Lp-LqESTIMATES FOR ELECTROMAGNETIC HELMHOLTZ
EQUATION.
ANDONI GARCIA
Abs ac . In space dimension n≥3, we conside he elec omagne ic Sch ¨odinge
Hamil onian H= (∇−iA(x))2−Vand he co esponding Helmhol z equa ion
(∇ − iA(x))2u+u−V(x)u= in Rn.
We ex end he well known Lp-Lqes ima es o he solu ion o he ee Helmhol z
equa ion o he case when he elec omagne ic hamil onian His conside ed.
1. In oduc ion
This pape is de o ed o he s udy o some es ima es o he Helmhol z equa ion
wi h elec omagne ic po en ial. A e y na u al ques ion is o ex end he known
esul s o he Helmhol z equa ion wi h cons an coe icien s o he case when we
conside he pe u bed Helmhol z equa ion by a po en ial. Ou goal will be o
ex end he well known Lp-Lqes ima es o he ee Helmhol z equa ion gi en in
[KRS], [CS], [Gu ] and [Gu 1] o he case when we pe u b he equa ion wi h
an elec omagne ic po en ial. Mo e p ecisely, condi ions on he elec ic and he
magne ic pa o he po en ial will be gi en in o de o ensu e ha he es ima es
emain ue. The Lp-Lqes ima es o he ee Helmhol z equa ion a e he ollowing:
(1.1) kukLq(Rn)≤Ck kLp(Rn),
whe e uis a solu ion o
(1.2) ∆u+ (τ±i)u=− τ, > 0.
The exponen s pand qin (1.1) ha e o e i y some speci ic condi ions ha will be
speci ied la e on. He e Ccan depend on τ,p,qand nand is independen o .
The in es iga ion o he es ima es (1.1) s a ed in [KRS], whe e he s udy o
uni o m Sobole es ima es o cons an coe icien second o de di e en ial ope a-
o s was accomplished. La e on, in [Gu ], [Gu 1] (See also [CS]) he ange o he
exponen s pand qwhe e he es ima es (1.1) hold was de e mined.
Mo eo e , in he pu ely elec ic case i.e., o Sch ¨odinge Hamil onians o he
ype ∆ + V(x), whe e V:Rn→Ris he elec ic po en ial, some posi i e esul s
we e gi en in [RV].
The e o e he aim o he pape is o p o e he co esponding es ima es (1.1) in
he case when he elec omagne ic Sch ¨odinge hamil onian is conside ed .
In he i s pa we will p o e ha he exis ing esul s o he ee Helmhol z
equa ion can be ex ended o he pe u bed equa ion i we impose p ecise decay
condi ions a in ini y o he elec ic and he magne ic po en ial. This can be done
wi hou assuming smallness, nei he o he elec ic pa , no o he magne ic pa .
As i will be shown, o he elec omagne ic case, he ange o he exponen s p
and qwhe e he es ima es (1.1) a e alid is no he same as he one o he ee
Da e: No embe 15, 2021.
2000 Ma hema ics Subjec Classi ica ion. 35J10, 35L05, 58J45.
Key wo ds and ph ases. dispe si e equa ions, Helmhol z equa ion, magne ic po en ial.
The au ho is suppo ed by he g an BFI06.42 o he Basque Go e nmen .
1
This is he accep ed manusc ip o he a icle ha appea ed in inal o m in Jou nal o Ma hema ical Analysis and
Applica ions 384(2) : 409-420 (2011), which has been published in inal o m a h ps://doi.o g/10.1016/
j.jmaa.2011.05.080. © 2011 Else ie unde CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-
nd/4.0/)
2 ANDONI GARCIA
case, hence in o de o go u he , we deal wi h he Helmhol z equa ion wi h pu ely
elec ic po en ial, ying o ob ain esul s in he same egion o boundedness o he
ee equa ion.
The e o e, we conside he elec omagne ic Sch ¨odinge hamil onian Ho he
o m
(1.3) H= (∇ − iA(x))2−V(x),
and he co esponding Helmhol z equa ion in dimensions n≥3, namely,
(1.4) (∇ − iA(x))2u+u−V(x)u= in Rn.
He e A: (A1, . . . , An) : Rn→Rnis he magne ic po en ial and V(x) : Rn→Ris
he elec ic po en ial. Since now on, we deno e by
∇A=∇ − iA, ∆A=∇2
A.
The magne ic po en ial Ais a ma hema ical cons uc ion which desc ibes he in-
e ac ion o pa icles wi h an ex e nal magne ic ield. The magne ic ield B, which
is he physically measu able quan i y, is gi en by
(1.5) B∈ Mn×n, B =DA −(DA) ,
i.e. i is he an i-symme ic g adien o he ec o ield A(o , in geome ical e ms,
he di e en ial dA o he 1- o m which is s anda dly associa ed o A). In dimension
n= 3 he ac ion o Bon ec o s is iden i ied wi h he ec o ield cu lA, namely
(1.6) B = cu lA× n = 3,
whe e he c oss deno es he ec o ial p oduc in R3.
One o he mos in e es ing ac s ela ed o he Lp-Lqes ima es o he elec o-
magne ic hamil onian is ha i seems ha in o de o conclude he boundedness o
he solu ion, one should be able o bound he i s o de e m ha appea s when he
hamil onian His expanded. Mo e conc e ely, when he e m (∇ − iA(x))2in (1.4)
is expanded, a i s o den e m, namely A· ∇, comes ou and i is well known ha
he e a e no Lp-Lqes ima es o he g adien o he solu ion o he ee Helmhol z
equa ion,
(1.7) ∆u+u=− in Rn.
We will p oceed in he ollowing way. Le us conside he modi ied Helmhol z
equa ion wi h elec omagne ic po en ial and ixed equency τ= 1. I eads as
ollows
(1.8) (∇ − iA(x))2u+ (1 ±i)u−V(x)u= in Rn, 6= 0.
Rema k 1.1.Fo con enience we will deal only wi h he case τ= 1, in con as
wi h he case o gene al τ > 0.
We will p o e he co esponding Lp-Lqes ima es, independen o , o he so-
lu ion o (1.8). The independence o will imply ha he esul emains ue o
he solu ion o (1.4). This can be seen in [IS].
Ou me hod is a mix u e o an a p io i es ima e and pe u ba i e a gumen s.
This is wha allows us o a oid smallness condi ions in he po en ials. Simila a gu-
men s ha e been used in he se ing o he ee Sch ¨odinge equa ion, as can be seen
in [BPST] and [DFVV]. Along he p oo ou basic ools will be he co esponding
Lp-Lqes ima es and a L2-local es ima e o he solu ion o he ee Helmhol z equa-
ion, oge he wi h an a p io i es ima e o he solu ion o he modi ied Helmhol z
equa ion wi h elec omagne ic po en ial (1.8).
Lp-LqESTIMATES FOR ELECTROMAGNETIC HELMHOLTZ EQUATION. 3
Be o e we desc ibe he esul s ha we a e going o use du ing ou p oo , le us
in oduce some basic no a ion. Fo :Rn→Cwe de ine he Mo ey-Campana o
no m as
(1.9) ||| |||2:= sup
R>0
1
RZ|x|≤R
| |2dx.
Mo eo e , we deno e, o j∈Z, he annulus C(j) by
C(j) = {x∈Rn: 2j≤ |x| ≤ 2j+1},
(1.10) N( ) := X
j∈Z 2j+1 ZC(j)
| |2dx!1/2
,
and we easily see he duali y ela ion
Z gdx ≤ |||g||| · N( ).
These no ms we e in oduced by Agmon and H¨o mande in [AH].
Du ing he exposi ion, he unca ed e sion o he no ms appea ing abo e will be
necessa y. We will deno e hem espec i ely by
(1.11) ||| |||2
0:= sup
R≥1
1
RZ|x|≤R
| |2dx,
(1.12) N0( ) := X
j≥0 2j+1 ZC(j)
| |2dx!1/2
.
Le us also deno e by L2
β(Rn), o β∈R, he Hilbe space o all unc ions such
ha (1 + |x|)β is squa e in eg able o e Rn. The no m in his space is deno ed by
k·kβ. T i ially we ha e ha i β > 1/2 and ∈L2
β(Rn), hen N0( )<+∞.
As we ha e said, pa o ou me hod is pe u ba i e, so in o de o be able o
s a , le us emind wha is known o he ee Helmhol z equa ion. Fi s ly, we a e
going o s a e he esul conce ning he Lp-Lqes ima es which appea s in [Gu ]
and [Gu 1]. Le be
A=n+ 3
2n,n−1
2n, A0=n+ 1
2n,n−3
2n
B=n2+ 4n−1
2n(n+ 1) ,n−1
2n, B0=n+ 1
2n,n2−2n+ 1
2n(n+ 1)
and ∆(n), he se o poin s o [0,1] ×[0,1] gi en by
(1.13) ∆(n) = 1
p,1
q∈[0,1]2:2
n+ 1 ≤1
p−1
q≤2
n,1
p>n+ 1
2n,1
q<n−1
2n.
The se ∆(n) is he apezium ABB0A0wi h he closed line segmen s AB and B0A0
emo ed (see Figu e 1).
4 ANDONI GARCIA
Figu e 1. ∆(n), n≥3.
Now, we a e in condi ions o ecall he exis ing esul o he Helmhol z equa ion
wi h cons an coe icien s.
Rema k 1.2.In [Gu ] (See also [Gu 1]), es ima es o he solu ion o he equa ion
pe u bed wi h gene als τ > 0 and > 0, a e gi en, namely,
(1.14) ∆u+ (τ+i)u=−F, τ, > 0.
The special case whe e he poin (1/p, 1/q) lies on he open segmen AA0and on he
duali y line 1/q = 1 −1/p in Figu e 1 was p e iously ob ained in [[KRS], Theo em
2.2 and 2.3 espec i ely].
Recall ha we will only deal wi h he case o ixed equency τ= 1. The Theo em
eads as ollows.
Theo em 1.1. Le u be a solu ion o
(1.15) ∆u+ (1 + i)u=−F, > 0.
Then, he e exis s a cons an C, independen o , such ha
(1.16) kukLq(Rn)=k(∆ + (1 + i))−1FkLq(Rn)≤CkFkLp(Rn)
when (1
p,1
q)∈∆(n),n≥3.
As we men ioned, ano he ool ha will be c ucial in he p oo is an L2-local
es ima e, which bounds he unca ed Mo ey-Campana o no m o he solu ion o
he ee equa ion, de ined in (1.11), in e ms o he Lpno m o he RHS da a. This
Theo em also appea s in [RV], [Gu ] and [Gu 1]. The s a emen is he ollowing.
Theo em 1.2. Le u be a solu ion o
(1.17) ∆u+ (1 + i)u=−F, > 0.
I
(i) n= 3 o 4and 1
n+1 ≤1
p−1
2<1
2, o
(ii) n≥5and 1
n+1 ≤1
p−1
2≤2
n,
Lp-LqESTIMATES FOR ELECTROMAGNETIC HELMHOLTZ EQUATION. 5
hen, he e exis s a cons an C, independen o , such ha
(1.18) sup
R≥11
RZBR
|u(x)|2dx1/2
≤CkFkLp(Rn).
The las esul conce ns an a p io i es ima e o he solu ion o he pe u bed
equa ion. I s a es ha , gi en p ecise condi ions, wi hou assuming smallness, on
he decay a in ini y o he he elec ic po en ial, he magne ic po en ial and he
adial de i a i e o he elec ic po en ial, we ha e a p ecise con ol o k∇Auk−(1+δ)
2
and kuk−(1+δ)
2. This esul appea s in [IS] .
The Theo em is he ollowing one.
Theo em 1.3. Le n≥3, and u∈C∞
0be a solu ion o
(∇ − iA(x))2u+ (1 ±i)u−V(x)u= , 6= 0.
Le us assume ha :
(V) V(x)can be decomposed as V(x) = V1(x) + V2(x), and he e exis s ic ly
posi i e cons an s Cand µsuch ha
(V1)
|V1(x)| ≤ C|x|−µ,(∂ V1)(x)≤C|x|−1−µ,|x| ≥ 1,
(V2)
|V2(x)| ≤ C|x|−1−µ,|x| ≥ 1,
(B)
|B(x)| ≤ C|x|−1−µ,|x| ≥ 1.
Choose δ > 0su icien ly small (so ha δ≤µ/2,δ < 1). Then, he e exis s a
cons an C=C(δ), which depends uni o mly in , such ha he ollowing a p io i
es ima e holds
(1.19) k∇Auk−(1+δ)
2+kuk−(1+δ)
2≤Ck k1+δ
2.
Rema k 1.3.Obse e ha , since he elec ic po en ial Vand magne ic po en ial
Amus sa is y he condi ions o he heo em, singula i ies a he o igin a e no
allowed.
Rema k 1.4.No ice ha he unique con inua ion p ope y holds o he di e en ial
ope a o H= (∇−iA(x))2−V(x), as can be seen in [R]. The assump ions (V), (V1),
(V2) and (B), oge he wi h his obse a ion implies ha he limi ing abso p ion
p inciple holds.
Rema k 1.5.The condi ions on he decay o he elec ic po en ial Vand he
magne ic ield Bgi en by (V1), (V2) and (B) espec i ely, a e su icien o us, due
we ha e ixed he equency τ= 1. I can be seen in [IS], ha he esul is ue
p o ided τand belong o he ollowing se deno ed by K
(1.20) K={k=τ+i ∈C/τ ∈(τ0, τ1), ∈(0, 1)},
whe e 0 < τ0< τ1<∞and 0 < 1<∞.
Hence, he c i ical case τ= 0 is excluded. This si ua ion equi es mo e decay
o bo h po en ials ( ypically hxi−(2+), > 0, o Band Vi n= 3 and hxi−2
o n≥4, whe e hxi= (1 + |x|2)1/2), in o de o ob ain a p io i es ima es o he
solu ion o he pe u bed equa ion, as can be seen in [F], whe e Mo ey -Campana o
ype es ima es, uni o m in , a e ob ained o τ≥0.
No e also ha his esul gi es an a p io i es ima e wi hou assuming smallness
nei he o he non epulsi e componen o he elec ic ield no o he apping
componen o he magne ic ield, de ined by Bτ:= x
|x|B.
6 ANDONI GARCIA
Once we ha e desc ibed all he ools which a e going o be used, i is necessa y
o in oduce he egion whe e we a e able o ex end he known esul s o he ee
Helmhol z equa ion o he case when elec omagne ic pe u ba ions a e conside ed.
Du ing he discussion, i will appea a sub egion o ∆(n), o n≥3, which will be
deno ed by ∆0(n), gi en by
(1.21) ∆0(n) = 1
p,1
q∈∆(n) : 1
n+ 1 ≤1
p−1
2,1
n+ 1 ≤1
2−1
q.
The se ∆0(n) is he solid iangle de e mined by he poin s Q,Q0and Q00 (see
Figu e 2).
This will be he egion o boundedness o he pe u bed Helmhol z equa ion.
Figu e 2. ∆0(n), n≥3.
Rema k 1.6.Fo he case o he pe u bed elec omagne ic equa ion we a e no
able o ob ain a posi i e esul o boundedness o he whole egion ∆(n), since we
ha e no con ol o he g adien e m, namely A·∇, ou side ∆0(n). Howe e , when
we se A≡0, and conside he elec ic hamil onian, he esul s can be ex ended
ou side ∆(n) by imposing mo e decay on V.
2. Elec omagne ic Helmhol z Equa ion.
In his sec ion we will gi e he p ecise s a emen and he p oo o he heo em,
whe e we ex end he known esul o he ee Helmhol z equa ion o he case when
elec omagne ic pe u ba ions a e conside ed. The basic heo ems which will be
used along he p oo we e gi en in he sec ion 1, as well as he basic no a ion. Fi s
we will announce he esul o he gene al elec omagne ic case and a e wa ds, by
se ing A≡0, he elec ic case will be ea ed by ex ending ou p e ious esul .
Le us s a by conside ing he solu ion o he Helmhol z equa ion wi h elec-
omagne ic po en ial ha sa is ies ei he he ingoing o he ou going Somme eld
adia ion condi ion. Fo n≥3, i eads,
(2.1) (∇ − iA(x))2u+u−V(x)u= in Rn,
whe e A: (A1, . . . , An) : Rn→Rnis he magne ic po en ial and V(x) : Rn→Ris
he elec ic po en ial.
Lp-LqESTIMATES FOR ELECTROMAGNETIC HELMHOLTZ EQUATION. 7
We will assume ha he magne ic po en ial Asa is ies he Coulomb gauge con-
di ion
(2.2) ∇ · A= 0.
We will p o e Lp-Lqes ima es o he solu ion o he equa ion (2.1). In o de
o do ha we will conside he solu ion o (2.1) as he solu ion o he modi ied
Helmhol z elec omagne ic equa ion,
(2.3) (∇ − iA(x))2u+ (1 ±i)u−V(x)u= in Rn, 6= 0,
ia limi ing abso p ion p inciple, by aking he limi o he solu ion o (2.3) when
goes o 0. We will ob ain he co esponding Lp-Lqes ima es, independen o ,
o he solu ion o (2.3), so hese will emain ue o he solu ion o (2.1). This is
gua an eed by he esul s appea ing in [IS].
The goal is o de e mine he egion o pand qwhe e he solu ion o (2.3) sa is ies
Lp-Lqes ima es, namely,
(2.4) kukLq(Rn)≤Ck kLp(Rn).
wi h Cindependen o .
The main esul o his sec ion is he ollowing.
Theo em 2.1. Le u be a solu ion o
(2.5) (∇ − iA(x))2u+ (1 ±i)u−V(x)u= in Rn, n ≥3, 6= 0.
Le Vand Asa is y (V),(V1),(V2)and (B)in Theo em 1.3, and suppose ha he e
exis cons an s C, µ > 0such ha
(2.6) |A(x)| ≤ C
(1 + |x|)1+µ,|V(x)| ≤ C
(1 + |x|)1+µ.
Then, he e exis s a cons an C, independen o , such ha
(2.7) kukLq(Rn)≤Ck kLp(Rn),
when 1
p,1
q∈∆0(n).
P oo .
Rema k 2.1.No ice ha he e a e no smallness assump ion nei he o he elec ic
po en ial Vno o he magne ic po en ial A. Also we bound he solu ion only
assuming sho - ange decay o Vand A. As we said in Rema k 1.3, singula i ies
a he o igin o Vand Aa e no conside ed.
S ep 1. I will be p o ed ha , whene e 1
p,1
q∈∆0(n) hen we ge he ollowing
(2.8) kukLq(Rn)≤Ck k1+δ
2.
Le ube a solu ion o (2.5). Since ∇ · A≡0, we can expand he e m (∇ − iA)2in
he ollowing o m
(2.9) (∇ − iA)2u= ∆u−2iA · ∇Au+|A|2u.
This is he key poin in o de o conside he elec omagne ic case as a pe u ba ion
o he ee equa ion. As can be seen, he e appea e ms o o de ze o and o de
one. So by passing e ms o he RHS, we ha e ha uis solu ion o he ollowing
equa ion
(2.10) ∆u+ (1 ±i)u= + 2iA · ∇Au− |A|2u+V u.
8 ANDONI GARCIA
Now we apply he esul coming om Theo em 1.2. By conside ing he dual es i-
ma e o (1.18), we ge ha , i 1
p,1
q∈∆0(n) i holds
kukLq(Rn)=k(∆ + (1 ±i))−1( + 2iA · ∇Au− |A|2u+V u)kLq(Rn)
(2.11)
≤C(N0( ) + N0(2iA · ∇Au) + N0(|A|2u) + N0(V u)).
wi h Cindependen o and N0de ined in (1.12).
Now we con inue by ea ing he e ms appea ing on he RHS o (2.11). Fi s
we deal wi h he e m N0(2iA · ∇Au). F om (2.6), we ge ha his e m can be
bounded as
N0(2iA · ∇Au)2=CX
j≥0
2j+1 ZC(j)
|A· ∇Au|2dx(2.12)
≤CX
j≥0
2jZC(j)
|A|2|∇Au|2dx
≤CX
j≥0
2j(δ−2µ)ZRn
|∇Au|2
(1 + |x|)1+δdx.
The e o e, we inally ha e
(2.13) N0(2iA · ∇Au)≤Ck∇Auk−(1+δ)
2.
Le us con inue wi h he e m N0(|A|2u). As be o e, om (2.6), we can ea
his e m as ollows
N0(|A|2u)2=X
j≥0
2j+1 ZC(j)
||A|2u|2dx(2.14)
≤CX
j≥0
2jZC(j)
|A|4|u|2dx
≤CX
j≥0
2j(−2+δ−4µ)ZRn
|u|2
(1 + |x|)1+δdx.
Hence, we ge
(2.15) N0(|A|2u)≤Ckuk−(1+δ)
2.
The las e m is N0(V u). Simila ly, we ob ain om (2.6)
N0(V u)2=X
j≥0
2j+1 ZC(j)
|V u|2dx(2.16)
≤CX
j≥0
2jZC(j)
|V|2|u|2dx
≤CX
j≥0
2j(δ−2µ)ZRn
|u|2
(1 + |x|)1+δ.dx
So, i e i ies
(2.17) N0(V u)≤Ckuk−(1+δ)
2.
Lp-LqESTIMATES FOR ELECTROMAGNETIC HELMHOLTZ EQUATION. 9
F om (2.11), (2.13), (2.15) and (2.17), we ge ha , whene e 1
p,1
q∈∆0(n), he
Lqno m o ucan be bounded as
kukLq(Rn)=k(∆ + (1 ±i))−1( + 2iA · ∇Au− |A|2u+V u)kLq(Rn)
(2.18)
≤C(N0( ) + N0(2iA · ∇Au) + N0(|A|2u) + N0(V u)).
≤CN0( ) + C1k∇Auk−(1+δ)
2+C2kuk−(1+δ)
2.
Now we emind he a p io i es ima e gi en by Theo em 1.3, which ensu es ha ,
unde he assump ions (V), (V1), (V2) and (B) o Vand A espec i ely, he e
exis s a cons an C, such ha he ollowing holds
(2.19) k∇Auk−(1+δ)
2+kuk−(1+δ)
2≤Ck k1+δ
2.
Rema k 2.2.The cons an Cwhich appea s he e depends uni o mly in .
So, om (2.18) and (2.19), we ge
(2.20) kukLq(Rn)≤CN0( ) + C1k k1+δ
2.
Mo eo e , i holds ha N0( ) can be bounded as
(2.21) N0( )≤Ck k1+δ
2.
This, oge he wi h (2.20) concludes ha , i uis a solu ion o (2.5), i e i ies
(2.22) kukLq(Rn)≤Ck k1+δ
2.
The e o e, we ge he desi ed es ima e.
S ep 2. By applying duali y o he las es ima e, we ge ha i 1
p,1
q∈∆0(n),
(2.23) kuk−(1+δ)
2≤Ck kLp(Rn)
Rema k 2.3.The adjoin ope a o is he one co esponding o ∓. Since we can do
he same a gumen o bo h signs, all he compu a ions a e jus i ied.
S ep 3. This is he inal s ep in he p oo . As we said in he in oduc ion he main
di icul y will be o handle he i s o de e m gi en by A· ∇Au, since he e a e
no Lp-Lqes ima es o he g adien o he solu ion o he ee Helmho z equa ion.
Ins ead o conside ing his no m ou a gumen will end up by ea ing k∇Auk−(1+δ)
2,
and his no m will be unde con ol in he egion ∆0(n).
Consequen ly, we ha e ha i 1
p,1
q∈∆0(n), om he Lp-Lqes ima es o he
solu ion o he ee equa ion, namely (1.16), gi en in Theo em 1.1, and p oceeding
as in he s ep 1 o he e ms 2iA · ∇Au,|A|2uand V u, we ge
kukLq(Rn)=k(∆ + (1 ±i))−1( + 2iA · ∇Au− |A|2u+V u)kLq(Rn)
(2.24)
≤Ck kLp(Rn)+C1(N0(2iA · ∇Au) + N0(|A|2u) + N0(V u)).
whe e Cand C1do no depend on .
Le us emind ha , om (2.6) i holds
N0(2iA · ∇Au)≤C1k∇Auk−(1+δ)
2,(2.25)
N0(|A|2u)≤C2kuk−(1+δ)
2,
N0(V u)≤C3kuk−(1+δ)
2.
The e o e, we ge
(2.26) kukLq(Rn)≤Ck kLp(Rn)+C1k∇Auk−(1+δ)
2+C2kuk−(1+δ)
2.
