^, ⁱⁿ â ^bounded ^domain Ω^. ^The L¹ ^framework

requires an appropriate notionof solution; in [35, 2℄ the notion of renormalized solution from

[11℄ was used. IntheOtto formulation,existeneof a(strong) boundary trae γu ôf uônΣ îs

notassumed;aBLNkindonditionisreformulatedin termsofweak normalboundarytraes of

ϕ(u)ândôf^theâssoiated^boundaryêntropy ûxesF(u;u^D, k)^(theêxisteneôf^the^weak^traes

isarelativelysimpleonsequeneofthefatthatuîsâ^Kruzhkovêntropy^solutionôf^the^salar

onservation lawinside (0, T)×Ω^). ^We^refer ^to ^[27, ^28, ^25℄ ^and ^to ^[41, ^35,^39℄ ^for ^details ^and

resultsrelatedtotheapproahofOtto.

1.3. Strong traes of entropy solutions on the boundary. Although the denition of

[27,28℄ andtheaforementionedgeneralizationswerearemarkablestepforwardin thestudy of

boundaryvalueproblemsforonservationlaws,itwaspossibletobypasstheuseofweaktraes

andtheassoiatedboundaryentropies'tehniques of[27,28℄. Indeed,forthesakeof simpliity

letusstartwiththefollowingux non-degenerayassumption:

(1.7) ∀ξ∈R^N\{0} ∀c∈R theLebesque measureoftheset{z|ξ·ϕ(z) =c} ^is^zero.

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Usingtheapproahofkinetisolutions(see[24,34℄),Vasseurin[40℄hasshownthatforϕ^regular

enough,

(1.8) under(1.7),anyL^∞ ^Kruzhkovêntropy^solutionⁱⁿ QT âdmitsâ^strong^trae γuônΣ.

Thenon-degenerayassumption(1.7)onϕ^is^typial^for^the"ompatiationproperties"inthe theoryofkinetisolutions,seePerthame[34℄andreferenestherein. AspointedoutbyVasseur,

(1.8) gives sense to the pointwiseBLN ondition (1.5) for general L^∞ ^entropy ^solutions, ^and

notonlyforsolutionsorrespondingtoBV ^data; ^thus^the^weak^trae^tehnique^of^Otto^[27,^28℄

is bypassed(yet for general(t, x)^-dependentûx ϕ^, ^the âpproah ôf ^[27, ^28℄ ^remains^the ^most

powerful; seein partiular theresultsofVallet [39℄). Furtherresultsin thespirit of (1.8) were

obtained by Kwon and Vasseur [23℄ for the ase N = 1 ^(see ^also ^[7, ^37℄ ^where ^we ^treat ^the

ase of aat boundary using ahint due to Panov). Tothe authors' knowledge, the strongest

generalizationof (1.8) isthe resultof Panov [32℄ obtainedusing thetehniqueof parametrized

families of H^-measures ^(see âlso^[30, ^33℄);^Panov^dropsâll ^regularityâssumptionôn ϕ^, ând, ⁱⁿ

asense,he alsodropsnon-degenerayassumptions ofthekind(1.7). Beause ofitsimportane

forourpaper,weshouldmakethelatterstatementmorepreise:

• ^(upon^rotatingâxesând^loalizingâroundâ^pointx^∗ ôf^the^boundary)

theboundary∂Ω^isrepresentedbythegraphofaLipshitz 2

funtion g^onW^,^i.e.,

∂Ω∩U ={(g(x^′), x^′)|x^′ ∈W}, Ω∩U ={(x0, x^′)|x0=y+g(x^′), x^′ ∈W, y∈(0, h)}

for some neighbourhood U ôf x^∗^, ^some neighbourhood V ôf ^zero ⁱⁿ R^N−1^, ând ^some h >0^;^further,^theûnitêxterior^normalêld

ν(g(x^′), x^′)

x^′∈W

islifted insideΩ∩U ^by

theformulaν(x0, x^′) = √ ¹

1+|∇g(x^′)|²

−1,∇g(x^′)

(theeld isonstantinx0∈[0, h)^);

• ^forx∈∂Ω∩U^,^onsider^the^singular^mapping Vϕν(x):r7→Rr

0 |ϕ^′(s)·ν(x)|ds^onR

(notiethatthemappingisindependentofx0^,ând ît^dependsônx^′ ontinuously)

• ^then^forânyu∈L^∞(QT)^thatîsâ^Kruzhkovêntropy^solutionⁱⁿQT^,^there êxists

(1.9) esslimy↓0Vϕν(x)

u(t, y+g(x^′), x^′)

=:

γVϕν(x)(u)

(t, x) ⁱⁿL¹((0, T)×W),

wherex:= (g(x^′), x^′)îsâ^generi^pointôfU∩∂Ω^;^reall^thatν(y+g(x^′), x^′)≡ν(g(x^′), x^′)^.

Statement (1.9)is atually are-interpretation of the loalization property that appears in the

proof [32, p.571℄ of Panov; we use it to give a sense to pointwise formulations of boundary

onditions,in thesameveinas Vasseurin [40℄. Ifforall ξ∈R^N\ {0} ^the^funtion r7→ϕ(r)·ξ

is non-onstant on any interval (this is a weaker versionof (1.7) typial for the tehnique of

parametrizedH^-measures,^see^[30,^32,^33℄),^thenVϕν(x0) îsânînvertible^funtion ^(whih^means

that strongtraeγu êxists). Îfϕîs ^notâBV ^funtion,ôneânûseânother ^singular^mapping

insteadofthemap r7→Rr

0 |ϕ^′(z)·ν(x0)|dz ^(whih^is^not^well^dened),^e.g., Vϕν(x)(r) =

Z r 0

1^lF(s)ds, F ^being^theûnionôfâll^theîntervals

wherethemap s7→ϕ(s)·ν(x)^does^not^vary.

Remark 1.1. By the denition of the singular mapping, Vϕν(x)(·) ^has ^the ^properties ^of ^being

monotonenon-dereasingandofbeingonstantonthesameintervalswhereϕν(x)(·)^is^onstant.

Therefore ϕ(r)·ν(x) = Φν(x)◦Vϕν(x) ^with ^some ^ontinuous ^funtion Φν(x) : R → R. As a onsequene of (1.9), there exists the strong traeγϕ(u)·ν(x) ^(with ^the ^same^meaning ^as ⁱⁿ

(1.9))whihisequaltoΦν(x)

γVϕν(x)(u)

.

2

WhilethesettingofPanov[32℄isC¹^regular^domains,^the^author^indiates^that^thegeneralizationtoLipshitz

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Inthesameway,oneanrepresenttheprojetionsonthediretionν(x)^of^thesemi-Kruzhkov entropyuxes

(1.10) q^±(u, k) := sign^±(u−k)

ϕ(u)−ϕ(k)

withthehelpofontinuousfuntionsQ^±_ν(x)(·,·)^of^two^variables:

(1.11) q^±(u, k)·ν(x) =Q^±_ν(x)

Vϕν(x)(u), Vϕν(x)(k) .

Heneforaoupleu,uˆôfêntropy^solutions,ît^follows^thatâ^strong^traeôfq^±(u,û)·ν(x)êxists

andanberepresentedasQ^±_ν(x)

γVϕν(x)(u), γVϕν(x)(ˆu)

. ThesameistruefortheKruzhkov

uxes:

(1.12)

q(u, k)·ν(x) =Qν(x)

Vϕν(x)(u), Vϕν(x)(k)

with q:=q⁺+q⁻, Qν(x):=Q⁺_ν(x)+Q⁻_ν(x).

1.4. Interpretation of a general boundary ondition. The Bardos-LeRoux-Nédéleon-

dition (1.4),(1.5) is generally reognized as the orret interpretation of the Dirihlet bound-

ary ondition; this is justied in partiular by onvergene of vanishing visosity ornumerial

approximations of the boundary value problem (see Vovelle [41℄), onsidered asquite natural

approximations. Observationsof visousor numerial boundary layers explain howthe formal

boundaryonditionu=u^D ônΣ^transformsînto^theêetive^boundaryôndition(1.4),(1.5) . The strong trae result of [40℄ was used by Bürger, Frid and Karlsen in [13℄ in order to

givesense totheformal zero-ux boundaryondition (in ourterminology,thisis theNeumann

boundaryonditionwithg≡0⁾ⁱⁿ^the^partiular^butîmportantâseϕ(0) = 0 =ϕ(1)^. Ûnder^this

assumptionandfor[0,1]^-valued^initial^data,^the^zero-ux^boundary^ondition^forut+divϕ(u) = 0

anbeunderstoodliterally (see[13℄)(inthesense thattheproblemis well-posedandsolutions

arelimitsofthevanishingvisosityapproximation).

Letusstressthatingeneral,alsoforthezero-uxboundaryonditionϕ(u)·ν = 0^a^boundary

layerwouldform in approximatesolutions,and thisformal zero-uxboundary ondition would

transformintosomedierenteetiveboundaryondition. Forasimpleexample,onsider the

zero-ux problem for the transport equation ut+ux = 0 ôn [0,1]^; âs ⁱⁿ ^[10℄, ârguing âlong

harateristisonesees thatthezero-uxondition(that readsu= 0 ^beauseϕ=Id⁾^at ^the

rightboundaryx= 1^must^be^merely^dropped.

It is the purpose of this paper to provide a naturalinterpretation for a generaldissipative

boundaryondition(formallygivenbyafamilyβ ^of^maximal ^monotone^graphsβ(t,x)(·)⁾^under

theform of an eetiveboundaryondition. Mostgenerally, this eetiveboundaryondition

anbewritten undertheform

(1.13) H^N^-a.e. ^onΣ^,^the^ouple

γVϕνu, ϕ(γVϕνu)·ν

liesinthegraphβ˜(t,x)(·)◦ Vϕν

−1

,

withβ˜^to^be^dened,^and^with^the^notationγVϕνu:=

γVϕν(x)(u) (t, x)^.

Tolarifytheesseneoftheondition(1.13),onsidertheasewhereVϕν(x)=Id^an^be^taken

(reallthat this isthe aseif (1.7) holds). Then (1.13) means that (γu)(t, x)∈ Domβ˜(t,x)(·)^;

andfromthedenitionofβ˜ⁱⁿ^Setion²^we^will^see^that^thisautomatiallyinludestheequality

β˜(t,x)(γu(t, x)) = ϕ(γu(t, x))·ν^. ^Thus ^the ôndition ϕ(u)·ν(x) = ˜β(t,x)(u) ôn Σ ân ^be

understoodliterally asapointwiseequality;thisis whywealliteetiveboundaryondition.

Notie that ondition (1.13) takes the form (γVϕνu)(t, x) ∈ Vϕν(I(t, x)) â.e. ôn Σ^, î.e., ît

presribessomeset I(t, x) ôf^possible^trae^valuesôf uôn^the ^boundary. ^Reall^that ^the ^BLN