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On he egula i y o e y weak solu ions o linea ellip ic equa ions in di e gence o m
© 2020 he Au ho s
Published e sion
La Manna, Domenico Angelo; Leone, Chia a; Schia a ella, Robe a
La Manna, D. A., Leone, C., & Schia a ella, R. (2020). On he egula i y o e y weak solu ions
o linea ellip ic equa ions in di e gence o m. Nodea: Nonlinea Di e en ial Equa ions and
Applica ions, 27(5), A icle 43. h ps://doi.o g/10.1007/s00030-020-00646-8
2020
Nonlinea Diffe . Equ. Appl.
(2020) 27:43
c
2020 The Au ho (s)
h ps://doi.o g/10.1007/s00030-020-00646-8
Nonlinea Diffe en ial Equa ions
and Applica ions NoDEA
On he egula i y o e y weak solu ions o
linea ellip ic equa ions in di e gence o m
Domenico Angelo La Manna, Chia a Leone and
Robe a Schia a ella
Abs ac . In his pape we conside a linea ellip ic equa ion in di e gence
o m

i,j
Dj(aij (x)Diu)=0 inΩ.(0.1)
Assuming he coefficien s aij in W1,n(Ω) wi h a modulus o con inui y
sa is ying a ce ain Dini- ype con inui y condi ion, we p o e ha any e y
weak solu ion u∈Ln
loc(Ω) o (0.1) is ac ually a weak solu ion in W1,2
loc (Ω).
Ma hema ics Subjec Classifica ion. P ima y 35-02, Seconda y 35B65.
Keywo ds. Ellip ic equa ions, Ve y weak solu ions.
1. In oduc ion
Le n≥2 and Ω ⊂Rnbe a bounded open se . In his pape we s udy egula i y
p ope ies o e y weak solu ions o he linea ellip ic equa ion

i,j
Dj(aij(x)Diu)=0 inΩ,(1.1)
whe e he ma ix-field A:Ω→Rn×n,A(x)=(aij(x))i,j, is ellip ic and
belongs o W1,n(Ω,Rn×n)∩L∞(Ω,Rn×n), i.e.
sup
i,j=1,...,n
aijW1,n(Ω) ≤M(1.2)
and
λ|ξ|2≤
i,j
aij(x)ξiξj≤Λ|ξ|2∀ξ=(ξ1,...,ξ
n)∈Rn,a.e. in Ω,(1.3)
o some posi i e cons an s λ, Λ,and M. Mo eo e , he ma ix Ais symme ic,
ha is aij =aji a.e. in Ω o all i, j ∈{1, ..., n}.
0123456789().: V,- ol
43 Page 2 o 23 D. A. L. Manna e al. NoDEA
Finally we assume ha he coefficien s (aij(x))i,j a e double-Dini con in-
uous in Ω, i.e. aij ∈C0(Ω) and
¯
AΩ( ):=
i,j
sup
x,y∈Ω
|x−y|≤
|aij(x)−aij(y)|, >0,
sa isfies
ˆdiam(Ω)
0
1
ˆ
0
¯
AΩ(s)
sds d < ∞.(1.4)
A common ype o double-Dini con inuous unc ions a e, o cou se, ω( )= α,
0<α≤1, hus an example o a ma ix-field Asa is ying (1.2) and (1.4)is
A∈W1,p(Ω,Rn×n), wi h p>n. On he o he hand, condi ion (1.4) occu s
no only o ω( )= α, bu mo e gene ally o ω( ) = logβ1
,β<−2.
Gi en a measu able ma ix A(x)=(aij(x))i,j sa is ying (1.3), a unc ion
u∈W1,2
loc (Ω) is called a weak solu ion o (1.1)i

i,j ˆaij(x)DiuDjϕdx=0,∀ϕ∈C∞
c(Ω).
The celeb a ed esul by De Gio gi in [5] s a es ha i uis a weak solu ion
o (1.1) hen uis locally H¨olde con inuous.
Subsequen ly, J. Se in p oduced in [14] a amous example, cons uc ing
an equa ion o he o m (1.1) which has a solu ion u∈W1,p(Ω), wi h 1 <
p<2, and u/∈L∞
loc(Ω). Se in conjec u ed ha i he coefficien s aij a e
locally H¨olde con inuous, hen any solu ion (in he sense o dis ibu ions)
u∈W1,1
loc (Ω) o (1.1) mus be a (usual) weak solu ion, i.e. u∈W1,2
loc (Ω).
Se in’s conjec u e was es ablished by Hage and Ross [9], and hen in ull
gene ali y by B ezis [2](seealso[1] o a ull p oo ) s a ing wi h u∈W1,1
loc (Ω),
o e en wi h u∈BVloc(Ω), i.e., u∈L1
loc(Ω) and i s de i a i es (in he sense
o dis ibu ions) being Radon measu es. Le us ema k ha in B ezis’s esul
he coefficien s aij, sa is ying (1.3), a e Dini con inuous unc ions in Ω. The
Dini con inui y o he coefficien s is op imal in some sense: o he uni ball B1
and con inuous coefficien s, Jin e al. [10] cons uc ed a solu ion (in he sense
o dis ibu ions) u∈W1,1
loc (B1) W1,p
loc (B1) o e e y p>1.
Fo A(x)=(aij(x))i,j sa is ying (1.2) and (1.3), we will conside a e y
weak solu ion u∈Ln
loc(Ω) o (1.1), namely

i,j ˆu(x)Di(aij(x)Djϕ(x)) dx =0,∀ϕ∈C∞
c(Ω),(1.5)
wi h n=n
n−1.
Rema k 1.1. I is no difficul o p o e ha he es unc ions ϕin (1.5)can
be aken in W2,n(Ω) ∩W1,∞(Ω), wi h supp ϕΩ. Indeed, one can a gue by
densi y o show ha gi en a unc ion ϕ∈W2,n(Ω) ∩W1,∞(Ω) wi h compac
suppo , we may find a sequence ϕk∈C∞
c(Ω) such ha ϕk→ϕs ongly in
W2,n and supkϕk1,∞<∞(so ha Dϕkcon e ges o Dϕ weakly* in L∞)
and hen aking he limi as kgoes o infini e in he Eq. (1.5) o ϕk.
NoDEA On he egula i y o e y weak solu ions Page 3 o 23 43
The main esul o he pape is he ollowing.
Theo em 1.2. Le ube a e y weak solu ion o (1.1),wi hA(x)=(aij(x))i,j
sa is ying (1.2),(1.3)and (1.4), hen ubelongs o W1,2
loc (Ω) and hus i is a
weak solu ion.
Rema k 1.3. I is wo h no ing ha , unde hypo heses (1.2) and (1.3), one
can conside a e y weak solu ion u∈Ln
loc(Ω) o (1.1), bu when dealing
wi h he egula i y p ope ies o usome ex a condi ions on he coefficien s
aij mus be conside ed. The coun e example cons uc ed in [10] p o ides in
ac con inuous coefficien s aij which belong also o W1,n(B1), showing ha
one can no expec a e y weak solu ion u∈Ln
loc(Ω) o be a weak solu ion in
W1,2
loc (Ω) unde jus condi ions (1.2) and (1.3). Fo he sake o comple eness,
we will p opose he example gi en in [10] in he “Appendix B”, unde lining
ha he cons uc ed coefficien s belong also o W1,n(B1).
On he o he hand, in Sec . 4we p opose an al e na i e o double Dini
con inuous coefficien s which again bypasses he coun e example. In pa icula ,
unde hypo heses (1.2) and (1.3) we conside a e y weak solu ion in Lq
loc(Ω),
wi h q>n
.
Rema k 1.4. In [15] Zhang and Bao deal wi h he case o e y weak solu ions
u∈L1
loc(Ω) o (1.5), in e p e ing he coefficien s as Lipschi z unc ions, due
o he assump ion made on he solu ions. Thus ou esul ep esen s a na u al
ex ension om hei esea ch.
2. No a ion and p elimina y esul s
We collec he e he main defini ions and no a ion and some use ul esul s ha
will be needed in he sequel.
2.1. No a ion
In he ollowing, we deno e by B (x)={y∈Rn:|y−x|< } he ball o adius
cen e ed a x.
We indica e by {e1,...e
n} he canonical basis o Rn.Gi enh∈R {0},
o a measu able unc ion ψ:Rn→Rand o =1,...,n, we in oduce he
no a ion
Δ
hψ:= ψ(x+he)−ψ(x)
h
o he inc emen al quo ien in he - h di ec ion. We ecall ha o e e y pai
o unc ions ϕ, ψ,weha e
Δ
h(ϕψ)=Δ

hϕψ+ϕ(x+he)Δ

hψ. (2.1)
The ollowing esul pe aining o diffe ence quo ien s o unc ions in
Sobole spaces is well known (see [8, P oposi ion 4.8] o example).
43 Page 4 o 23 D. A. L. Manna e al. NoDEA
Theo em 2.1. Le p>1;i ψ∈W1,p(Ω), hen Δ
hψ∈Lp(Ω) o any ΩΩ
sa is ying h<dis (Ω,∂Ω)
2, and we ha e
Δ
hψLp(Ω)≤DψLp(Ω).
I ψ∈Lp(Ω) and he e exis s L≥0such ha , o e e y h<dis (Ω,∂Ω),
=1,...,n, we ha e
Δ
hψLp(Ω)≤L,
hen ψ∈W1,p(Ω),DψLp(Ω)≤Land Δ
hψ→Dψin Lp(Ω)as h→0.
Finally, gi en p>1, we deno e by p=p
p−1 he conjuga e exponen o p.
2.2. Dini con inuous unc ions
We say ha a con inuous unc ion on Ω is Dini con inuous i he modulus
o con inui y ¯
Ω:[0,diam(Ω)] →R+defined by
¯
Ω( ):= sup
x,y∈Ω
|x−y|≤
| (x)− (y)|
sa isfies
ˆdiam(Ω)
0
¯
Ω( )
d < ∞.
We also deno e by CD(Ω) he space o Dini con inuous unc ions; i u ns
ou o be a Banach space equipped wi h he ollowing no m:
 CD(Ω) :=  ∞+ˆdiam(Ω)
0
¯
Ω( )
d ,
whe e ·
∞is he usual uni o m no m.
Le us ema k ha by he uni o m con inui y, any unc ion in CD(Ω) may
be ex ended up o he bounda y o Ω wi h he same modulus o con inui y.
Mo eo e ,
C0,α(Ω) ⊆CD(Ω),
o any 0 <α≤1, whe e C0,α(Ω) deno es he space o H¨olde con inuous
unc ions.
The space CD
c(Ω) will deno e he se o unc ions in CD(Ω) wi h compac
suppo in Ω.
Lemma 2.2. The space C∞
c(Ω) is dense in CD
c(Ω).
P oo . Le ∈CD
c(Ω) ha we ex end o ze o on Rn Ω and se ε(x)=
(ρε∗ )(x), whe e ρεis a s anda d mollifie . Then, i εis sufficien ly small,
ε∈C∞
c(Ω); we will p o e ha
ε→ in CD(Ω).(2.2)
I is easily seen ha εuni o mly con e ges o in Ω, hus in o de o
p o e (2.2) we will jus show ha
ˆdiam(Ω)
0
( − ε)Ω( )
d →0,

NoDEA On he egula i y o e y weak solu ions Page 5 o 23 43
as ε ends o 0. Obse e ha
( − ε)Ω( )= sup
x,y∈Ω
|x−y|<
{| ε(x)− (x)− ε(y)+ (y)|} ≤ ¯
Ω( )+(¯
ε)Ω( )
and
(¯
ε)Ω( )= sup
x,y∈Ω
|x−y|<
{| ε(x)− ε(y)|}
=sup
x,y∈Ω
|x−y|<
ˆρε(z)( (x−z)− (y−z)) dz
≤ˆρε(z)¯
Ω( )dz =¯
Ω( ),
which oge he yield
( ε− )Ω( )≤2¯
Ω( ).
On he o he hand, since ( ε− )Ω→0 poin wise, he domina ed con e gence
heo em implies
ˆdiam(Ω)
0
( ε− )Ω( )
→0,
which concludes he p oo o (2.2). 
Rema k 2.3. The p e ious esul ensu es ha CD
c(Ω) is a sepa able space,
no ing ha C1
c(Ω) is sepa able wi h espec o he usual no m  1,∞:=
|α|≤1Dα ∞,C1
c(Ω) ⊆CD
c(Ω) and ¯
Ω( )≤ D ∞, o e e y ∈C1
c(Ω).
Lemma 2.4. Le , ε,and gbelonging o CD(Ω) such ha εcon e ges o
in CD; hen g εcon e ges o g in CD.
P oo . As be o e, i is enough o p o e he con e gence o he semino m since
he uni o m con e gence is immedia e. Then, w i ing he defini ion o he
modulus o con inui y, we ha e
g( ε− ) Ω( )= sup
x,y∈Ω
|x−y|<
{|g(x)( ε(x)− (x)) −g(y)( ε(y)− (y))|}
≤sup
x,y∈Ω
|x−y|<
{|g(x)||( ε(x)− (x)) −( ε(y)− (y))|}
+sup
x,y∈Ω
|x−y|<
{|g(x)−g(y)|| (y)− ε(y)|}
≤g∞( − ε)Ω( )+¯gΩ( ) − ε∞.
(2.3)
Hence,
ˆdiam(Ω)
0
[g( ε− )]Ω( )
d ≤g∞ˆdiam(Ω)
0
( − ε)Ω( )
d
+ − ε∞ˆdiam(Ω)
0
¯gΩ( )
d ,
43 Page 6 o 23 D. A. L. Manna e al. NoDEA
which goes o ze o as ε ends o ze o. 
2.3. C1-Dini egula i y o solu ions o di e gence o m ellip ic equa ions wi h
Dini-con inuous coefficien s
Fo he p oo o ou esul , we will need he ollowing ex ension o he Schaude
egula i y heo y o ellip ic equa ions in di e gence o m wi h Dini con inuous
coefficien s (see [12, Theo em 1.1] and [6, Theo em 1.3], see also [11]whichis
inclusi e o he pa abolic case.). Fo he Lp- egula i y heo y we e e o [7],
whe e he gene al case o VMO coefficien s is ea ed (see also [13, Theo em
5.5.3 (a)] o [3, Theo em 2.2. Chap e 10] o he case o con inuous coeffi-
cien s).
Theo em 2.5. Fo Ω⊂Rn,le aij sa is y (1.3)and (1.4); we conside =
( 1,
2,...,
n)wi h j∈C∞
c(Ω) o all j∈{1,...,n}. Assume ha u∈H1(Ω)
is a weak solu ion o he equa ion

i,j
Dj(aijDiu)=
j
Dj jin Ω(2.4)
Then u∈C1,D(Ω), o any bounded open se Ω,ΩΩ.
Mo eo e , le ΩaC1,1bounded open subse o Rn,le aij sa is y (1.2)
and (1.3),andle j∈Lp(Ω), o e e y j∈{1,...,n},wi h1<p<∞, hen
he e exis s a unique solu ion u∈W1,p
0(Ω) o he p oblem

i,j ˆΩ
aijDiuDjϕdx=
jˆΩ
jDjϕdx ∀ϕ∈W1,p
0(Ω),
and
uW1,p(Ω) ≤C
j
 jLp(Ω) (2.5)
holds, whe e Cdepends on n, λ, Λ,p,∂Ω,AW1,n(Ω,Rn×n).
Rema k 2.6. The fi s conclusion o Theo em 2.5 comes wi h an es ima e o
he Dini modulus o con inui y o Du in ol ing he Dini modulus o con i-
nui y o aij and j. Ac ually, in [12, Theo em 1.1] and in [6, Theo em 1.3]
only he con inui y o Du is p o ed and hese esul s a e ob ained wi h a
weake assump ion on he coefficien s aij . Assuming (1.4) o he coefficien s
we a e able o p o e also he Dini con inui y o he g adien o he solu ion. In
“Appendix A” we will esume in b oad e ms he p oo o [12, Theo em 1.1],
de eloping i in o de o ge he needed Dini con inui y esul .
2.4. C2- egula i y o solu ions o non di e gence o m ellip ic equa ions wi h
Dini-con inuous coefficien s
Le us fi s ecall he W2,p-sol abili y o he Di ichle p oblem o non di e -
gence ellip ic equa ions wi h discon inuous coefficien s (see [4, Theo em 4.2
and Theo em 4.4]).
Theo em 2.7. The Di ichle p oblem
⎧
⎨
⎩

i,j
aij(x)Diju= a.e.in Ω
u=0 on ∂Ω
(2.6)
NoDEA On he egula i y o e y weak solu ions Page 7 o 23 43
whe e Ωis a C1,1smoo h and bounded subse o Rn, ∈Lp(Ω) wi h 1<p<
∞,andaij sa is ies (1.2)and (1.3), admi s a unique solu ion u∈W2,p(Ω) ∩
W1,p
0(Ω) and
||u||W2,p(Ω) ≤C||u||Lp(Ω) +|| ||Lp(Ω),(2.7)
whe e he cons an Cdepends on n, p, λ, Λ,∂Ω,AW1,n(Ω,Rn×n).
The nex esul specifies es ima e (2.7); i s p oo is qui e s anda d bu
we p e e o w i e i o he sake o comple eness.
P oposi ion 2.8. Suppose uis a solu ion o he ellip ic Di ichle p oblem (2.6)
wi h aij, ,p and Ωas abo e. Then
||u||W2,p(Ω) ≤C|| ||Lp(Ω).(2.8)
P oo . Le
L=L=
i,j
aijDij ,sup
i,j
||aij||W1,n(Ω) ≤2M, λ|ξ|2≤
i,j
aij(x)ξiξj≤Λ|ξ|2;
ha ing in mind Theo em 2.7, i we p o e ha o any ope a o L∈Land o
any ∈Lp(Ω), he solu ion uo
Lu = a.e. in Ω
u=0 on∂Ω,
sa isfies
||u||Lp(Ω) ≤C|| ||Lp(Ω),
we a e done. Suppose i is no he case, hen his is equi alen o say ha o
e e y N∈N, he e exis s an ope a o LN=i,j aN
ij Dij ∈Land a unc ion
N∈Lp(Ω) such ha he co esponding solu ion uN o he Di ichle p oblem
LNuN= Na.e. in Ω
uN=0 on∂Ω,
sa isfies
||uN||Lp(Ω) >N|| N||Lp(Ω).(2.9)
Le us define N=uN/uNLp(Ω) and gN= N/uNLp(Ω),so ha N
sol es (2.6) wi h LNand gN.By heW2,p es ima e (2.7),
|| N||W2,p(Ω) ≤C|| N||Lp(Ω) +||gN||Lp(Ω)<C1+ 1
N,
whe e Cdoes no depend on Nand hence,
|| N||W2,p(Ω) ≤C. (2.10)
Thus Nis a p ecompac sequence: up o a non elabeled subsequence, we can
suppose Nu
∗weakly in W2,p(Ω), o some u∗∈W2,p(Ω), mo eo e u∗∈
W2,p(Ω) ∩W1,p
0(Ω). Simila ly, we can also say ha , o e e y i, j =1,...,n,
43 Page 8 o 23 D. A. L. Manna e al. NoDEA
aN
ij a
∗
ij weakly in W1,n(Ω) and aN
ij →a∗
ij s ongly in Lq(Ω) ∀1≤q<∞.
Thus, he ope a o L∗=i,j a∗
ijDij belongs o Land o ϕ∈Lp(Ω) we ha e
ˆΩ
(LN N−L∗u∗)ϕdx

≤
n

i,j=1 ˆΩ
(aN
ij −a∗
ij)∂2 N
∂xi∂xj
ϕ
dx +ˆΩ
a∗
ijϕ∂2 N
∂xi∂xj
−∂2u∗
∂xi∂xjdx
≤C
n

i,j=1
(aN
ij −a∗
ij)ϕLp(Ω) +
n

i,j=1 ˆΩ
a∗
ijϕ∂2 N
∂xi∂xj
−∂2u∗
∂xi∂xjdx.
The e o e, LN Ncon e ges weakly in Lp(Ω) o L∗u∗. On he o he hand, using
(2.9), we ha e
||gN||Lp(Ω) <1
N.
Passing o he limi in he equa ion sa isfied by N, we disco e ha he limi
u∗∈W2,p(Ω) ∩W1,p
0(Ω) sa isfies L∗u∗= 0 a.e. in Ω. By he uniqueness
p ope ies o he solu ions o (2.6), i ollows ha u∗=0.Thus Ncon e ges
o ze o and he a gumen becomes con adic o y since  NLp(Ω) =1. 
In [6, Theo em 1.5] i is shown ha solu ions o ellip ic equa ions in
non di e gence o m wi h ze o Di ichle bounda y condi ions a e C2up o he
bounda y when he leading coefficien s a e Dini con inuous unc ions.
Theo em 2.9. Assume ha Ωis a C2,1smoo h and bounded open subse o Rn,
∈CD(Ω) and aij sa is ies (1.2),(1.3),and(1.4).Le u∈W2,2(Ω)∩W1,2
0(Ω)
be a solu ion o he Di ichle p oblem
⎧
⎨
⎩

i,j
aij(x)Diju= a.e. in Ω
u=0 on ∂Ω,
(2.11)
hen u∈C2(Ω).
Rema k 2.10. The assump ion in [6] abou he coefficien s is weake hen (1.4),
since hey assume ha he modulus o con inui y
˜
AΩ( ):=
i,j
sup
x∈Ω
−
ˆB (x)∩Ω
|aij(y)−(aij)B (x)∩Ω|dy
wi h (aij)B (x)∩Ω=−
ˆB (x)∩Ω
aij, sa isfies
ˆ0
˜
AΩ( )
d < ∞.
NoDEA On he egula i y o e y weak solu ions Page 15 o 23 43
A he end, he es ima es p o ed o I1,I
2and I3lead o

j
μj,ξ
j≤C Lp(Ω,Rn),
as well

j
μjξ, j≤C Lp(Ω,Rn).
Since is an a bi a y smoo h unc ion in Lp(Ω,Rn), we conclude

j
μjξLp(Ω)≤C,
which means, using a fini e co e ing a gumen , ha μjis a unc ion in Lp
loc(Ω)
and hen u∈W1,p
loc (Ω), since, o e e y ϕ∈C∞
c(Ω) and o hsmall enough,
we ha e
ˆΔj
huϕdx=ˆuΔj
−hϕdx;
passing o he limi as h→0, we de i e
μj,ϕ=ˆϕμjdx =−ˆuDjϕdx.
Since u∈W1,p
loc (Ω), B ezis’s esul implies ha uis a weak solu ion o
he equa ion (1.5), i.e. ou s a emen . 
4. Sobole coefficien s
As poin ed ou in he In oduc ion, e y weak solu ions in Ln
loc(Ω) associa ed
o coefficien s in W1,n(Ω) a e no weak solu ions, due o he coun e example
ound in [10]. The quo ed e e ences on his p oblem ha e sugges ed us o
conside Sobole coefficien s wi h a modulus o con inui y sa is ying he double
Dini condi ion.
On he o he hand, ano he way o ge a ound he coun e example is
o deal wi h e y weak solu ions in Lq
loc(Ω), wi h q>n
. The esul is he
ollowing.
Theo em 4.1. Le u∈Lq
loc(Ω),q>n
, be a e y weak solu ion o (1.1),wi h
A(x)=(aij (x))i,j sa is ying (1.2)and (1.3), hen ubelongs o W1,2
loc (Ω) and
hus i is a weak solu ion.
P oo . The p oo es s on a duali y and a boo s ap a gumen .
S ep 1 We claim ha u∈W1,qn
q−n
loc (Ω).
We p oceed as in he S ep 1 o he p oo o Theo em 1.2 o a i e o (3.2).
Now we es ima e he six e ms Im.WeuseH¨olde ’s inequali y and P oposi ion
2.8 o ge

43 Page 16 o 23 D. A. L. Manna e al. NoDEA
|I1|≤DηL∞(Rn)AW1,n(Ω,Rn×n)uLq(Ω0)D 
L
qn
q−n(Ω0,Rn)
≤Cw
L
qn
q−n(Ω0)
,
|I2|≤AW1,n (Ω,Rn×n)uLq(Ω0)D2 
L
qn
q−n(Ω0,Rn×n)
≤Cw
L
qn
q−n(Ω0)
,
|I3|,|I4|≤ΛuLq(Ω0)DηL∞(Rn,Rn)D2 Lq(Ω0,Rn×n)≤CwLq(Ω)≤Cw
L
qn
q−n(Ω)
,
|I5|≤ΛuLq(Ω0)D2ηL∞(Rn,Rn×n)D Lq(Ω0,Rn)≤CwLq(Ω)≤Cw
L
qn
q−n(Ω)
,
and finally, as o (3.3),
|I6|≤Cw
L
qn
q−n(Ω)
+CwLq(Ω)≤Cw
L
qn
q−n(Ω)
.
So, a guing as in he S ep 1 o Theo em 1.2, we deduce
ˆΩ
wΔ
hudx

≤Cw
L
qn
q−n(Ω)
,
which in u n implies, hanks also o Theo em 2.1, ha u∈W1,qn
q−n
loc (Ω).
Le us no e ha hanks o his, he equa ion sa isfied by umay be ew i en
as

i,j ˆaij(x)DiuDjϕdx=0,(4.1)
whe e he es unc ions ϕcan be aken in W1,qn
q−n(Ω) wi h compac suppo .
On he o he hand, he summabili y o he solu ion uis no imp o ed by i s
belonging o his Sobole space, since qn
q−n=qn
qn−q+nand he Sobole
conjuga e o qn
qn−q+nis q.
S ep 2 We p o e ha u∈W1,q
loc (Ω).
As in he S ep 2 o he p oo o Theo em 1.2, o j∈{1,...,n}le
=( 1,...,
n) wi h j∈C∞
c(Ω) be such ha

j
|| j||Lq(Ω)≤1.
Fo e e y p>1, le ∈W1,p
0(Ω) be he weak solu ion o he p oblem

i,j ˆaijDi Djϕdx=
jˆDjϕ jdx ∀ϕ∈W1,p
0(Ω).(4.2)
By Theo em 2.5 we ha e in pa icula ha
|| ||W1,q(Ω)≤C|| ||Lq(Ω,Rn).
NoDEA On he egula i y o e y weak solu ions Page 17 o 23 43
As be o e, we ake BR/2⊂BR⊂Ωa pai o concen ic balls cen e ed
a x0∈Ωandweconside ξ(x)=ξ(|x−x0|) a smoo h unc ion such ha
ξ( )=1 o ∈[0,R/2] and ξ( )=0 o ≥R.Wecanchooseϕ= ξ in
(4.1)andϕ=uξ as es unc ion in (4.2), so ha

i,j ˆaijDiuDj ξdx+
i,j ˆaijDiuDjξ dx =0,
and

i,j ˆaijDi Djuξdx+
i,j ˆaijDi Djξudx
=
jˆ jDjuξdx+
jˆ jDjξudx.
Sub ac ing he wo equa ions and using he symme y o aij we ge

jˆ jDjuξdx=−
jˆ jDjξudx +
i,j ˆaijDi Djξudx
−
i,j ˆaijDiuDjξ dx =I1+I2+I3.
We es ima e he h ee e ms Im.Weha e
|I1|≤uLq(Ω)DξL∞(BR,Rn) Lq(Ω,Rn)≤C Lq(Ω,Rn),
|I2|≤ΛDξL∞(BR,Rn)uLq(Ω)D Lq(Ω,Rn)≤C Lq(Ω,Rn),
and finally
|I3|≤Λu
W1,(qn
q−n)
(Ω)
 
L
qn
q−n(Ω)
≤C W1,q(Ω),
whe e he las inequali y de i es om he ac ha he Sobole conjuga e o
qis qn
q−n. To sum up we ha e ob ained

jˆ jξDjudx
≤C Lq(Ω,Rn),
as well
ξDuLq(Ω,Rn)≤C,
and, using a fini e co e ing a gumen , his implies ha u∈W1,q
loc (Ω). Le
us obse e ha his Sobole egula i y imp o es he summabili y o u.In
pa icula , u∈Lq∗
loc(Ω), whe e q∗is he Sobole conjuga e o q.
S ep 3 We claim ha i q>n hen uis a weak solu ion.
By he p e ious s ep, we deduce ha i q>n hen he solu ion uis in
L∞
loc(Ω). A his poin , i is no difficul o p o e, a guing as in S ep 1, ha
u∈W1,n
loc (Ω).
S ep 4 We p o e ha u∈L∞
loc(Ω).
43 Page 18 o 23 D. A. L. Manna e al. NoDEA
We jus obse ed ha i q>nwe a e done. Le us conside now q≤n.
The solu ion uis in W1,q
loc (Ω) and by he Sobole ’s embedding u∈Lq∗
loc(Ω),
whe e q∗=qn
n−qi q<nand any numbe g ea e hen 1 i q=n. A guing
exac ly as in he S ep 2 we de i e ha u∈W1,q∗
loc (Ω), which in u n implies
ha uis in L∞
loc(Ω) i q∗>n. We al eady no iced in S ep 3 ha his gi es he
desi ed esul . Le us obse e ha i q=n,q∗is any numbe g ea e hen 1
and so his can be chosen g ea e hen n, while i q<n,q∗>nis equi alen
o q>n
2. We can i e a e his p ocedu e. Gi en q>n
=n
n−1a e (a mos )
n−1 imes we deduce ha uis locally bounded.
By S ep 3 he locally boundedness o he solu ion gi es he desi ed esul .

Acknowledgemen s
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wi hin he CRUI-CARE Ag eemen . Open access unding p o ided by Uni-
e si `a degli S udi di Napoli Fede ico II wi hin he CRUI-CARE Ag eemen .
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P obabili `a e le lo o Applicazioni (GNAMPA) o INdAM. The fi s au ho was
pa ially suppo ed by he Academy o Finland G an 314227. The esea ch o
R. S. has been unded by PRIN P ojec 2017JFFHSH.
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Appendix A: The C1-Dini egula i y o solu ions o di e gence
o m ellip ic equa ions wi h Dini-con inuous coefficien s
As announced in Rema k 2.6, we will speci y he modulus o con inui y o
he g adien o solu ions o (2.4) in he p oo o [12, Theo em 1.1]. We will
conside only he main poin s o he p oo e e ing o he es o [12]. The se
Ω is supposed o be he ball B4cen e ed a 0 and Ω=B1. The imp o emen
ega ds P oposi ion 1.1 in [12]: o he sake o comple eness we will ske ch he
p oo , modi ying he o iginal when needed.
NoDEA On he egula i y o e y weak solu ions Page 19 o 23 43
P oposi ion A.1. Fo B4⊂Rn,n≥1,le aij, de ined on B4, sa is y (1.3)
and (1.4)and le =( 1,
2,...,
n)wi h j∈C∞
c(B4) o all j∈{1,...,n}.
Assume ha u∈H1(B4)is a weak solu ion o (2.4), hen he e exis a∈R
and b∈Rnsuch ha
−
ˆB
|u(x)−(a+b·x)|dx ≤ δ( )[uL2(B2)+ C1(B2)],∀ ∈(0,1),(A.1)
whe e δ( ), depending on n, λ, Λ,and on he modulus o con inui y o aij and
, is a mono onically inc easing posi i e unc ion de ined on (0,1) sa is ying
ˆ1
0
δ( )
d < +∞.
Rema k A.2. As shown in [12, P oposi ion 1.2], δ( ) will be he modulus o
con inui y o Du.
P oo . The p oo is ca ied ou o = 0. We use he same no a ion o [12],
deno ing by ϕ he modulus o con inui y such ha
−
ˆB
|A−A(0)|21
2
≤ϕ( ),
whe e A=(aij)i,j. Obse e ha in ou case, assuming (1.4), ϕ( ) has he
ollowing o m
−
ˆB
|A−A(0)|21
2
≤−
ˆB
¯
AB4(|x|)2dx1
2
≤C1
nˆ
0
¯
AB4(ρ)2ρn−1dρ1
2
=: ϕ( ),
which is double-Dini con inuous since ϕ( )≤¯
AB4( ), and sa isfies
max
/2≤s≤ ϕ(s)≤μϕ( ),
wi h μ>1. As in [12], by induc ion, one will find, o k≥0, wk∈H1(B3/4k+1 )
such ha

i,j
Dj(aij(0)Diwk)=0 inB3/4k+1 ,(A.2)
wkL2(B2/4k+1 )≤C4−k(n+2)
2ϕ(4−k),DwkL∞(B1/4k+1 ,Rn)Cϕ(4−k),
(A.3)
D2wkL∞(B1/4k+1 ,Rn×n)≤C4kϕ(4−k),(A.4)
u−
k

j=0
wjL2(B1/4k+1 )≤4−(k+1)(n+2)
2ϕ(4−(k+1)),(A.5)
and
wkL∞(B1/4k+1 )≤C4−kϕ(4−k),(A.6)
see [12, (14), (15), (16), (17), and (18) o P oposi ion 1.1]. He e and in he
sequel Cwill deno e a uni e sal cons an .
43 Page 20 o 23 D. A. L. Manna e al. NoDEA
Fo x∈B1/4k+1 , using (A.3), (A.4), (A.6) and Taylo expansion,
|
k

j=0
wj(x)−
∞

j=0
wj(0) −
∞

j=0
Dwj(0) ·x|
≤
∞

j=k+1
(|wj(0)|+|Dwj(0)||x|)+
k

j=0
D2wjL∞(B1/4k+1 ,Rn×n)|x|2
≤C
∞

j=k+1
(4−jϕ(4−j)+ϕ(4−j)|x|)+C
k

j=0
4jϕ(4−j)|x|2
≤C4−(k+1) ˆ4−k
0
ϕ( )
d +C|x|2ˆ1
|x|
2
ϕ( )
2d .
(A.7)
We hen de i e om (A.5)and heabo e,usingH¨olde ’s inequali y, ha
ˆB1/4k+1
|u(x)−
∞

j=0
wj(0) −
∞

j=0
Dwj(0) ·x|dx
≤
k

j=0
wj(x)−
∞

j=0
wj(0) −
∞

j=0
Dwj(0) ·xL1(B1/4k+1 )
+u−
k

j=0
wj(x)L1(B1/4k+1 )
≤C4−(k+1)(n+1) ˆ4−k
0
ϕ( )
d +Cˆ1/4k+1
0
ρn+1 ˆ1
ρ
2
ϕ( )
2d dρ
+C4−(k+1)(n+1)ϕ(1/4k+1).
(A.8)
P oposi ion A.1 ollows om he abo e wi h a=
∞

j=0
wj(0), b=
∞

j=0
Dwj(0) ·x,
and
δ( )≃ˆ
0
ϕ(s)
sds +1
n+1 ˆ
0
ρn+1 ˆ1
ρ
2
ϕ(s)
s2ds dρ +ϕ( ),
he symbol ≃s anding o = up o a cons an . I emains o p o e ha δ( )
is a Dini modulus o con inui y. Thanks o assump ion (1.4), i occu s i we
show he Dini con inui y o he second e m in he p e ious sum. I yields
1
n+1 ˆ
0
ρn+1 ˆ1
ρ
2
ϕ(s)
s2ds dρ ≤ ˆ1
2
ϕ(s)
s2ds +ˆ
0
ϕ(ρ)
ρdρ,
so ha , in eg a ing by pa s,
ˆ0ˆ1
2
ϕ(s)
s2ds d = ˆ1
2
ϕ(s)
s2ds0
+ˆ0
ϕ(
2)
/4d .

NoDEA On he egula i y o e y weak solu ions Page 21 o 23 43
I is easy o see ha lim →0 ´1
2
ϕ(s)
s2ds = 0, and hus he hesis ollows by he
Dini con inui y o ϕ.
Appendix B: The coun e example
To cons uc he example, one fi s conside s, o ∈(0,1) and o β>1, he
unc ion
α( )= −βn
(n−1) log  0
+β(β+1)
(n−1) log2 0
,
o some 0>1. One akes hen A(x)=(aij(x))i,j defined by
aij(x)=δij +α(|x|)δij −xixj
|x|2,
wi h 0la ge enough so ha α≥−
1
2,Abeing hen uni o mly ellip ic.
Le us check now ha A∈W1,n(B1,Rn×n). Simple compu a ion gi es

∂aij
∂x
|α(|x|)|+|α(|x|)|1
|x|,
o e e y i, j,  =1,...,n ( he symbol s and o ≤up o a cons an ). On
he o he hand
|α(|x|)|≃ 1
|x|log2 0
|x|+1
|x|log3 0
|x|,
which in u n implies

∂aij
∂x
1
|x|log  0
|x|+1
|x|log2 0
|x|+1
|x|log3 0
|x|1
|x|log  0
|x|,
i 0is big enough. Thus, he belonging o A o W1,n(B1,Rn×n) is p o ided
by he es ima e
ˆB1
∂aij
∂x
n
dx ˆB1
1
|x|nlogn 0
|x|dx ≃ˆ1
0
1
logn 0
d < +∞.
Wi h such an A, which is con inuous bu no Dini-con inuous, in [10] he
au ho s cons uc a solu ion o (1.1) (in he sense o dis ibu ions) u∈
W1,1
loc (B1) W1,p
loc (B1) o e e y p>1. In pa icula , le us obse e ha such a
solu ion belongs o Ln
loc(B1).
Re e ences
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ions (wi h an appendix by Ha¨ım B ezis). J. Func . Anal. 257, 2124–2158 (2009)
[2] B ezis, H.: On a conjec u e o J. Se in. Rend. Lincei Ma . Appl. 19, 335–338
(2008)
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[3] Chen, Y.Z., Wu, L.C.: Second O de Ellip ic Equa ions and Ellip ic Sys ems.
T ansla ions o Ma hema ical Monog aphs, ol. 174. Ame ican Ma hema ical
Socie y, P o idence (1998)
[4] Chia enza, F., F asca, M., Longo, P.: W2,p- sol abili y o he Di ichle p oblem
o non di e gence ellip ic equa ions wi h VMO coefficien s. T ans. Am. Ma h.
Soc. 336, 841–853 (1993)
[5] De Gio gi, E.: Sulla diffe enziabili `a e l’anali ici `a delle es emali degli in eg ali
mul ipli egola i. Mem. Accad. Sci. To ino Cl. Sci. Fis. Ma . Na . 3, 25–43 (1957)
[6] Dong, H., Escau iaza, L., Kim, S.: On C1,C2, and weak ype-(1,1) es ima es
o linea ellip ic ope a o s, pa II. Ma h. Ann. 370, 447–489 (2018)
[7] Di Fazio, G.: Lpes ima es o di e gence o m ellip ic equa ions wi h discon in-
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[8] Giaquin a, M., Ma inazzi, L.: An In oduc ion o he Regula i y Theo y o
Ellip ic Sys ems. Ha monic Maps and Minimal G aphs. Edizioni della No male,
Pisa (2012)
[9] Hage , R.A., Ross, J.: A egula i y heo em o linea second o de ellip ic di e -
gence equa ions. Ann. Sc. No m. Sup. Pisa 23, 283–290 (1971)
[10] Jin, T., Maz’ya, V., Scha ingen, J.V.: Pa hological solu ions o ellip ic p oblems
in di e gence o m wi h con inuous coefficien s. C. R. Acad. Sci. Pa is Se . I 347,
773–778 (2009)
[11] Kuusi, T., Mingione, G.: New pe u ba ion me hods o nonlinea pa abolic
p oblems. J. de Ma h. Pu es e Appl. 98(4), 390–427 (2012)
[12] Li, Y.: On he C1 egula i y o solu ions o di e gence o m ellip ic sys ems wi h
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NoDEA On he egula i y o e y weak solu ions Page 23 o 23 43
Domenico Angelo La Manna
Depa men o Ma hema ics and S a is ics
Uni e si y o Jy askyl¨a
P.O. Box 35 (MaD)40014 Jy askyla
Finland
e-mail: [email p o ec ed]
Chia a Leone and Robe a Schia a ella
Dipa imen o di Ma ema ica e Applicazioni “R. Caccioppoli”
Uni e si `a di Napoli Fede ico II
Via Cin ia
80126 Naples
I aly
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Robe a Schia a ella
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Recei ed: 10 Ma ch 2020.
Accep ed: 3 July 2020.