Equivalence of viscosity and weak solutions for a p-parabolic equation
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Equi alence o iscosi y and weak solu ions o a p-pa abolic equa ion
© 2021 The Au ho (s)
Published e sion
Sil akoski, Ja kko
Sil akoski, J. (2021). Equi alence o iscosi y and weak solu ions o a p-pa abolic equa ion.
Jou nal o E olu ion Equa ions, 21(2), 2047-2080. h ps://doi.o g/10.1007/s00028-020-00666-y
2021
J. E ol. Equ.
© 2021 The Au ho (s)
h ps://doi.o g/10.1007/s00028-020-00666-y
Jou nal o E olu ion
Equa ions
Equi alence o iscosi y and weak solu ions o a p-pa abolic
equa ion
Ja kko Sil akoski
Abs ac . We s udy he ela ionship o iscosi y and weak solu ions o he equa ion
∂ u−Δpu= (Du),
whe e p>1and ∈C(RN)sa is ies sui able assump ions. Ou main esul is ha bounded iscosi y
supe solu ions coincide wi h bounded lowe semicon inuous weak supe solu ions. Mo eo e , we p o e he
lowe semicon inui y o weak supe solu ions when p≥2.
1. In oduc ion
A classical solu ion o a pa ial di e en ial equa ion is a smoo h unc ion ha sa is-
ies he equa ion poin wise. Since many equa ions ha appea in applica ions admi no
such solu ions, a mo e gene al class o solu ions is needed. One such class is he ex en-
si ely s udied dis ibu ional weak solu ions de ined by in eg a ion by pa s. Ano he
is he celeb a ed iscosi y solu ions based on gene alized poin wise de i a i es. When
bo h classes o solu ions can be meaning ully de ined, i is na u ally c ucial ha hey
coincide. This has been p o usely s udied s a ing om [10]. In [12], he equi alence
o solu ions was p o ed o he pa abolic p-Laplacian. The objec i e o he p esen
wo k is o p o e his equi alence in a di e en way while also allowing he equa ion
o depend on a i s -o de e m. To he bes o ou knowledge, he p oo is new e en
in he homogeneous case, a leas when 1 <p<2.
Mo e p ecisely, we s udy he pa abolic equa ion
∂ u−Δpu= (Du), (1.1)
whe e 1 <p<∞and ∈C(RN)sa is ies a ce ain g ow h condi ion, o de ails
see Sec . 2. We show ha bounded iscosi y supe solu ions o (1.1) coincide wi h
bounded lowe semicon inuous weak supe solu ions. Mo eo e , we p o e he lowe
Ma hema ics Subjec Classi ica ion: 35K92, 35J60, 35D40, 35D30, 35B51
Keywo ds: Compa ison p inciple, G adien e m, Pa abolic p-Laplacian, Viscosi y solu ion, Weak
solu ion.
J. Sil akoski J. E ol. Equ.
semicon inui y o weak supe solu ions in he ange p≥2 unde sligh ly s onge
assump ions on .
To show ha iscosi y supe solu ions a e weak supe solu ions, we apply he ech-
nique in oduced by Julin and Juu inen [11]. In con as o [12], we do no employ he
uniqueness machine y o iscosi y solu ions. Ins ead, ou s a egy is o app oxima e a
iscosi y supe solu ion uby i s in -con olu ion uε. I is s aigh o wa d o show ha
uεis s ill a iscosi y supe solu ion in a smalle se . This and he poin wise p ope -
ies o he in -con olu ion imply ha uεis also a weak supe solu ion in he smalle
se . Fu he mo e, i ollows om Caccioppoli’s es ima es ha uεcon e ges o uin
a sui able Sobole space. I hen emains o pass o he limi o see ha uis a weak
supe solu ion.
To show ha weak supe solu ions a e iscosi y supe solu ions, we apply he a gu-
men om [12] ha is based on he compa ison p inciple o weak solu ions. Howe e ,
we could no ind a e e ence o compa ison p inciple o Eq. (1.1). The e o e, we
gi e a de ailed p oo o such a esul .
To p o e he lowe semicon inui y o weak supe solu ions, we adap he s a egy
o [17]. Fi s , we p o e es ima es o he essen ial sup emum o a subsolu ion using
Mose ’s i e a ion echnique. Then, we use hose es ima es o deduce ha a supe solu-
ion is lowe semicon inuous a i s Lebesgue poin s.
The equi alence o iscosi y and weak solu ions o he p-Laplace equa ion and
i s pa abolic e sion was i s p o en in [12]. A di e en p oo in he ellip ic case
was ound in [11]. Recen ly he equi alence o solu ions has been s udied o a ious
equa ions. These include he no malized p-Poisson equa ion [1], a non-homogeneous
p-Laplace equa ion [22] and he no malized p(x)-Laplace equa ion [25]. Mo eo e ,
in [24] he equi alence is shown o he adial solu ions o a pa abolic equa ion. We
also men ion ha an unpublished e sion o [18] applies [11] o ske ch he equi alence
o solu ions o (1.1) in he homogeneous case when p≥2.
Compa ison p inciples o quasilinea pa abolic equa ions ha e been s udied by
se e al au ho s. In [13], compa ison is p o en o ∂ u−Δpu+ (u,x, )=0 when
p>2 and is a con inuous unc ion such ha | (u,x, )|≤g(u) o some g∈C1.
The homogeneous case o he p-pa abolic equa ion is conside ed also in [16] and he
gene al equa ion ∂ u−di A(x, ,Du)=0in[15]. Equa ions wi h g adien e ms
a e s udied o example in [2], whe e compa ison p inciple is shown o he equa ion
∂ u−Δpu−|Du|β=0 when p>2 and β>p−1. In he ecen pape s [4,5], bo h
posi i e esul s and coun e examples a e p o ided o he compa ison, s ong compa i-
son, and maximum p inciples o he equa ion ∂ u−Δpu−λ|u|p−2u− (x, )=0.
Fu he mo e, acco ding o [3], he equa ion ∂ u−Δpu=q(x)|u|αcan admi mul i-
ple solu ions wi h ze o bounda y alues when 0 <α<1.
The pape is o ganized as ollows. Sec ion 2con ains he p ecise de ini ions o weak
and iscosi y solu ions. In Sec . 3, we show ha weak supe solu ions a e iscosi y
supe solu ions, and he con e se is shown in Sec . 4. Finally, he lowe semicon inui y
o weak supe solu ions is conside ed in Sec . 5.
Equi alence o iscosi y and weak solu ions
2. P elimina ies
The symbols Ξand Ωa e ese ed o bounded domains in RN×Rand RN,
espec i ely. Fo 1< 2, we de ine he cylinde Ω 1, 2:= Ω×( 1, 2)and i s pa a-
bolic bounda y ∂pΩ 1, 2:= (Ω ×{ 1})∪(∂Ω ×( 1, 2]). Mo eo e , o T>0we
se ΩT:= Ω0,T.
The Sobole space W 1,p(Ω) con ains he unc ions u∈Lp(Ω) o which he dis-
ibu ional g adien Du exis s and belongs in Lp(Ω). I is equipped wi h he no m
uW1,p(Ω) := uLp(Ω) +DuLp(Ω) .
A Lebesgue measu able unc ion u:Ω 1, 2→Rbelongs o he pa abolic Sobole
space L p( 1, 2;W1,p(Ω)) i u(·, )∈W1,p(Ω) o almos e e y ∈( 1, 2)and he
no m
Ω 1, 2
|u|p+|Du|pdz1
p
is ini e. By dz, we mean in eg a ion wi h espec o space and ime a iables, i.e.,
dz =dx d . In eg al a e age is deno ed by
−
ΩT
udz:= 1
|ΩT|ΩT
udz.
G ow h condi ion
Unless o he wise s a ed, he unc ion ∈C(RN)is assumed o sa is y he g ow h
condi ion
| (ξ)|≤C (1+|ξ|β) o all ξ∈RN,(G1)
whe e C >0 and 1 ≤β<p.
De ini ion 2.1. (Weak solu ion) A unc ion u:Ξ→Ris a weak supe solu ion o
(1.1)inΞi u∈Lp( 1, 2;W1,p(Ω)) whene e Ω 1, 2Ξ, and
Ξ
−u∂ ϕ+|Du|p−2Du ·Dϕ−ϕ (Du)dz≥0
o all non-nega i e es unc ions ϕ∈C∞
0(Ω 1, 2).Fo weak subsolu ions, he inequal-
i y is e e sed and a unc ion is a weak solu ion i i is bo h a supe - and subsolu ion.
To de ine iscosi y solu ions o (1.1), we se o all ϕ∈C2wi h Dϕ= 0
Δpϕ:= |Dϕ|p−2Δϕ +p−2
|Dϕ|2D2ϕDϕ, Dϕ.
J. Sil akoski J. E ol. Equ.
De ini ion 2.2. (Viscosi y solu ion) A lowe semicon inuous and bounded unc ion
u:Ξ→Ris a iscosi y supe solu ion o (1.1)inΞi whene e ϕ∈C2(Ξ) and
(x0, 0)∈Ξa e such ha
⎧
⎪
⎪
⎨
⎪
⎪
⎩
ϕ(x0, 0)=u(x0, 0),
ϕ(x, )<u(x, )when (x, )= (x0, 0),
Dϕ(x, )= 0 when x= x0,
hen
lim sup
(x, )→(x0, 0)
x=x0∂ ϕ(x, )−Δpϕ(x, )− (Dϕ(x, ))≥0.
An uppe semicon inuous and bounded unc ion u:Ξ→Ris a iscosi y subsolu ion
o (1.1)inΞi whene e ϕ∈C2(Ξ) and (x0, 0)∈Ξa e such ha
⎧
⎪
⎪
⎨
⎪
⎪
⎩
ϕ(x0, 0)=u(x0, 0),
ϕ(x, )>u(x, )when (x, )= (x0, 0),
Dϕ(x, )= 0 when x= x0,
hen
lim in
(x, )→(x0, 0)
x=x0∂ ϕ(x, )−Δpϕ(x, )− (Dϕ(x, ))≤0.
A unc ion ha is bo h a iscosi y sub- and supe solu ion is a iscosi y solu ion.
I a unc ion ϕis like in he de ini ion o iscosi y supe solu ion, we say ha ϕ
ouches u om below a (x0, 0). The limi supe io in he de ini ion is needed because
he ope a o Δpis singula when 1 <p<2. When p≥2, he ope a o is degene a e
and he limi supe io disappea s.
3. Weak solu ions a e iscosi y solu ions
We show ha bounded, lowe semicon inuous weak supe solu ions o (1.1) a e is-
cosi y supe solu ions when 1 <p<∞and ∈C(RN)sa is ies he g ow h condi ion
(G1). One way o p o e his kind o esul s is by applying he compa ison p inciple
[12]. Howe e , we could no ind he compa ison p inciple o Eq. (1.1) in he li e a u e
and he e o e we p o e i i s . To his end, we i s p o e compa ison Lemmas 3.2
and 3.3 o locally Lipschi z con inuous . The local Lipschi z con inui y allows us
o abso b he i s -o de e ms in o he e ms ha appea due o he p-Laplacian, see
S ep 2 in p oo o Lemma 3.2. To deal wi h gene al , we ake a locally Lipschi z
con inuous app oximan δsuch ha − δL∞(RN)<δ/4T. Then, o sub- and
supe solu ions uand , we conside he unc ions
uδ:= u−δ
T− /2and δ:= +δ
T− /2.
Equi alence o iscosi y and weak solu ions
These unc ions will be sub- and supe solu ions o (1.1) whe e is eplaced by δ.
Since δis locally Lipschi z con inuous, i ollows om Lemmas 3.2 and 3.3 ha
uδ≤ δ. Le ing δ→0 hen yields ha u≤ .
Fo simila compa ison esul s, see [2, P oposi ion 2.1] and [13]. See also Chap e s
3.5 and 3.6 in [23] o he ellip ic case. A mino di e ence in ou esul s is ha ins ead
o equi ing ha bo h he subsolu ion and he supe solu ion ha e uni o mly bounded
g adien s, we only equi e his o he subsolu ion.
To p o e he compa ison p inciple, we need o use a es unc ion ha depends
on he supe solu ion i sel . Howe e , supe solu ions do no necessa ily ha e a ime
de i a i e. One way o deal wi h his is o use molli ica ions in he ime di ec ion. Fo
a compac ly suppo ed ϕ∈Lp(ΩT), we de ine i s ime-molli ica ion by
ϕ(x, )=R
ϕ(x, −s)ρ(s)ds,
whe e ρis a s anda d molli ie whose suppo is con ained in (−, ). Then, ϕ
has ime de i a i e and ϕ→ϕin Lp(ΩT). Fu he mo e, he ime-molli ica ion o a
supe solu ion sa is ies a egula ized equa ion in he sense o he ollowing lemma.
Lemma 3.1. Le ∈L∞(ΩT)be a weak supe solu ion (subsolu ion) o (1.1)in ΩT.
Then, we ha e
ΩT
− ∂ ϕ+|D |p−2D
·Dϕ−ϕ( (D ))dz≥(≤)0 (3.1)
o all ϕ∈W1,p(ΩT)∩L∞(ΩT)wi h compac suppo in ΩT. Mo eo e , i he
s onge g ow h condi ion
| (ξ)|≤C 1+|ξ|p−1(G2)
holds, hen he assump ion ϕ∈L∞(ΩT)is no needed.
Obse e ha in he abo e lemma ϕis in he usual Sobole space W1,p(ΩT)so i
has a weak ime de i a i e ∂ ϕ∈Lp(ΩT). To p o e he lemma, one i s assumes ha
ϕis smoo h. Then, es ing he weak o mula ion o (1.1) wi h ϕ, changing a iables
and using Fubini’s heo em yields (3.1). The gene al case ollows by app oxima ing
ϕin W1,p(ΩT)wi h he s anda d molli ica ion. We omi he de ails.
Lemma 3.2. Le 1<p<2and le be locally Lipschi z. Le u, ∈L∞(ΩT),
espec i ely, be weak sub- and supe solu ions o (1.1)in ΩT. Assume ha o all
(x0, 0)∈∂pΩT
ess lim sup
(x, )→(x0, 0)
u(x, )≤ess lim in
(x, )→(x0, 0) (x, ).
Suppose also ha Du ∈L∞(ΩT). Then, u ≤ a.e. in ΩT.
J. Sil akoski J. E ol. Equ.
P oo . (S ep 1) Le l>0 and se w:= (u− −l)+. Le also s∈(0,T). We wan
o use w·χ[0,s]as a es unc ion, bu since i is no smoo h, we mus pe o m molli-
ica ions. Le h>0 and de ine
ϕ:= η(u− −l)+,
whe e
η( )=⎧
⎪
⎪
⎨
⎪
⎪
⎩
1, ∈(0,s−h],
(− +s+h)/2h, ∈(s−h,s+h),
0, ∈[s+h,T).
The unc ion ϕis compac ly suppo ed and belongs in W1,p(ΩT). The e o e, by
Lemma 3.1 we ha e
ΩT
−(u− )∂ ϕdz
≤ΩT|D |p−2D
−|Du|p−2Du·Dϕ+ϕ (Du)− (D )dz.
(3.2)
We use he linea i y o con olu ion and in eg a ion by pa s o elimina e he ime
de i a i e. We ob ain
ΩT
−(u− )∂ ϕdz
=−ΩT
(u− )(u− −l)+∂ η+η(u− )∂ (u− −l)+dz
=−ΩT
(u− −l)((u− −l))+∂ η+l(u− −l)+∂ η
+η(u− −l)∂ (u− −l)++lη∂ (u− −l)+dz
=−ΩT
((u− −l))2
+∂ η+1
2η∂ ((u− −l))2
+dz
=−
1
2ΩT
((u− −l))2
+∂ ηdz
→
→0−1
2ΩT
(u− −l)2
+∂ ηdz.
Mo eo e , by he Lebesgue di e en ia ion heo em o a.e. s∈(0,T)i holds
−1
2ΩT
(u− −l)2
+∂ ηdz=1
4hs+h
s−hΩ
w2(x, )dxd →
h→0
1
2Ω
w2(x,s)dx.
The e ms a he igh -hand side o (3.2) con e ge simila ly. Hence, o a.e. s∈(0,T)
we ha e
Equi alence o iscosi y and weak solu ions
1
2Ω
w2(x,s)dx
≤Ωs
| (Du)− (D )|wdz−Ωs|Du|p−2Du −|D |p−2D ·Dwdz
=: I1−I2.(3.3)
(S ep 2) We seek o abso b some o I1in o I2so ha we can conclude om G önwall’s
inequali y ha w≡0 almos e e ywhe e. Since is locally Lipschi z con inuous,
he e a e cons an s M≥max(2DuL∞(ΩT),1)and L=L(M)such ha
| (ξ) − (η)|≤L|ξ−η|when |ξ|,|η|<M.(3.4)
We deno e Ω+
s:= {x∈Ωs:w≥0},
A:= Ω+
s∩{|D |<M}and B:= Ω+
s∩{|D |≥M}.
Obse e ha in Bwe ha e by he g ow h condi ion (G1), choice o Mand he assump-
ion ha β≥1
| (Du)|≤C (1+|Du|β)≤C (M+Mβ)≤2C Mβ≤2C |D |β(3.5)
and
| (D )|≤C (1+|D |β)≤2C |D |β.(3.6)
I ollows om (3.4), (3.5), (3.6) and Young’s inequali y ha
I1≤A
L|Du −D |wdz+B
(| (Du)|+| (D )|)wdz
≤A
L|Du −D |wdz+B
4C |D |βwdz
≤A
|Du −D |2+C(, L)w2dz+B
|D |
βp
β+C(, p,β,L,C )w
p
p−βdz
≤A
|Du −D |2dz+B
|D |pdz+C(, p,β,L,C ,wL∞)Ωs
w2dz,
(3.7)
whe e in he las s ep we used ha p
p−β>2 o es ima e
Ωs
wp/(p−β) dz=Ωs
wp/(p−β)−2w2dz≤wp/(p−β)−2
L∞(ΩT)Ωs
w2dz.
Using he ec o inequali y
|a|p−2a−|b|p−2b·(a−b)≥(p−1)|a−b|21+|a|2+|b|2p−2
2,(3.8)
J. Sil akoski J. E ol. Equ.
which holds when 1 <p<2[19, p98], we ge
I2=Ωs|Du|p−2Du −|D |p−2D ·Dwdz
≥(p−1)Ω+
s
|Du −D |2
1+|Du|2+|D |22−p
2
dz
≥(p−1)A
|Du −D |2
1+M2+M22−p
2
dz+(p−1)B
(|D |−|Du|)2
3|D |22−p
2
dz
≥C(p,M)A
|Du −D |2dz+(p−1)B|D |−1
2M2
3|D |22−p
2
dz
≥C(p,M)A
|Du −D |2dz+(p−1)B1
2|D |2
3|D |22−p
2
dz
=C(p,M)A
|Du −D |2dz+C(p)B
|D |pdz,(3.9)
whe e C(p,M), C(p)>0. Combining he es ima es (3.7) and (3.9), we a i e a
I1−I2≤(−C(p,M))A
|Du −D |2dz+(−C(p))B
|D |pdz
+C0Ωs
w2dz,
whe e C0=C(, p,β,L,C ,wL∞). Recalling (3.3) and aking small enough
yields
Ω
w2(x,s)dx≤2C0Ωs
w2dz.
Since his holds o a.e. s∈(0,T), G önwall’s inequali y implies ha w≡0a.e.in
ΩT. Finally, le ing l→0 yields ha u− ≤0a.e.inΩT.
Lemma 3.3. Le p ≥2and le be locally Lipschi z. Le ∈L∞(ΩT)be a weak
supe solu ion o (1.1)and le u ∈L∞(ΩT)be a weak subsolu ion o
∂ u−Δpu− (Du)≤−δin ΩT
o some δ>0. Assume ha o all (x0, 0)∈∂pΩT
ess lim sup
(x, )→(x0, 0)
u(x, )≤ess lim in
(x, )→(x0, 0) (x, ).
Suppose also ha Du ∈L∞(ΩT). Then, u ≤ a.e. in ΩT.
Equi alence o iscosi y and weak solu ions
− (Duε, j)dz
≤lim
j→∞ Ξε
−uε, j∂ ϕ+Duε, jp−2Duε, j·Dϕ−ϕ (Duε, j)dz.
We in end o use Fa ou’s lemma a he le -hand side and domina ed con e gence a
he igh -hand side. Once we e i y hei assump ions, we a i e a he inequali y
Ξε
ϕ∂ uε−Δpuε− (Duε)dz≤Ξε
−uε∂ ϕ
+|Duε|p−2Duε·Dϕ−ϕ (Duε)dz.
The le -hand side is non-nega i e since by Lemma 4.7 he in -con olu ion uεis s ill
a iscosi y supe solu ion in Ξε. Consequen ly uεis a weak supe solu ion in Ξεas
desi ed. I emains o jus i y ou use o Fa ou’s lemma and he domina ed con e gence
heo em. I ollows om Rema k 4.8 ha uε, j,∂ uε, jand Duε, ja e uni o mly
bounded by some cons an M>0 in he suppo o ϕwi h espec o j. This jus i ies
ou use o he domina ed con e gence heo em. Obse e hen ha since φjis conca e,
we ha e D2uε, j≤C(q,ε,u)I. Hence,
∂ uε, j−Duε, jp−2Δuε, j+(p−2)
Duε, j2D2uε, jDuε, j,Duε, j− (Duε, j)
≥−M−C(q,ε,u)Mp−2(N+p−2)−sup
|ξ|≤M
| (ξ)|.
The in eg and a he le -hand side o (4.1) is he e o e bounded om below wi h
espec o j, jus i ying ou use o Fa ou’s lemma.
Nex , we conside he singula case 1 <p<2. We canno di ec ly epea he p e i-
ous p oo because Δpuεno longe has a clea meaning a he poin s whe e Duε=0.
To deal wi h his, we conside he egula ized e ms
Δp,δu:= δ+|Du|2p−2
2Δu+p−2
δ+|Du|2Δ∞u,(4.2)
whe e Δ∞u=D2uDu,Du.
Lemma 4.2. Le 1<p<2. Le u be a bounded iscosi y supe solu ion o (1.1)in
Ξ. Then, uεis a weak supe solu ion o (1.1)in Ξε.
P oo . (S ep 1) Le ϕ∈C∞
0(Ξε)be a non-nega i e es unc ion. We se
φ(x, ):= uε(x, )−C(q,ε,u)|x|2+ 2,
whe e C(q,ε,u)is he semi-conca i y cons an o uεin Ξε. Then, by Rema k 4.8 we
can app oxima e φby smoo h conca e unc ions φjso ha φj,∂
φj,Dφj,D2φj→
φ,∂ φ, Dφ, D2φa.e. in Ξε. We de ine
uε, j(x, ):= φj(x, )+C(q,ε,u)|x|2+ 2.
J. Sil akoski J. E ol. Equ.
Le δ∈(0,1). Since uε, jis smoo h and ϕis compac ly suppo ed in Ξε, we calcula e
ia in eg a ion by pa s
Ξε
ϕ∂ uε, j−δ+Duε, j2p−2
2Δuε, j+p−2
δ+Duε, j2Δ∞uε, j− (Duε, j)dz
=Ξε
ϕ∂ uε, j−ϕdi δ+Duε, j2p−2
2Duε, j−ϕ (Duε, j)dz
=Ξε
−uε, j∂ ϕ+δ+Duε, j2p−2
2Duε, j·Dϕ−ϕ (Duε, j)dz.
Recalling he sho hand Δp,δ de ined in (4.2), we deduce om he abo e ha
lim in
j→∞ Ξε
ϕ∂ uε, j−Δp,δuε, j− (Duε, j)dz
≤lim
j→∞ Ξε
−uε, j∂ ϕ+δ+Duε, j2p−2
2Duε, j·Dϕ−ϕ (Duε, j)dz.(4.3)
We use Fa ou’s lemma a he le -hand side and he domina ed con e gence a he igh -
hand side. Once we e i y hei assump ions, we a i e a he auxilia y inequali y
Ξε
ϕ∂ uε−Δp,δuε− (Duε)dz
≤Ξε
−uε∂ ϕ+δ+|Duε|2p−2
2Duε·Dϕ−ϕ (Duε)dz.(4.4)
Nex , we e i y he assump ions o Fa ou’s lemma and he domina ed con e gence
heo em. By Rema k 4.8, he unc ions uε, j,∂ uε, jand Duε, ja e uni o mly
bounded by some cons an M>1 in he suppo o ϕwi h espec o j. Hence,
he assump ions o he domina ed con e gence heo em a e sa is ied. Obse e hen
ha he conca i y o φjimplies ha D2uε, j≤C(q,ε,u)I. Thus, he in eg and a he
le -hand side o (4.3) has a lowe bound independen o jwhen Duε, j=0. When
Duε, j= 0, we ha e
∂ uε, j−δ+Duε, j2p−2
2Δuε, j+p−2
δ+Duε, j2Δ∞uε, j− (Duε, j)
=∂ uε, j−δ+Duε, j2p−2
2
δ+Duε, j2Duε, j2Δuε, j+p−2
Duε, j2Δ∞uε, j+δΔuε, j
− (Duε, j)
≥−∂ uε, j−δ+Duε, j2p−2
2
δ+Duε, j2C(q,ε,u)Duε, j2(N+p−2)+δN− (Duε, j)
≥−∂ uε, j−C(q,ε,u)δ+Duε, j2p−2
2(2N+p−2)− (Duε, j)
≥−M−C(q,ε,u)δ p−2
2(2N+p−2)−sup
|ξ|≤M
| (ξ)|,
Equi alence o iscosi y and weak solu ions
so ha ou use o Fa ou’s lemma is jus i ied.
(S ep 2) We le δ→0 in he auxilia y inequali y (4.4). Since uεis Lipschi z con in-
uous, he domina ed con e gence heo em implies
lim in
δ→0Ξε
ϕ∂ uε−Δp,δuε− (Duε)dz
≤Ξε
−uε∂ ϕ+|Duε|p−2Duε·Dϕ−ϕ (Duε)dz.(4.5)
Applying Fa ou’s lemma (we e i y assump ions a he end), we ge
lim in
δ→0Ξε
ϕ∂ uε−Δp,δuε− (Duε)dz
≥Ξε
lim in
δ→0ϕ∂ uε−Δp,δuε− (Duε)dz
=Ξε∩{Duε=0}
lim in
δ→0ϕ∂ uε−Δp,δuε− (Duε)dz
+Ξε∩{Duε=0}
lim in
δ→0ϕ(∂ uε−δp−2
2Δuε− (0)) dz
=Ξε∩{Duε=0}
ϕ∂ uε−Δpuε− (Duε)dz
+Ξε∩{Duε=0}
ϕ(∂ uε− (0))dz≥0,(4.6)
whe e he las inequali y ollows om Lemma 4.7 since uεis wice di e en iable
almos e e ywhe e. Combining (4.5) and (4.6), we ind ha uεis a weak supe solu ion
in Ξε. I emains o e i y he assump ions o Fa ou’s lemma, i.e., ha he in eg and
a he le -hand side o (4.5) has a lowe bound independen o δ. When Duε=0, his
ollows di ec ly om he inequali y
D2uε≤q−1
ε|Duε|
q−2
q−1I,
which holds by Lemma 4.6. When Duε= 0, we ecall ha by Lipschi z con inui y
∂ uεand Duεa e uni o mly bounded in Ξε, and es ima e
−δ+|Duε|2p−2
2Δuε+p−2
δ+|Duε|2Δ∞uε
=−
δ+|Duε|2p−2
2
δ+|Duε|2|Duε|2Δuε+p−2
|Duε|2Δ∞uε+δΔuε
≥−
δ+|Duε|2p−2
2
δ+|Duε|2
(q−1)
ε|Duε|
q−2
q−1+2(N+p−2)+|Duε|
q−2
q−1δN
≥−δ+|Duε|2p−2
2(q−1)
ε|Duε|
q−2
q−1(2N+p−2)
J. Sil akoski J. E ol. Equ.
≥−|Duε|p−2+q−2
q−1(q−1)
ε(2N+p−2)
≥−Duεp−2+q−2
q−1
L∞(Ξε)
(q−1)
ε(2N+p−2),
whe e we used ha p−2+q−2
q−1>0. Hence, he assump ions o Fa ou’s lemma hold.
I uεis he sequence o in -con olu ions o a iscosi y supe solu ion o (1.1), hen
by nex Caccioppoli’s inequali y he sequence Duεcon e ges weakly in Lp
loc(Ξ) up
o a subsequence. Howe e , we need s onge con e gence o pass o he limi unde
he in eg al sign o
Ξ
−uε∂ ϕ+|Duε|p−2Duε·Dϕ−ϕ (Duε)dz≥0.
Fo his end, we show in Lemma 4.4 ha Duεcon e ges in L
loc(Ξ) o all 1 < <p.
Lemma 4.3. (Caccioppoli’s inequali y) Le 1<p<∞. Assume ha u is a locally
Lipschi z con inuous weak supe solu ion o (1.1)in Ξ. Then, he e is a cons an C =
C(p,β,C )such ha o any es unc ion ξ∈C∞
0(Ξ) we ha e
Ξ
ξp|Du|pdz≤CΞ
M2∂ ξp+Mp|Dξ|p+(M
p
p−β+M)ξ pdz,
whe e M =uL∞(sp ξ).
P oo . Since uis locally Lipschi z con inuous, he unc ion ϕ:= (M−u)ξpis an
admissible es unc ion. Tes ing he weak o mula ion o (1.1) wi h ϕyields
Ξ
ξp|Du|pdz≤Ξ
u∂ ϕ+pξp−1(M−u)|Du|p−1|Dξ|+ϕ (Du)dz.(4.7)
We ha e by in eg a ion by pa s
Ξ
u∂ ϕdz=Ξ
−ξpu∂ u+u(M−u)∂ ξpdz
=Ξ
−1
2ξp∂ u2+u(M−u)∂ ξpdz
=Ξ
1
2u2∂ ξp+u(M−u)∂ ξpdz≤Ξ
CM2∂ ξpdz.
By Young’s inequali y,
Ξ
pξp−1(M−u)|Du|p−1|Dξ|dz≤Ξ
1
4ξp|Du|pdz+C(p)Ξ
Mp|Dξ|pdz.
Equi alence o iscosi y and weak solu ions
Using he g ow h condi ion (G1) and Young’s inequali y, we ge
Ξ
ϕ (Du)dz
≤Ξ
(M−u)ξpC 1+|Du|βdz
=Ξ
C (M−u)ξp−βξβ|Du|β+C (M−u)ξ pdz
≤Ξ
1
4ξp|Du|p+C(p,β,C )(M−u)
p
p−βξp+C (M−u)ξpdz
≤Ξ
1
4ξp|Du|p+C(p,β,C )M
p
p−β+Mξpdz.
Combining hese es ima es wi h (4.7) and abso bing he e ms wi h Du o he le -hand
side yields he desi ed inequali y.
The p oo o Lemma 4.4 is based on ha o Lemma 5 in [20], see also Theo em 5.3
in [15]. Fo he con enience o he eade , we gi e he ull de ails.
Lemma 4.4. Le 1<p<∞. Suppose ha ujis a sequence o locally Lipschi z
con inuous weak supe solu ions o (1.1)such ha u j→uinL
p
loc(Ξ). Then, Du j
is a Cauchy sequence in L
loc(Ξ) o any 1< <p.
P oo . Le UΞand ake a cu o unc ion θ∈C∞
0(Ξ) such ha 0 ≤θ≤1 and
θ≡1inU.Fo δ>0, we se
wjk =⎧
⎪
⎪
⎨
⎪
⎪
⎩
δ, uj−uk>δ,
uj−uk,uj−uk≤δ,
−δ, uj−uk<−δ.
Then, he unc ion (δ −wjk)θ is an admissible es unc ion wi h a ime de i a i e
since i is Lipschi z con inuous. Since ujis a weak supe solu ion, es ing he weak
o mula ion o (1.1) wi h (δ −wjk)θ yields
0≤Ξ
−uj∂ ((δ −wjk)θ) +Du jp−2Du j·D((δ −wjk)θ) −(δ −wjk)θ (Du j)dz
=Ξ
−θDu jp−2Du j·Dwjk +(δ −wjk)Du jp−2Du j·Dθ−(δ −wjk)θ (Du j)
+uj∂ (wjk)θ −(δ −wjk)uj∂ θdz.
Since wjk≤δand Dwjk =χ{|uj−uk|<δ}Du j−Duk, he abo e becomes
{|uj−uk|<δ}
θDu jp−2Du j·Du j−Dukdz
≤Ξ
2δDu jp−1|Dθ|+2δθ (Du j)+uj∂ (wjk)θ +2δuj|∂ θ|dz.
J. Sil akoski J. E ol. Equ.
Since ukis a weak supe solu ion, he same a gumen s as abo e bu es ing his ime
wi h (δ +wjk)θ yield he analogous es ima e
{|uj−uk|<δ}
−θ|Duk|p−2Duk·Du j−Dukdz
≤Ξ
2δ|Duk|p−1|Dθ|+2δθ | (Duk)|−uk∂ wjkθ+2δ|uk||
∂ θ|dz.
Summing up hese wo inequali ies, we a i e a
{|uj−uk|<δ}
θDu jp−2Du j−|Duk|p−2Duk·Du j−Dukdz
≤2δΞ
|Dθ|Du jp−1+|Duk|p−1dz+2δΞ
θ (Du j)+| (Duk)|dz
+Ξ
(uj−uk)∂ wjkθdz+2δΞuj+|uk||∂ θ|dz
=: I1+I2+I3+I4.(4.8)
We p oceed o es ima e hese in eg als. Deno ing M:= supjujL∞(sp θ) <∞,we
ha e by he Caccioppoli’s inequali y Lemma 4.3
sup
jsp θDu jpdz≤C(p,β,C ,θ,M). (4.9)
The es ima e (4.9) and Hölde ’s inequali y imply ha
I1≤δC(p,β,C ,θ,M).
To es ima e I2, we also use he g ow h condi ion (G1) and he assump ion β<p.We
ge
I2≤2δΞ
θC (2+Du jβ+|Duk|β)dz≤δC(p,β,C ,θ,M).
The in eg al I3is es ima ed using in eg a ion by pa s and ha wjk≤δ
I3=Ξ
θ(uj−uk)∂ wjkdz=Ξ
1
2θ∂ w2
jk dz=Ξ
−1
2w2
jk∂ θdz≤δC(θ, M).
Fo he las in eg al, we ha e di ec ly I4≤δC(θ, M). Combining hese es ima es wi h
(4.8), we a i e a
{|uj−uk|<δ}
θDu jp−2Du j−|Duk|p−2Duk·Du j−Dukdz≤δC0,
(4.10)
whe e C0=C(p,β,C ,θ,M).I 1<p<2, Hölde ’s inequali y and he algeb aic
inequali y (3.8) gi e he es ima e ( ecall ha 1 < <pand θ≡1inU)
U∩{|uj−uk|<δ}Du j−Duk dz
Equi alence o iscosi y and weak solu ions
≤U∩{|uj−uk|<δ}1+Du j2+|Duk|2 (2−p)
2(2− )dz2−
2
·U∩{|uj−uk|<δ}Du j−Duk2
1+Du j2+|Duk|22−p
2
dz
2
≤C(p,β, ,C ,θ,M)
·{|uj−uk|<δ}
θDu jp−2Du j−|Duk|p−2Duk·Du j−Dukdz
2
,
whe e in he las inequali y we also used (4.9) wi h he knowledge (2−p)
(2− )≤p(2−p)
2−p=
p.
I p≥2, Hölde ’s inequali y and he algeb aic inequali y (3.12)imply
U∩{|uj−uk|<δ}Du j−Duk dz
≤Ξ
1dzp−
pU∩{|uj−uk|<δ}Du j−Dukpdz
p
≤C(p, ){|uj−uk|<δ}
θDu jp−2Du j−|Duk|p−2Duk·Du j−Dukdz
p
.
Hence, (4.10) leads o
U∩{|uj−uk|<δ}Du j−Duk dz≤δ
max(2,p)C(p,β, ,C ,θ,M).
On he o he hand, Hölde ’s and Tchebyshe ’s inequali ies wi h (4.9)imply
U∩{|uj−uk|≥δ}Du j−Duk dz
≤U∩uj−uk≥δp−
pU∩{|uj−uk|≥δ}Du j−Dukpdz
p
≤δ −puj−ukp−
Lp(U)C(p,β, ,C ,θ,M).
So we a i e a
UDu j−Duk dz≤(δ
max(2,p)+δ −puj−ukp−
Lp(U))C(p,β, ,C ,θ,M).
Taking i s small δ>0 and hen la ge j,k, we can make he igh -hand side a bi a ily
small.
Now we a e eady o p o e he main esul o his sec ion which s a es ha bounded
iscosi y supe solu ions a e weak supe solu ions.
Theo em 4.5. Le 1<p<∞. Le u be a bounded iscosi y supe solu ion o (1.1)
in Ξ. Then, u is a weak supe solu ion o (1.1)in Ξ.
J. Sil akoski J. E ol. Equ.
P oo . Fix a non-nega i e es unc ion ϕ∈C∞
0(Ξ) and ake an open cylinde
Ω 1, 2Ξsuch ha sp ϕΩ 1, 2.Le ε>0 be so small ha Ω 1, 2Ξε. Then,
Lemma 4.2 implies ha uεis a weak supe solu ion o (1.1)inΞε. The e o e, by he
Caccioppoli’s inequali y Lemma 4.3,Duεis bounded in Lp(Ω 1, 2). Hence, Duεcon-
e ges weakly in Lp(Ω 1, 2)up o a subsequence. Since also uε→uin Lp(Ω 1, 2)by
domina ed con e gence and he ac ha uε→upoin wise in Ω 1, 2, i ollows ha
u∈Lp( 1, 2;W1,p(Ω)).
Since uεis a weak supe solu ion, i emains o show ha up o a subsequence
lim
ε→0Ω 1, 2
uε∂ ϕ+|Duε|p−2Duε·Dϕdz=Ω 1, 2
u∂ ϕ+|Du|p−2Du ·Dϕdz
(4.11)
and
lim
ε→0Ω 1, 2
ϕ (Duε)dz=Ω 1, 2
ϕ (Du)dz.(4.12)
Since uε→uin Lp(Ω 1, 2)and Duε→Du in L (Ω 1, 2) o any 1 < <pby
Lemma 4.4, he claim (4.11) ollows by applying he ec o inequali y (see [19, pp.
95–96])
|a|p−2a−|b|p−2b≤22−p|a−b|p−1when p<2,
2−1|a|p−2+|b|p−2|a−b|when p≥2.
To show (4.12), le M≥1 and w i e using he g ow h condi ion (G1)
Ω 1, 2
| (Duε)− (Du)|dz
≤{|Duε|<M}
| (Duε)− (Du)|dz+{|Duε|≥M}
C (2+|Duε|β+|Du|β)dz
=: I1+I2.
Then, by Hölde ’s inequali y
I2=C {|Duε|≥M}
2|Duε|p
|Duε|p+|Duε|p
|Duε|p−β+|Du|β|Duε|p−β
|Duε|p−βdz
≤C 2
Mp+1
Mp−βDuεp
Lp(Ω 1, 2)+C
1
Mp−βDuβ
Lp(Ω 1, 2)Duεp−β
Lp(Ω 1, 2)
≤1
Mp−βC(p,β,C ,DuLp(Ω 1, 2),sup
ε
DuεLp(Ω 1, 2)).
On he o he hand, we ha e | (Duε)− (Du)|→0a.e.inΩ 1, 2up o a subse-
quence and he in eg and in I1is domina ed by an in eg able unc ion since he g ow h
condi ion (G1) implies
| (Duε)− (Du)|≤C (2+|M|β+|Du|β)when |Duε|<M.
Equi alence o iscosi y and weak solu ions
Hence, o any M≥1, we ha e I1→0asε→0 by he domina ed con e gence
heo em. By aking i s la ge M≥1 and hen small ε>0, we can make I1+I2
a bi a ily small.
The es o his sec ion is de o ed o he p ope ies o he in -con olu ion. The ac s
in he ollowing lemma a e well known, see, e.g., [6,11,14]o [24].
Lemma 4.6. Assume ha u :Ξ→Ris lowe semicon inuous and bounded. Then,
uεhas he ollowing p ope ies.
(i) We ha e uε≤uinΞand uε→u poin wise as ε→0.
(ii) Deno e (ε) := qεq−1oscΞu1
q, (ε) := (2εoscΞu)1
2.Fo (x, )∈RN+1,
se
Ξε:= (x, )∈Ξ:B (ε)(x)×( − (ε), + (ε)) Ξ.
Then, o any (x, )∈Ξε he e exis s (xε, ε)∈B (ε)(x)×[ − (ε), + (ε)]
such ha
uε(x, )=u(xε, ε)+|x−xε|q
qεq−1+| − ε|2
2ε.
(iii) The unc ion uεis semi-conca e in Ξεwi h a semi-conca i y cons an depending
only on u, q and ε.
(i ) Assume ha uεis di e en iable in ime and wice di e en iable in space a
(x, )∈Ξε. Then,
∂ uε(x, )= − ε
ε,
Duε(x, )=(x−xε)|x−xε|q−2
εq−1,
D2uε(x, )≤q−1
ε|Duε|
q−2
q−1I.
Nex , we show ha he in -con olu ion o a iscosi y supe solu ion o (1.1) is s ill
a supe solu ion in he smalle se Ξε. Since he in -con olu ion is “ la enough,” ha
is, since q>p/(p−1), he in -con olu ion essen ially cancels he singula i y o he
p-Laplace ope a o . This allows us o ex ac in o ma ion on he ime de i a i e a
hose poin s o di e en iabili y whe e Duε anishes.
Lemma 4.7. Le 1<p<∞. Le u be a bounded iscosi y supe solu ion o (1.1)in
Ξ. Then, he in -con olu ion uεis also a iscosi y supe solu ion o (1.1)in Ξε.
Mo eo e , i uεis di e en iable in ime and wice di e en iable in space a (x, )∈
Ξεand Duε(x, )=0, hen ∂ uε(x, )− (0)≥0.
P oo . Assume ha ϕ ouches uε om below a (x, )∈Ξε.Le (xε, ε)belikein he
p ope y (ii) o Lemma 4.6. Then,
ϕ(x, )=uε(x, )=u(xε, ε)+|x−xε|q
qεq−1+| − ε|2
2ε,(4.13)
J. Sil akoski J. E ol. Equ.
ϕ(y,τ)≤uε(y,τ)≤u(z,s)+|y−z|q
qεq−1+|τ−s|2
2ε o all (y,τ),(z,s)∈Ξ.
(4.14)
Se
ψ(z,s):= ϕ(z+x−xε,s+ − ε)−|x−xε|q
qεq−1−| − ε|2
2ε.
Then, ψ ouches u om below a (xε, ε)since by (4.13)
ψ(xε, ε)=ϕ(x, )−|x−xε|q
qεq−1−| − ε|2
2ε=u(xε, ε)
and selec ing (y,τ)=(z+x−xε,s+ − ε)in (4.14)gi es
ψ(z,s)=ϕ(z+x−xε,s+ − ε)−|x−xε|q
qεq−1−| − ε|2
2ε≤u(z,s).
Since uis a iscosi y supe solu ion, i ollows ha
0≤lim sup
(z,s)→(xε, ε)
z=xε∂sψ(z,s)−Δpψ(z,s)− (Dψ(z,s))
=lim sup
(z,s)→(x, )
z=x∂sϕ(z,s)−Δpϕ(z,s)− (Dϕ(z,s)),
and he i s claim is p o en. To p o e he second claim, assume ha uεis di e en iable
in ime and wice di e en iable in space a (x, )∈Ξεand Duε(x, )=0. By he
p ope y (i ) in Lemma 4.6,weha ex=xε, so ha
uε(x, )=u(x, ε)+| − ε|2
2ε.
Hence, by he de ini ion o in -con olu ion
u(y,s)+|x−y|q
qεq−1+| −s|2
2ε≥uε(x, )=u(x, ε)+| − ε|2
2ε o all (y,s)∈Ξ.
A anging he e ms as
u(y,s)≥u(x, ε)−|x−y|q
qεq−1−| −s|2
2ε+| − ε|2
2ε=: φ(y,s),
we see ha he unc ion φ ouches u om below a (x, ε). Since uis a iscosi y
supe solu ion and Dφ(y,s)= 0 when y= x,weha e
lim sup
(y,s)→(x, ε)
y=x∂sφ(y,s)−Δpφ(y,s)− (Dφ(y,s))≥0.
Equi alence o iscosi y and weak solu ions
Using Lemma 5.2 on he subsolu ion := θ+u, we ge he es ima e
ess sup
BσR(x0)×( 0−σpT, 0)
u≤CΛT
Rp−
BR(x0)×( 0−T, 0)
(θ+u)p−2+δdz1
δ
≤CΛ1
δT
Rpθp−2+δ1
δ
+CΛ1
δT
Rp−
BR(x0)×( 0−T, 0)
up−2+δdz1
δ
,
whe e
T
Rpθp−2+δ=T1−p−2+δ
p−1R−p+p(p−2+δ)
p−1=T1−δRp(δ−1)1
p−1=Rp
Tδ−1
p−1
.
Taking σ=1/2 now yields he desi ed inequali y.
Lemma 5.4. Assume ha p ≥2and ha (0)=0. Le u be a weak subsolu ion o
(1.1)in Ω 1, 2. Then, u+=max(u,0)is also a weak subsolu ion.
P oo . Fix a non-nega i e es unc ion ζ∈C∞
0(Ω 1, 2). We es he egula ized
equa ion in Lemma 3.1 wi h min {k(u)+,1}ζ. Then, by simila a gumen s as in he
p oo o Lemma 5.1 we ge he es ima e
Ω 1, 2
min {ku+,1}(−u∂ ζ+|Du|p−2Du ·Dζ−ζ (Du)) dz
≤−1
2kΩ 1, 2
(min {ku+,1})2∂ ζdz−k{0<ku<1}
ζ|Du|pdz.
Le ing k→∞ his implies
{u>0}
−u∂ ζ+|Du|p−2Du ·Dζ−ζ (Du)dz≤0.
Since (0)=0 and u+∂ ζ=0=Du+a.e. in {u≤0}, we ge ha
Ω 1, 2
−u+∂ ζ+|Du+|p−2Du+·Dζ−ζ (Du+)dz≤0.
Theo em 5.5. Assume ha p ≥2,(G2)holds and ha (0)=0. Suppose ha
u is a weak supe solu ion o (1.1)in Ξ. Le u∗deno e he lowe semicon inuous
egula iza ion o u, ha is,
u∗(x, ):= ess lim in
(y,s)→(x, )u(y,s):= lim
R→0ess in
BR(x)×( −Rp, +Rp)u.
Then, u =u∗almos e e ywhe e.
J. Sil akoski J. E ol. Equ.
P oo . Fo all M∈N, we de ine he cylinde s
QM
R(x, ):= BR(x)×( −MRp, +MRp).
We deno e by EM he se o Lebesgue poin s wi h espec o he basis {QM
R}, ha is,
EM:= (x, )∈Ξ:lim
R→0−
QM
R(x, )
|u(x, )−u(y,s)|p−1
2dyds=0.
Then, EM⊂EM+1so ha
E:=
M∈N
EM=E1.
Mo eo e , we ha e |E|=|Ξ|, which ollows om [26, p. 13] by a simple a gumen ,
see o example [8,p.54].
We now claim ha i (x0, 0)∈E, hen
u(x0, 0)≤ess lim in
(x, )→(x0, 0)u(x, ). (5.11)
We make he coun e assump ion
u(x0, 0)−ess lim in
(x, )→(x0, 0)u(x, )=ε>0.
Le R0be a adius such ha
ess lim in
(x, )→(x0, 0)u(x, )−ess in
Q1
R(x0, 0)
u≤ε/2
o all 0 <R≤R0. Fo such R,weha e
u(x0, 0)−ess in
Q1
R(x0, 0)
u≥ε/2.(5.12)
We se := (u(x0, 0)−u)+. Since (x0, 0)∈E, we ind o any M∈Na adius
R1=R1(M)such ha
−
QM
R1(x0, 0)
p−1
2dxd ≤−
QM
R1(x0, 0)
|u(x0, 0)−u|p−1
2dxd ≤1
M2
.(5.13)
On he o he hand, by Lemma 5.4 he unc ion is a weak subsolu ion o
∂ +Δp −g(D ) ≤0,
whe e g(ξ) =− (−ξ). Obse e also ha he cylinde QM
R1(x0, 0)sa is ies he con-
di ion (5.10) since Rp
1/(MRp
1)≤1. Hence, we may apply Theo em 5.3 wi h δ=3/2
and hen use (5.13) o ge
ess sup
QM
(R1)/2(x0, 0)
≤CRp
1
Rp
1M1
3(p−1)
+CRp
1M
Rp
1
−
QM
R1(x0, 0)
p−1
2dxd 2
3
Equi alence o iscosi y and weak solu ions
≤C
M3(p−1)+CM·1
M22
3
≤C1
M1
3
.
Now we i s ix Mso la ge ha C/M1
3≤ε/4 and his will also ix R1. Then, we
ake R∈(0,R0]so small ha Q1
R(x0, 0)⊂QM
(R1)/2(x0, 0). Then, (5.12) leads o a
con adic ion since
ε/4≥ess sup
QM
(R1)/2(x0, 0)
≥ess sup
Q1
R(x0, 0)
≥u(x0, 0)−ess in
Q1
R(x0, 0)
u≥ε/2.
Hence, (5.11) holds and we ha e
u(x0, 0)≤ess lim in
(x, )→(x0, 0)u(x, )≤lim
R→0−
Q1
R
u(x, )dxd =u(x0, 0).
Thus, u∗=ualmos e e ywhe e and i is easy o show ha u∗is lowe semicon inuous.
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Ja kko Sil akoski
Depa men o Ma hema ics and S a is ics
Uni e si y o Jy äskylä
P.O. Box 35
40014 Jy askyla
Finland
E-mail: [email p o ec ed]
Accep ed: 19 Decembe 2020