Elec onic Jou nal o Di e en ial Equa ions, Vol. 2020 (2020), No. 69, pp. 1–19.
ISSN: 1072-6691. URL: h p://ejde.ma h. xs a e.edu o h p://ejde.ma h.un .edu
H¨
OLDER CONTINUITY FOR VECTOR-VALUED MINIMIZERS
OF QUADRATIC FUNCTIONALS
JOSEF DANˇ
Eˇ
CEK, EUGEN VISZUS
Abs ac . In his a icle we gi e a su icien condi ion o in e io e e ywhe e
H¨olde con inui y o weak minimize s o a class o quad a ic unc ionals wi h
coe icien s Aαβ
ij (·, u) belonging o he V MO-class, uni o mly wi h espec o
u∈RN, and con inuous wi h espec o u. The condi ion is global. I is
ypical o he unc ionals belonging o he class ha he con inui y moduli
o hei coe icien s become slowly g owing su icien ly a om ze o. Some
ea u es o he main esul a e illus a ed by examples.
1. In oduc ion
The aim o his a icle is o s udy he in e io e e ywhe e egula i y o unc ions
minimizing a ia ional in eg als
A(u; Ω) = ZΩ
Aαβ
ij (x, u)DαuiDβujdx (1.1)
whe e u: Ω →RN,N > 1, Ω ⊂Rn,n≥3 is a bounded open se , x= (x1, . . . , xn)∈
Ω, u(x) = (u1(x), . . . , uN(x)), Du ={Dαui},Dα=∂/∂xα,α= 1, . . . , n,i=
1, . . . , N.
Th oughou he whole ex we use he summa ion con en ion o e epea ed
indices. We call a unc ion u∈W1,2(Ω,RN) is a minimize o he unc ional
A(u; Ω) i and only i A(u; Ω) ≤ A( ; Ω) o e e y ∈W1,2(Ω,RN) such ha
u− ∈W1,2
0(Ω,RN). Fo mo e in o ma ion see [5, 10].
On he unc ional Awe assume:
(i) Aαβ
ij =Aβα
ji ,Aαβ
ij a e con inuous unc ions in u∈RN o e e y x∈Ω and
he e exis s M > 0 such ha Pi,j,α,β |Aαβ
ij (x, u)| ≤ M, o all x∈Ω, and
all u∈RN.
(ii) (ellip ici y) The e exis s ν > 0 such ha
Aαβ
ij (x, u)ξi
αξj
β≥ν|ξ|2,∀x∈Ω,∀u∈RN,∀ξ∈RnN .(1.2)
(iii) (oscilla ion o coe icien s) The e exis s a eal unc ion ωcon inuous on
[0,∞), which is bounded, nondec easing, conca e, ω(0) = 0 and such ha
2010 Ma hema ics Subjec Classi ica ion. 35J60.
Key wo ds and ph ases. Quad a ic unc ionals; minimize s; egula i y; Mo ey spaces.
c
2020 Texas S a e Uni e si y.
Submi ed Ap il 4, 2019. Published July 2, 2020.
1
2 J. DANˇ
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CEK, E. VISZUS EJDE-2020/69
o all x∈Ω and u, ∈RN
X
i,j,α,β |Aαβ
ij (x, u)−Aαβ
ij (x, )| ≤ ω(|u− |).(1.3)
We se ω∞= lim →∞ ω( )≤2M.
(i ) Fo all u∈RN,Aαβ
ij (·, u)∈V MO(Ω) (uni o mly wi h espec o u∈RN).
Assump ions (i) and (ii) allow us o conclude ha i u∈W1,2(Ω,RN) is a minimize
o (1.1) hen o any admissible unc ion ∈W1,2(Ω,RN)
ZΩ|Du|2dx ≤M
νZΩ|D |2dx . (1.4)
Conce ning he assump ion (iii) i is wo h o poin ou (see [5, p.169]) ha o
uni o mly con inuous coe icien s Aαβ
ij he e exis s a eal unc ion ωsa is ying he
assump ion (iii) and, ice e sa, (iii) implies he uni o m con inui y o coe icien s
and absolu e con inui y o ωon [0,∞).
In his pape we will conside he con inuous unc ion
ω( ) = (ω0( ) o 0 ≤ < 0, 0≥0
ω1( )≤ω∞, o 0≤ < ∞(1.5)
whe e ω0is an a bi a y con inuous, conca e, nondec easing unc ion, inc easing on
a neighbou hood o ze o such ha ω0(0) = 0 and he poin 0and he unc ion ω1
a e chosen in such a way ha ωp ese es i s con inui y and conca i y on [0,∞).
Wi h espec o (i ) i is wo h o ecall ha since he space o con inuous
unc ions is a p ope subse o V MO, he con inui y o coe icien s Aαβ
ij =Aαβ
ij (x, u)
wi h espec o xis no supposed. In he linea case, when he coe icien s Aαβ
ij =
Aαβ
ij (x) belong o C0,γ(Ω) he egula i y o minimize s o unc ionals as (1.1) is
well unde s ood (see [5, Tho ems 3.1, 3.2 on p.87, 88]). These esul s we e la e
gene alized o he case whe e he abo e coe icien s a e in V MO, hence possibly
discon inuous (see [4, 19] and e e ences he ein).
I is well known ha e en in he con inuous case he dependence o coe icien s
Aαβ
ij on uleads o weake egula i y esul s o minimize s. In dimension n≥3
he e a e examples o ec o ial quad a ic unc ionals (N > 1) wi h analy ic coe -
icien s Aαβ
ij =Aαβ
ij (u) whose minimize s a e discon inuous (see [10, p. 317], [11]).
Fo he analy ic coe icien s Aαβ
ij =Aαβ
ij (x, u) see coun e example in [18]. These
examples indica e ha , in gene al, only pa ial egula i y esul s can be achie ed
o minimize s o ec o ial unc ionals. Fo de ailed in o ma ion on his opic we
e e o sou ces [5]-[10] o classic esul s and o [13, 15, 19] o ecen esul s.
Besides he pa ial egula i y esul s, a ew e e ywhe e egula i y esul s we e
ob ained o some special ypes o ec o ial unc ionals (see [10, 15]). Ou pape
deals jus wi h he las men ioned ype o egula i y esul s. In he ecen pape s
[1, 3] condi ions gua an eeing he local H¨olde con inui y o minimize s o unc ional
(1.1) in Ω a e gi en. Because he pape [3] ex ends he esul s o [1], we men ion
only [3] in mo e de ail. Main esul s o he pape [3] a e s a ed in wo heo ems.
The i s o hem e e s ha i a quan i y exp essed by means o pa ame e s ω∞/ν
and M/ν is small enough, he minimize s o (1.1) a e egula . This esul is no
e y su p ising bu , mo eo e , an uppe bound (al hough p obably no op imal) o
he abo e men ioned quan i y is designed. In a case when he men ioned condi ion
EJDE-2020/69 H ¨
OLDER CONTINUITY 3
is no ul illed a su icien condi ion o egula i y o minimize s o unc ional (1.1)
is s a ed as well. A basic ad an age o he second condi ion in he pape [3] is,
ha i admi s ( o su icien ly big ellip ici y cons an ν) an a bi a y g ow h o he
con inui y modulus ω=ω( ) when is nea by ze o. He e i is need ul o no e ha
he second condi ion wo ks likewise when νis small bu , in his case, he modulus o
con inui y ωhas o g ow slowly enough. A disad an age o he condi ion is i s ”local
cha ac e ”, analogous o he egula i y condi ions in pa ial egula i y heo y. The
p esen pape essen ially ex ends esul s o [1] and [3]. He e we s udy he egula i y
o a ia ional in eg als, coe icien s o which sa is y (iii) wi h modulus o con inui y
gi en by (1.5). Toge he wi h mo e delica e es ima es and ca e ul designing o some
pa ame e s in p oo , i allows us o s a e he egula i y condi ion p ese ing all he
ad an ages o he p e ious men ioned condi ions om [1, 3] and, mo eo e , he
condi ion is o mula ed much simple and mo e exac ly han he p e ious ones in
[1, 3]. Consequen ly, i imp o es he possibili y o immedia e applica ion (i is well
isible mainly in he case o he Di ichle p oblem - see Rema k 1.4 below). I is
wo h o men ion ha he egula i y condi ion (exp essed by (1.6), (1.7), (1.8)) has,
compa ed o ha one om [3, Thm. 2], global ea u es. The me hods o p o ing he
main esul s a e based on hose ha we e de eloped in he classic pa ial egula i y
heo y ( see o example [5, 10]), bu hey a e essen ially modi ied. In Rema k 4.2
i is shown ha , in a case o spli coe icien s, joining he esul s o his pape wi h
hose om [12], we a e able o gua an ee he egula i y o minimize s o (1.1) in Ω.
Now we can o mula e he main esul .
Theo em 1.1. Le Ω0⊂⊂ Ω,n−2≤ϑ < n be gi en and he coe icien s Aαβ
ij o
he unc ional (1.1) sa is y (i), (ii), (iii) and (i ). The e exis s a posi i e cons an
Msuch ha i he minimize uo he unc ional (1.1) sa is ies he condi ion
1
|Ω|1−2/n ZΩ|Du|2dy ≤1
M2(1.6)
hen ubelongs o C0,(ϑ−n+2)/2(Ω0,RnN )when ϑ>n−2and o BMO(Ω0,RnN )
when ϑ=n−2. He e
M= sup
0< <∞e
Ψω( )
ε−e
Ψω( 0)
ε
− 0
and e
Ψω( 0)
ε≤2n+2pC2.(1.7)
Rema k 1.2. In he o egoing o mula he unc ion e
Ψ(u) = ue(u/2√µ)2/(2µ−1) ( o
u he p ope ies o e
Ψ see (2.1) below), 0≥0 ( 0is he pa ame e om he
de ini ion o ω, see (1.5)), ε=ω∞/Cρ
µ,Cµ= (µ/((p−1)e))µ, he cons an s µ≥6
and ρ > 1/p a e such ha
Cρp−1
µ≥K C2p
1C(p+1)/2
2Lpϑ/(n−ϑ)ω∞
νp|Ω|1−2/n
(2d)n−2(p−1)/2
(1.8)
in he case when he coe icien s Aαβ
ij depend only on u. He e p > 1 is om
Lemma 2.9, K= 2(n+11+(n+3)ϑ/(n−ϑ))p−(2n+5) κ1−p
n,Lis he cons an om Lemma
2.7 below, C1,C2a e he cons an s om Lemma 2.9 and 2.10 espec i ely, d=
dis (Ω0, ∂Ω)/2>0 and he symbol |·|s ands o he n-dimensional Lebesgue
measu e (κnis he Lebesgue measu e o he uni ball in Rn).
I Aαβ
ij =Aαβ
ij (x, u) hen, o mally, he cons an Kon he igh -hand side o (1.8)
is subs i u ed by 2K(he e, as i is isible a he end o he p oo o Theo em 1.1,
4 J. DANˇ
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CEK, E. VISZUS EJDE-2020/69
he mul iplie 2 could be subs i u ed by ano he one, bigge han 1). I is impo an
o elease ha he dependence o he coe icien s Aαβ
ij on a iable x ends o he
choice d= min{R0,dis (Ω0, ∂Ω)/2}( o de ini ion o R0see (3.25) below) and so d
and, consequen ly, he alue o he cons an Cρp−1
µ om (1.8) depend on ”VMO-
quali y” o x-dependence o coe icien s Aαβ
ij as well. B oadly speaking, he bigge
R0is, he be e egula i y esul one can ob ain.
Rema k 1.3. I is easily seen ha ins ead o he assump ion (i ) in he o egoing
Theo em 1.1 one can suppose he coe icien s Aαβ
ij o he unc ional (1.1) o be o
BMO-class wi h sui able small BMO semi-no ms (see (3.25) below).
Rema k 1.4. I is a consequence o he es ima e (1.4) ha i u∈W1,2(Ω,RN),
men ioned in he o egoing heo em, is such ha u−g∈W1,2
0(Ω,RN) o some
g∈W1,2(Ω,RN) ( he Di ichle p oblem o unc ional (1.1)), hen he le -hand
side o (1.6) can be eplaced by he e m
M
ν|Ω|1−2/n ZΩ|Dg|2dy .
The egula i y heo em, we o mula ed abo e, can be illus a ed wi h wo samples
o he unc ion ω, de ined by (1.5), o which we gi e es ima es o he pa ame e
M. B oadly speaking, i he coe icien s o he unc ional sa is y (iii) wi h some ω
gi en below and (1.8) is ul illed, we ha e he egula i y.
Example 1.5. Le
ω( ) = 






ω0( ) o 0 ≤ < 0,
ω∞ln 1 + eε/ω∞−1
γ
0 γ o 0≤ ≤ 1,0< γ ≤1,
ω∞ o > 1
(1.9)
whe e ω0is an a bi a y con inuous, conca e, nondec easing unc ion such ha
ω0(0) = 0 and he poin s 0, 1a e chosen so ha ωis con inuous and conca e
on [0,∞). I we pu ε=ω∞/Cρ
µin (1.9) hen he igh -hand side o (1.6) can be
chosen in he o m (see Appendix o mo e in o ma ion)
1
M2= 0
10C
2
2µ−1ρ
µ
min n1,3C
2
2µ−1ρ
µ
eC
2
2µ−1ρ
µo2.(1.10)
He e µ≥6, ρ > 1/p and 0>0.
Example 1.6. Le
ω( ) = 2ω∞
πa c an 
Cτ
µ o 0 ≤ < ∞(1.11)
hen he cons an om (1.6) can ha e he o m (in his case 0= 0, see Appendix
as well)
1
M2=Cτ−ρ
µ
eCρ
µ
2√µ2
2µ−12
.(1.12)
He e τ > ρ > 1/p,µ≥6 sa is y (1.8) and e
Ψ(ω( 0)/ε) = 0.
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OLDER CONTINUITY 5
2. P elimina ies
I x∈Rnand is a posi i e eal numbe , we se B (x) = {y∈Rn:|y−x|< },
Ω (x) = Ω ∩B (x). Deno e by
ux, =1
|Ω (x)|ZΩ (x)
u(y)dy =−
ZΩ (x)
u(y)dy
he mean alue o he unc ion u∈L1(Ω,RN) o e he se Ω (x) whe e he symbol
|·|deno es he n-dimensional Lebesgue measu e. Mo eo e , we se φ( ) = φ(x, ) =
RB (x)|Du(y)|2dy,U =U (x) = 2−nφ(x, ) o B (x)⊂Ω. Beside he s anda d
space C∞
0(Ω,RN), H¨olde space C0,α(Ω,RN) and Sobole spaces Wk,p(Ω,RN),
Wk,p
0(Ω,RN) we use Mo ey spaces Lq,λ(Ω,RN) (see, e.g. [5, 14]). We will deno e
by Xloc(Ω,RN) he space o all unc ions which belong o X(e
Ω,RN) o any bounded
subdomain e
Ω wi h smoo h bounda y which is compac ly embedded in Ω.
We ecall a de ini ion o V MO - spaces and a ew p ope ies o Mo ey spaces.
We se o ∈L1(Ω), 0 <a<∞
Na( , Ω) := sup
x∈Ω, <a −
ZΩ (x)| (y)− x, |dy.
De ini ion 2.1 (see [20]).A unc ion ∈L1(Ω) is said o belong o BMO(Ω) i
Ndiam Ω( , Ω) <∞.
A unc ion ∈L1(Ω) is said o belong o V MO(Ω) i
lim
a→0Na( , Ω) = 0.
P oposi ion 2.2. Fo a bounded domain Ω⊂Rnwi h he Lipschi z bounda y, o
q∈(1,∞)and 0<λ<µ<∞we ha e he ollowing:
(a) Lq,µ(Ω,RN)⊂Lq,λ(Ω,RN).
(b) I u∈W1,2
loc (Ω,RN)and Du ∈L2,λ
loc (Ω,RnN ),n−2< λ < n hen u∈
C0,(λ−n+2)/2(Ω,RN).
(c) I u∈W1,2
loc (Ω,RN)and Du ∈L2,n−2
loc (Ω,RnN ) hen u∈BMOloc(Ω,RN).
(d) Lq,n(Ω,RN)is isomo phic o he L∞(Ω,RN).
(e) L∞(Ω,RN)$BMO(Ω,RN).
Le now Φ, Ψ be a pai o complemen a y Young unc ions
Φ(u) = ulnµ
+(au) o u≥0,
Ψ(u)≤Ψ(u) = 1
aue(u
2√µ)2/(2µ−1) =1
ae
Ψ(u) o u≥0(2.1)
whe e a > 0, µ≥2 a e cons an s, and
ln+(au) = (0 o 0 ≤u < 1/a,
ln(au) o u≥1/a. (2.2)
Then he Young inequali y o Φ and Ψ eads
u ≤Φ(u) + Ψ( ), u, ≥0.(2.3)

6 J. DANˇ
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Lemma 2.3 ([21, p.37]).Le φ: [0,∞)→[0,∞)be a non dec easing unc ion
which is absolu ely con inuous on e e y closed in e al o ini e leng h, φ(0) = 0. I
w≥0is measu able and l( ) = {y∈Rn:w(y)> } hen
ZRn
φ◦w dy =Z∞
0|l( )|φ0( )d .
Lemma 2.4. Le ≥0,b > 0,µ > 0and q > 1be a bi a y. Then
lnµ
+(b )≤Cµbq−1 q(2.4)
whe e Cµ=µ
(q−1)e µ.
Fo a p oo o he abo e lemma, calcula e sup lnµ
+(b )
q−1; ∈(0,∞). The nex
Lemma is aken om [1, Lemma 6].
Lemma 2.5. Le A,R0≤R1be posi i e numbe s, n−2≤ϑ<n,ηa nonnega i e
and nondec easing unc ion on (0,∞). Then he e exis 0,cposi i e so ha o
any nonnega i e, nondec easing unc ion φde ined on [0,2R1]and sa is ying wi h
(B1+B2η(U2R0)) ∈[0, 0] he inequali y
φ(σ)≤Aσ
Rn+1
21 + Aσ
Rn[B1+B2η(U2R)]φ(2R) (2.5)
o all σ,Rsuch ha 0< σ < R ≤R0, i holds
φ(σ)≤cσϑφ(2R0),∀σ: 0 < σ ≤R0.(2.6)
Rema k 2.6. No e ha we can ake
0=1
2(2n+1A)ϑ
n−ϑ
, c =(2n+1A)1
n−ϑ
2R0ϑ.
Lemma 2.7 ([5, p.78]).Gi en he sys em
−DαAαβ
ij Dβuj= 0, i = 1, . . . , N
whe e Aαβ
ij a e cons an s sa is ying (i) and (ii). The e exis s a cons an L=
L(n, N, M/ν)≥1such ha o e e y weak solu ion u∈W1,2(Ω,RN), o e e y
x∈Ωand 0< σ ≤R≤dis (x, ∂Ω) he ollowing es ima e holds,
ZBσ(x)|Du(y)|2dy ≤Lσ
RnZBR(x)|Du(y)|2dy .
Rema k 2.8. No e ha
L=c(n, N)M
ν2k, k = 1 + n
2
and o n= 3 and N= 2 i holds
L < 104M
ν4.(2.7)
One o he ools o he p oo o ou main esul is he ollowing e e se H¨olde
inequali y ha is s anda d in ou se ing .
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OLDER CONTINUITY 7
Lemma 2.9 (see [5, 10]).Le u∈W1,2(Ω,RN)be a minimum o he unc ional
(1.1) unde he assump ions (i) and (ii). Then Du ∈L2p
loc(Ω,RnN ) o some p > 1
and he e exis s a cons an C1=C1(n, N, M/ν)such ha o all balls B2R(x)⊂Ω,
−
ZBR(x)|Du|2pdy1/2p≤C1−
ZB2R(x)|Du|2dy1/2.
Le x0be any ixed poin o Ω, 0 < R ≤dis (x0, ∂Ω). We se
Aαβ
ij (ux0,R)x0,R =−
ZBR(x0)
Aαβ
ij (y, ux0,R)dy .
Asolu ion o he sys em
DαAαβ
ij (ux0,R)x0,RDβ j= 0 in BR(x0),
−u∈W1,2
0(BR(x0),RN)
(2.8)
posses he ollowing p ope y.
Lemma 2.10 (see [5, 6, 10]).Le ∈W1,2(BR(x0),RN)be a solu ion o (2.8) wi h
u∈W1,2p(BR(x0),RN),p≥1. Then
ZBR(x0)|D |2pdy ≤C2ZBR(x0)|Du|2pdy.
He e C2:= C2(M/ν).
Rema k 2.11. Re ising p oo s o Lemmas 2.9 and 2.10 one can see ha he
cons an s om he o egoing es ima es depend inc easingly on M/ν. Mo eo e ,
in a case p= 1, he cons an C2 om Lemma 2.10 can be compu ed as C2=
21+(M/ν)2.
In he p oo o Theo em 1.1 we use an inequali y which is a consequence o he
Na anson’s Lemma (see e.g. [17, pg. 262]). I eads as ollows.
Lemma 2.12 (see [2, Lemma 3.7]).Le : [a, ∞)→Rbe a nonnega i e unc ion
which is in eg able on [a, b] o all a < b < ∞and
N= sup
0<h<∞
1
hZa+h
a
( )d < ∞.
Le g: [a, ∞)→Rbe an a bi a y nonnega i e, non-inc easing and in eg able
unc ion. Then R∞
a ( )g( )d exis s and
Z∞
a
( )g( )d ≤ N Z∞
a
g( )d .
The nex wo p oposi ions will be used in he p oo o Theo em 1.1.
P oposi ion 2.13. Le u∈W1,2(Ω,RN)be a minimize o he unc ional (1.1)
unde he assump ions (i) and (ii). Then o e e y ball B2R(x)⊂Ω, a bi a y
cons an s b > 0,µ≥2and he cons an p > 1 om Lemma 2.9 we ha e
ZBR(x)|Du|2lnµ
+(b|Du|2)dy ≤2−nC2p
1Cµb−
ZB2R(x)|Du|2dyp−1ZB2R(x)|Du|2dy
whe e C1is he cons an om Lemma 2.9.
The abo e p oposi ion is a s aigh o wa d consequence o Lemmas 2.4 and 2.9.
8 J. DANˇ
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P oposi ion 2.14. Le ∈W1,2(BR(x0),RN)be a weak solu ion o (2.8) whe e
u∈W1,2(Ω,RN)be a minimize o he unc ional (1.1) unde he assump ions (i)
and (ii). Then o ball B2R(x0)⊂Ω, a bi a y cons an s b > 0,µ≥2and he
cons an p > 1 om Lemma 2.9 we ha e
ZBR(x0)|D |2lnµ
+b|D |2dx
≤2−nC2p
1C2Cµb−
ZB2R(x0)|Du|2dxp−1ZB2R(x0)|Du|2dx
(2.9)
whe e C2is he cons an om Lemma 2.10.
The p oo o he abo e p oposi ion is a consequence o Lemmas 2.4, 2.10 and
2.9.
3. P oo o Theo em 1.1
We di ide he p oo in o wo pa s. In he i s pa o he p oo we assume ha
he coe icien s Aαβ
ij o he unc ional (1.1) depend only on u, and he second pa
we conside he p oo o he heo em in i s ull gene ali y.
Case Aαβ
ij =Aαβ
ij (u).We se φ( ) = φ(x, ) = RB (x)|Du|2dy and U =U (x) =
2−nφ(x, ) o B (x)⊂Ω. Now le xbe any ixed poin o Ω0⊂Ω, dis (Ω0, ∂Ω) =
2d > 0, B2R(x)⊂Ω, 0 < R ≤dand be a minimize o he ozen unc ional
A0( ;BR(x)) = ZBR(x)
Aαβ
ij (uR)Dα iDβ jdy
among all he unc ions in W1,2(BR(x),RN) aking he alues uon ∂BR(x).
F om he Eule equa ion o and om Lemma 2.7 we ha e
ZBσ(x)|D |2dy ≤Lσ
RnZBR(x)|D |2dy, o 0 < σ ≤R. (3.1)
Pu w=u− . I is clea ha w∈W1,2
0(BR(x),RN). Using (3.1) by s anda d
a gumen s we ob ain
ZBσ(x)|Du|2dy
≤21+2Lσ
RnZBR(x)|Dw|2dy + 4Lσ
RnZBR(x)|Du|2dy.
(3.2)
EJDE-2020/69 H ¨
OLDER CONTINUITY 9
Now we es ima e he i s in eg al on he igh -hand side o (3.2). F om [7,
Lemma 2.1] we ha e
ZBR(x)|Dw|2dy ≤2
νA0(u;BR(x)) −A0( ;BR(x))
≤2
νnZBR(x0)Aαβ
ij (uR)−Aαβ
ij (u)DαuiDβujdx
+ZBR(x0)Aαβ
ij ( )−Aαβ
ij (uR)Dα iDβ jdx
+A(u;BR(x0)) −A( ;BR(x0)) o
=2
ν{I+II +A(u;BR(x)) −A( ;BR(x))}
≤2
ν(I+II).
(3.3)
No e ha A(u;BR(x))−A( ;BR(x)) ≤0, since uis a minimize . Now we es ima e
e ms Iand II om (3.3).
Assump ion (iii) and he Young inequali y (2.3) gi e
|I| ≤ ZBR(x)
ω(|u−uR|)|Du|2dy
≤ZBR(x)
Φε|Du|2dy +ZBR(x)
Ψ1
εω(|u−uR|)dy
=I1+I2.
(3.4)
By P oposi ion 2.13 we ha e
I1=εZBR(x)|Du|2lnµ
+aε|Du|2dy
≤ε2−nC2p
1Cµaε −
ZB2R(x)|Du|2dyp−1φ(2R).
(3.5)
Acco ding o Lemma 2.3 (see (2.1) as well) we ha e
I2=ZBR(x)
Ψ1
εω(|u−uR|)dy =1
aZ∞
0
d
d e
Ψω( )
εmR( )d =1
ae
I2(3.6)
whe e mR( ) = |{y∈BR(x) : |u(y)−uR|> }|.
16 J. DANˇ
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and he modulus o con inui y ωis gi en in Example 1.6. This is a sample o unc-
ional, egula i y p ope ies o which could be well unde s ood h ough Theo em
1.1.
Example 4.4. To comple e eade ’s no ion o p ac ical consequences o he e-
sul s o mula ed in Theo em 1.1, we gi e wo cha s o possible alues o he basic
pa ame e s appea ing in he heo em. The i s cha co esponds o he unc ion
ωde ined by (1.11) and he second one co esponds o (1.9). Fo he simplici y,
we pu Ω = BR(0) ⊂R3and Ω0=BR/2(0) (we use he same deno a ion as in
Example 4.3). Choosing in he p e ious example a= 16π b we ha e M/ν = 10.6,
P=ω∞/ν = 0.1, C1= 104,C2= 102, om (2.7) we ob ain L= 1.2·108, by means
o Rema k 2.8 we ha e 0= 2.3·10−6. In his case he unc ion ωis de ined by
(1.11) and choosing p= 1.5, ϑ= 1.05 we can p esen he ollowing cha .
ν= 1030 1040 1050 1060 1070
ω∞= 1029 1039 1049 1059 1069
ω(ω∞)≈1081027 1044 1059 1069
1≈1051 1051 1055 1059 1065
eal alue 1
M2≈1041061091014 1022
es ima e 1
M2by means o (1.12) ≈1021041071013 1021
ρ= 1.32 1.27 1.25 1.14 1.1
τ= 2 1.9 1.9 1.7 1.7
µ= 21 22 23 26.5 28.5
whe e 1is he poin o which ω( 1)=0.95 ·ω∞.
In he case when he unc ion ωis de ined by (1.9), o he o egoing pa ame e s
we ob ain he ollowing cha .
ω∞= 1030 1040 1050 1060 1070
0= 1071010 1013 1016 1019
ω( 0)≈1 1010 1019 1030 1040
ω(ω∞)≈1015 1028 1042 1057 1070
1≈1056 1060 1062 1066 1068
eal alue 1
M2≈1011 1017 1022 1028 1035
es ima e 1
M2by means o (1.10) ≈10 1071011 1018 1024
ρ= 1.51 1.51 1.51 1.5 1.49
γ= 0.61 0.61 0.62 0.62 0.62
µ= 17.7 17.9 18 18 18.1
whe e 1is he poin o which ω( 1) = ω∞. We no e ha o abo e men ioned
pa ame e s he second condi ion om (1.7) is sa is ied.
5. Appendix
We gi e es ima es o he cons an M om (1.7) whe e ωis de ined by Examples
1.5 and 1.6.
e
Ψω( )
ε−e
Ψω( 0)
ε
− 0
=d
d e
Ψω( )
ε| =ξ
=ω0(ξ)
εh1 + 2
2µ−11
2õ
ω(ξ)
ε2
2µ−1ie1
2õ
ω(ξ)
ε2
2µ−1,
o 0< ξ < ≤ 1.

EJDE-2020/69 H ¨
OLDER CONTINUITY 17
(a) Es ima e o M ela ed o he unc ion ω om Example 1.5. He e we conside
µ≥6, ρ > 1/p, 0 < γ < 1, 0>0, Cµ>1.
M= sup
0< < 1e
Ψω( )
ε−e
Ψω( 0)
ε
− 0
= sup
0< < 1ω0( )
εe(1
2õ
ω( )
ε)2
2µ−1h1 + 2
2µ−11
2õ
ω( )
ε2
2µ−1i
= sup
0< < 1γCρ
µ
e1/Cρ
µ−1
 γ
0+e1/Cρ
µ−1 γ γ−1eCρ
µ
2√µln 1+ e1/Cρ
µ−1
γ
0
γ 2
2µ−1
×h1 + 2
2µ−1Cρ
µ
2√µln 1 + e1/Cρ
µ−1
γ
0
γ 2
2µ−1i
≤sup
0< < 1γCρ
µ
e1/Cρ
µ−1
 γ
0+e1/Cρ
µ−1 γsup
0< < 1 γ−1eCρ
µ
2√µln 1+ e1/Cρ
µ−1
γ
0
γ 2
2µ−1
×h1 + 2
2µ−1sup
0< < 1Cρ
µ
2√µln 1 + e1/Cρ
µ−1
γ
0
γ 2
2µ−1i
=S1S21 + 2
2µ−1S3.
(5.1)
The es ima es o S1,S2and S3a e as ollows.
S1≤γCρ
µ
e1/Cρ
µ−1
γ
0≤γ(e −1)
γ
0
,∀ 0≤ ≤ 1;
S2≤sup
0< < 1
e1
õ(
0)γ2
2µ−1
1−γ.
I we de ine
( ) = e1
õ(
0)γ2
2µ−1
1−γ, ∈(0,∞),
hen he s anda d me hod o di e en ial calculus gi es us he es ima e
S2≤max{ ( 0), ( 1)} ≤ max ne1
√µ2
2µ−1
1−γ
0
,e2Cρ
µ
√µ2
2µ−1
C
1−γ
γρ
µ 1−γ
0o
≤1
1−γ
0
max n3,e2Cρ
µ
√µ2
2µ−1
C
1−γ
γρ
µo.
Finally,
S3≤Cρ
µ
2√µ2
2µ−1,∀ 0≤ ≤ 1.
18 J. DANˇ
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CEK, E. VISZUS EJDE-2020/69
Inse ing he abo e es ima es in o (5.1), we ob ain
M ≤ γ(e −1)
01 + 2
2µ−1Cρ
µ
2√µ2
2µ−1max n3,e2Cρ
µ
√µ2
2µ−1
C
1−γ
γρ
µo
≤10C
2
2µ−1ρ
µ
0
max n1,eC
2
2µ−1ρ
µ
3C
1−γ
γρ
µo.
(5.2)
The e m e
Ψ(ω( 0)
ε) om he de ini ion o Mwe can es ima e as
e
Ψω( 0)
ε=ω( 0)
εeω( 0)
2√µ ε 2/(2µ−1)
≤e1
2√µ2
2µ−1≤3,∀ 0>0.
(b) Es ima e o M o ω om Example 1.6:
M ≤ eCρ
µ
2√µ2
2µ−1
Cτ−ρ
µ
, τ > ρ > 1
p(5.3)
and e
Ψ(ω( 0)/ε) = 0.
Acknowledgemen s. E. Viszus was suppo ed by he esea ch p ojec Slo ak
G an Agency No. 1/0078/17 and No. 1/0358/20.
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Jose Danˇ
eˇ
cek
Vˇ
SB - Technical Uni e si y o Os a a, FEECS, Depa men o Applied Ma hema ics,
17. lis opadu 15/2172, 70833 Os a a-Po uba, Czech Republic
Email add ess:[email p o ec ed]
Eugen Viszus
Depa men o Ma hema ical Analysis and Nume ical Ma hema ics, Facul y o Ma he-
ma ics, Physics and In o ma ics Comenius Uni e si y, Mlynsk´
a dolina, 84248 B a isla a,
Slo ak Republic
Email add ess:[email p o ec ed]