TOWARDS AN L 1
-THEORY FOR VECTOR-VALUED ELLIPTIC
BOUNDARY VALUE PROBLEMS
ROBERTDENK, MATTHIAS HIEBER,ANDJAN PR
USS
1. Introduction
Vector-valuedellipticandparabolicboundaryvalueproblemssubjecttogeneralboundarycon-
ditions have been investigated recently in [DHP01] in the L
p
-context for 1 < p < 1. One
ofthemain goalsof this paperwasto deduce amaximal L
p
-regularityresultforthe solution
of the parabolic initial boundaryvalue problem. A classical reference in the elliptic context
are thecelebratedpapersof Agmon, Douglis and Nirenberg [ADN59]. Forfurther references
and informationon the scalarand vector-valuedcase we referto the [Ama01] and thelist of
referencesgivenin [DHP01].
Vector-valued elliptic and parabolicproblems on all of R n
we considered rst by Amann
[Ama01]onalargescaleoffunction spaces,includingL
1 (R
n
;E). HereE denotesanarbitrary
Banach space. He proved in particular that the L
1
-realization of such problems generates
an analytic C
0
-semigroup provided the top-order coeÆcients of the underlying operators are
uniformlybounded andHoldercontinuous.
Inthis note, weconsider vector-valued boundaryvalueproblems with constant coeÆcients
in the L
1
-setting for a half space. Following the approach described in [DHP01], we assume
the Lopatinskii-Shapiro to be true; we then obtaina representation of the solution u of the
elliptic problem by integraloperators which allows to deduce a-prioriestimates for u in the
L
1 (R
n+1
+
;E)-norm. HereEdenotesagainanarbitraryBanachspace. Theseestimatesimplyin
particularthattheL
1 (R
n+1
+
;E)-realizationofanellipticboundaryvalueproblemwithconstant
coeÆcientsin the half space R n+1
+
generatesan analytic C
0
-semigroupson L
1 (R
n+1
+
;E). For
dierentapproachesand resultswithvariablecoeÆcientsin thescalar-valuedcasewereferto
Amann[Ama83],Di Blasio[DiB91],Guidetti[Gui93]andTanabe[Tan97],Section5.4.
2. Elliptic Problems onL 1
(R n
;E)
Throughoutthissection,letE beaBanachspace. Followingthenotionof[Ama01]orSection
5of[DHP01],wecallahomogeneousB(E)-valuedpolynomialA()ofdegreem2N parameter-
elliptic ifthereis anangle2[0;)suchthat thespectrum(A())satises
(A())
forall2R n
; jj=1:
(1)
Wethencall
A
:=inff: (1)holdsg= sup
jj=1
jarg(A())j
theangle of ellipticity ofA. ForD= i(@
1
;:::;@
n
)wecall A(D)= P
jj=m a
D
parameter
elliptic,ifitssymbolA()isparameter-elliptic.
AssumenowthatA(D)isaparameter-ellipticoperatorwithangleof ellipticity
A
. Itwas
proved in Theorem 5.2 and Corollary5.3 of [DHP01] that for >
A
and k 2 N there are
1
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constantsc
;k , C
;k
suchthatthesolution
of
u+A(D)u=Æ
0
satisestheestimate
jD
(x)jC
;k jj
n+k
m 1
p n
m;k (c
;k
jxj); x2R n
;jargj ;jj=k;
(2)
wherep n
m;k
isgivenby
p n
m;k (r)=
Z
1
0
s n 2
(1+s) m k 1
e r(1+s)
ds:
Wedene theL
1 (R
n
;E)-realizationA ofA(D)bymeansofA=A
0 ,where
[A
0
u](x)=A(D)u(x); x2R n
; u2D(A
0 )=W
m
1 (R
n
;E):
TherstassertionofthefollowingpropositionisaspecialcaseofTheorem5.10of[Ama01].
Theproofgivenbelowisbasedonestimatesonthefundamental solution. Viaestimate(2)we
alsoobtainsomeinformationonthedomainofA, whichwill beimportantin thefollowing.
Proposition 2.1. Let n;m2N, E be aBanach space, a
2B(E) andsupposethat A(D) is
parameterelliptic with angle ofellipticity
A
<
2
. Then A generatesan analytic semigroup
onL
1 (R
n
;E)of angle
2
A
andwehave
W m
1 (R
n
;E)D(A)W m 1
1 (R
n
;E):
Proof. Obviously,A hasdensedomain. If f 2L
1 (R
n
;E), choose asequence f
n 2C
1
0 (R
n
;E)
such thatf
n
!f inL
1 (R
n
;E). For2
, >
A
,wehaveu
n :=
f
n 2W
m
1 (R
n
;E)
aswell asu
n
+A(D)u
n
=f
n
. Since u
n
!u=
f in L
1 (R
n
;E)asn !1, wesee that
u 2 D(A) and u+Au = f. This shows that +A is invertible for each 2
with
(+A) 1
f =
f. Thusby(2) andYoung'sinequalityweobtain
W m 1
1 (R
n
;E)D(A)D(A
0 )=W
m
1 (R
n
;E):
Furthermore,estimate(2)yields
(A) and
j(+A) 1
j
B (L
1 (R
n
;E))
M
;
foreach>
A .
3. EllipticProblems ina HalfSpace
Inthis sectionweconsider boundaryvalueproblemsoftheform
u+A(D)u = f; in R n+1
+
B
j
(D)u = g
j
; on @R
n+1
+
;j=1;:::;m:
onahalfspace. HereA(D)aswellasB
j
(D)forj2f1;:::;mgaredierentialoperatorswith
operator-valuedcoeÆcients. WealsoassumethatA(D)andB
j
(D)consistonlyoftheprincipal
parts,i.e.
A(D) = X
jj=2m a
D
B
j
(D) = X
jj=m b
j D
;
wherem
j
2f0;:::;2m 1g, a
2B(E)andb
j
2B(E)forj 2f1;:::;mg. Inthefollowing
weassumethatA(D)isparameter-ellipticwithangleofellipticity
A
2[0;),i.e. thereexists
2[0;)suchthat
(A())
; 2R n+1
; jj=1 (3)
and
A
isdenedastheinmumofallsatisfying(3). HereA()isthesymbolofA(D)dened
by
A()= X
jj=2m a
; 2R n+1
;a
2B(E):
(4)
WesupposethatthefollowingLopatinskii-ShapiroConditionholdstrue:
Lopatinskii-ShapiroCondition:
Foreach 0
2R n
and2
withj 0
j+jj6=0,theproblem
v(y)+A(
0
;D
y
)v(y) = 0; y>0;
B
j (
0
;D
y
)v(0) = g
j
; j=1;:::;m
admitsauniquesolutionu2C
0 (R
+
;E)foreach(g
1
;:::;g
m )
T
2E m
.
Inthefollowingweareinterestedin theL
1
-theoryoftheaboveproblem; morespecically,
for>
A
weconsiderthefollowingproblem:
Given 2
, f 2 L
1 (R
n +1
+
;E) and g
j 2 W
2m m
j
1
(R n+1
+
;E) for j 2 f1;:::;mg, nd
u2W 2m
1 (R
n+1
+
;E)whichsatises
u+A(D)u = f; in R n+1
+ (5)
B
j
(D)u = g
j
; on @R
n+1
+
;j=1;:::;m:
(6)
To this end, forA(D) and B
j
(D)dened asabove, wedene an operator A in L
1 (R
n+1
+
;E)
associated to the boundary value problem (5) and (6) with g
j
= 0 for all j 2 f1;:::;mg
bymeans of A =A
min
, where A
min : D(A
min )! L
1 (R
n+1
+
;E) m
j=1 W
2m mj
1
(R n+1
+
;E) is
denedby
D(A
min
) := W 2m
1 (R
n+1
+
;E)
A
min u :=
0
B
B
B
@ A(D)
B
1 (D)
.
.
.
B
m (D)
1
C
C
C
A u:
Moreover,wesetg:=(g
1
;:::;g
m )
T
. For2
,f 2L
1 (R
n+1
+
;E)andg
j 2W
2m mj
1
(R n+1
+
;E)
forj=1;:::;m,ourboundaryvalueproblem canrewrittenas
Jv+Av=
f
g
; (7)
T
Theorem3.1. Let A(D) be aparameter-elliptic operator oforder2mandangle of ellipticity
A
. Let >
A
. For j 2 f1;:::;mg let B
j
(D) be boundary operators of order m
j
< 2m.
AssumethattheLopatinskii-Shapiroconditionholds. LetEbeaBanachspace,f 2L
1 (R
n+1
+
;E)
and g
j 2W
2m mj
1
(R n+1
+
;E) for j =1;:::;m. Let 2
and let A be dened as above.
Then thereexistsauniquefunction u2W 2m 1
1 (R
n+1
+
;E)\D(A)satisfying
Ju+Au=
f
g
: (8)
Moreover, uisgiven by
u=P(+A
R n+1)
1
E
0 f+
m
X
j=1 R
j
f+
m
X
j=1 S
j
g
j
; (9)
where R j
and S
j
are kernel operators as dened in Propositions 6.8 and 6.9 of [DHP01].
Furthermore, there existsaconstantC >0suchthat for 0jj2m 1and 2
we
have
j 1
jj
2m
D
uj
L
1 (R
n+1
+
;E)
C[jfj
L
1 (R
n+1
+
;E) +
m
X
j=1
j( +jj 1
m
) 2m m
j
2
g
j j
L
1 (R
n+1
+
;E)
+ m
X
j=1
j( +jj 1
m
) 2m m
j 1
2
D
y g
j j
L1(R n+1
+
;E) ]:
Proof. ThefollowingproofisamodicationoftheargumentsgivenintheproofofTheorem6.10
of[DHP01]totheL
1
-situation. Indeed,notethatunderthegivenassumptionstheproblem(8)
hasauniquesolutionu2W 2m 1
1 (R
n+1
+
;E)\D(A)ifandonlyifudened asin(9)belongsto
W 2m 1
1 (R
n+1
+
;E)\D(A).
Givenf 2L
1 (R
n+1
+
;E),itfollowsfromProposition2.1(anditsproofin case>=2)that
thersttermontherighthandsideof(9)belongstotherequiredregularityclass.
Inordertotreatthesecond term,noticethatbyProposition6.6of[DHP01]thekernelk R;j
ofR j
satisesfor0jj2m 1and2
anestimateoftheform
jD
K R;j
(;y;y 0
)j
L 1
(R n
;B (E)) Cjj
2m+jj+1
2m
p n+1
2m+n;jj (cjj
1
2m
(y+y 0
)); y;y 0
>0:
(10)
The abovekernelestimates allowus to deriveL 1
-estimates for thesecond termon the right
hand sideof (9) viathefollowingsimplelemma on L 1
-continuity ofintegraloperatorsacting
inhalf spaces.
Lemma3.2. LetT beanintegral operator inL
1 (R
n+1
+
;E)of the form
(Tf)(x;y)= 1
Z
0 Z
R n
k(x x 0
;y;y 0
)f(x 0
;y 0
)dx 0
dy 0
; x2R n
;y>0;
wherek:R n
R
+ R
+
!B(E)isameasurablefunction. If
sup
y 0
>0 1
Z
0
jk(;y;y 0
)j
1
dy=:M <1;
thenT 2L(L
1 (R
n+1
+
;E))andjTj
L(L
1 (R
n+1
;E))
M.
Theproofof Lemma3.2 isconsists onlyofan applicationof Young'sandHolder'sinequality.
Combiningestimate(10)withLemma3.2 itfollowsthat
j 1
jj
2m
D
R j
fj
L1(R n+1
+
;E) Cjfj
L1(R n+1
+
;E) :
Similarly,byProposition6.8of[DHP01] thekernelk S;j
ofS
j
satisesfor0jj2m 1
and2
anestimateoftheform
jD
K S;j
(;y
0
)j
L 1
(R n
;B (E)) Cjj
2m+jj+1
2m
p n+1
2m+n;jj (cjj
1
2m
y 0
); y 0
>0:
Again,togetherwithLemma 3.2,thisestimateimpliesthat
j 1
jj
2m
D
S j
g
j j
L1(R n+1
+
;E)
C (
m
X
j=1
j( +jj 1
m
) 2m m
j
2
g
j j
L1(R n+1
+
;E)
+ m
X
j=1
j( +jj 1
m
) 2m m
j 1
2
D
y g
j j
L
1 (R
n+1
+
;E) ):
Thisprovestheassertion.
Foraparameter-ellipticoperator A(D)of order2mand angle ofellipticity
A
, wedenethe
L
1 (R
n+1
+
;E)-realizationA 0
B
oftheboundaryvalueproblem(8)withg=0as
A 0
B
u := A(D)u (11)
D(A 0
B
) := fu2W 2m
1 (R
n+1
+
;E);B
j
(D)u=0 forall j=1;:::;mg (12)
andset
A
B :=A
0
B :
ItfollowsfromTheorem3.1that (+A
B
)isinvertibleforall2
with>
A
andthat
(+A
B )
1
=P(+A
R n+1)
1
E
0 +
m
X
j=1 R
j
:
Theorem3.1alsoimpliesthat P
(A)andthatj(+A
B )
1
jMfor2
with
>
A
. Wethushavethefollowingcorollary.
Corollary3.3. Suppose that
A
<
2
. Then A
B
generates an analytic C
0
-semigroup on
L
1 (R
n+1
+
;E).
References
[ADN59] Agmon,S.,Douglis,A.,Nirenberg,L.: Estimatesneartheboundaryforsolutionsofellipticpartial
dierentialequationssatisfyinggeneralboundaryconditions,I.Comm.PureAppl.Math.22(1959),
623{727.II.Comm.PureAppl.Math.17(1964),35-92.
[Ama83] Amann, H.: Dualsemigroups and second orderlinear elliptic boundary valueproblems. Israel J.
Math.45(1983),225-254.
[Ama01] Amann,H.:Ellipticoperatorswithinnite-dimensionalstatespaces.J.Evol.Equ.1(2001),143-188.
[DHP01] Denk,R.,Hieber, M.,Pruss,J.: R-boundedness,Fouriermultipliersand problemsofelliptic and
[DiB91] DiBlasio, G.: Analytic semigroupsgenerated byelliptic operatorsinL 1
and parabolicequations.
OsakaJ.Math.28(1991),367-384.
[Gui93] Guidetti,D.:OnellipticsystemsinL 1
.OsakaJ.Math.30(1993),397-429.
[Tan97] Tanabe, H.: FunctionalAnalytic Methodsfor PartialDierential Equations.Marcel Dekker, New
York,1997.
Universi
atRegensburg,Naturwissenschaftliche
Fakult
at I -Mathematik,D-93040Regensburg,
Germany
E-mailaddress: robert.denk@mathematik.u ni- rege nsb urg. de
Technische
Universit
atDarmstadt,FachbereichMathematik,Schlossgartenstr. 7,D-64289Darm-
stadt,Germany
E-mailaddress: hieber@mathematik.tu-dar mst adt. de
Martin-Luther-Universit
atHalle-Wittenberg,FachbereichMathematikundInformatik,Institut
f
urAnalysis,Theodor-Lieser-Strae5,D-06120Halle,Germany
E-mailaddress: anokd@volterra.mathemati k.u ni-h all e.de
