Holder continuity for vector-valued minimizers of quadratic functionals

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 Eˇ 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 Eˇ 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. EJDE-2020/69 H ¨ 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 Eˇ CEK, E. VISZUS EJDE-2020/69 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 . EJDE-2020/69 H ¨ 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 Eˇ CEK, E. VISZUS EJDE-2020/69 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 Eˇ CEK, E. VISZUS EJDE-2020/69 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 2√µ ω(ξ) ε2 2µ−1ie1 2√µ 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 2√µ ω( ) ε)2 2µ−1h1 + 2 2µ−11 2√µ 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 √µ( 0)γ2 2µ−1 1−γ. I we de ine ( ) = e1 √µ( 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 Eˇ 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. Re e ences [1] J. Danˇeˇcek, E. Viszus; In e io C0,γ - egula i y o ec o - alued minimize s o quasilinea unc ionals. Nonlinea Anal., 74 (2011), 5274–5285. [2] J. Danˇeˇcek, E. Viszus; Regula i y on he in e io o he g adien o weak solu ions o non- linea second-o de ellip ic sys ems. Elec on. J. Di . Equa ions, 2013, 121 (2013), 1–17. [3] J. Danˇeˇcek, E. 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Nauka, Moscow (1974), (in Rus- sian). [18] J. Neˇcas, J. S a ´a; P incipio di massimo pe i sis emi elli ici quasilinea i non diagonali. Boll. Unione Ma . I al., (4)6 (1972), 1–10. [19] M. A. Ragusa, A. Tachikawa; Pa ial egula i y o he minimize s o quad a ic unc ionals wi h V MO coe icien s. J. London Ma h. Soc., (2),72 (2005), 609–620. [20] D. Sa ason; Func ions o anishing mean oscilla ion. T ans. Ame . Ma h. Soc., 207 (1975), 391–405. [21] W. P. Zieme , Weakly di e en iable unc ions. Sp inge -Ve lag, Heidelbe g, 1989. 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]