Holder continuity for vector-valued minimizers of quadratic functionals

Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 69, pp. 1–19. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu H¨ OLDER CONTINUITY FOR VECTOR-VALUED MINIMIZERS OF QUADRATIC FUNCTIONALS JOSEF DANˇ Eˇ CEK, EUGEN VISZUS Abstract. In this article we give a sufficient condition for interior everywhere H¨older continuity of weak minimizers of a class of quadratic functionals with coefficients Aαβ ij (·, u) belonging to the V MO-class, uniformly with respect to u∈RN, and continuous with respect to u. The condition is global. It is typical for the functionals belonging to the class that the continuity moduli of their coefficients become slowly growing sufficiently far from zero. Some features of the main result are illustrated by examples. 1. Introduction The aim of this article is to study the interior everywhere regularity of functions minimizing variational integrals A(u; Ω) = ZΩ Aαβ ij (x, u)DαuiDβujdx (1.1) where u: Ω →RN,N > 1, Ω ⊂Rn,n≥3 is a bounded open set, x= (x1, . . . , xn)∈ Ω, u(x) = (u1(x), . . . , uN(x)), Du ={Dαui},Dα=∂/∂xα,α= 1, . . . , n,i= 1, . . . , N. Throughout the whole text we use the summation convention over repeated indices. We call a function u∈W1,2(Ω,RN) is a minimizer of the functional A(u; Ω) if and only if A(u; Ω) ≤ A(v; Ω) for every v∈W1,2(Ω,RN) such that u−v∈W1,2 0(Ω,RN). For more information see [5, 10]. On the functional Awe assume: (i) Aαβ ij =Aβα ji ,Aαβ ij are continuous functions in u∈RNfor every x∈Ω and there exists M > 0 such that Pi,j,α,β |Aαβ ij (x, u)| ≤ M, for all x∈Ω, and all u∈RN. (ii) (ellipticity) There exists ν > 0 such that Aαβ ij (x, u)ξi αξj β≥ν|ξ|2,∀x∈Ω,∀u∈RN,∀ξ∈RnN .(1.2) (iii) (oscillation of coefficients) There exists a real function ωcontinuous on [0,∞), which is bounded, nondecreasing, concave, ω(0) = 0 and such that 2010 Mathematics Subject Classification. 35J60. Key words and phrases. Quadratic functionals; minimizers; regularity; Morrey spaces. c 2020 Texas State University. Submitted April 4, 2019. Published July 2, 2020. 1 2 J. DANˇ Eˇ CEK, E. VISZUS EJDE-2020/69 for all x∈Ω and u,v∈RN X i,j,α,β |Aαβ ij (x, u)−Aαβ ij (x, v)| ≤ ω(|u−v|).(1.3) We set ω∞= limt→∞ ω(t)≤2M. (iv) For all u∈RN,Aαβ ij (·, u)∈V MO(Ω) (uniformly with respect to u∈RN). Assumptions (i) and (ii) allow us to conclude that if u∈W1,2(Ω,RN) is a minimizer of (1.1) then for any admissible function v∈W1,2(Ω,RN) ZΩ|Du|2dx ≤M νZΩ|Dv|2dx . (1.4) Concerning the assumption (iii) it is worth to point out (see [5, p.169]) that for uniformly continuous coefficients Aαβ ij there exists a real function ωsatisfying the assumption (iii) and, viceversa, (iii) implies the uniform continuity of coefficients and absolute continuity of ωon [0,∞). In this paper we will consider the continuous function ω(t) = (ω0(t) for 0 ≤t<t0, t0≥0 ω1(t)≤ω∞,for t0≤t < ∞(1.5) where ω0is an arbitrary continuous, concave, nondecreasing function, increasing on a neighbourhood of zero such that ω0(0) = 0 and the point t0and the function ω1 are chosen in such a way that ωpreserves its continuity and concavity on [0,∞). With respect to (iv) it is worth to recall that since the space of continuous functions is a proper subset of V MO, the continuity of coefficients Aαβ ij =Aαβ ij (x, u) with respect to xis not supposed. In the linear case, when the coefficients Aαβ ij = Aαβ ij (x) belong to C0,γ(Ω) the regularity of minimizers of functionals as (1.1) is well understood (see [5, Thorems 3.1, 3.2 on p.87, 88]). These results were later generalized to the case where the above coefficients are in V MO, hence possibly discontinuous (see [4, 19] and references therein). It is well known that even in the continuous case the dependence of coefficients Aαβ ij on uleads to weaker regularity results for minimizers. In dimension n≥3 there are examples of vectorial quadratic functionals (N > 1) with analytic coefficients Aαβ ij =Aαβ ij (u) whose minimizers are discontinuous (see [10, p. 317], [11]). For the analytic coefficients Aαβ ij =Aαβ ij (x, u) see counterexample in [18]. These examples indicate that, in general, only partial regularity results can be achieved for minimizers of vectorial functionals. For detailed information on this topic we refer to sources [5]-[10] for classic results and to [13, 15, 19] for recent results. Besides the partial regularity results, a few everywhere regularity results were obtained for some special types of vectorial functionals (see [10, 15]). Our paper deals just with the last mentioned type of regularity results. In the recent papers [1, 3] conditions guaranteeing the local H¨older continuity of minimizers of functional (1.1) in Ω are given. Because the paper [3] extends the results of [1], we mention only [3] in more detail. Main results of the paper [3] are stated in two theorems. The first of them refers that if a quantity expressed by means of parameters ω∞/ν and M/ν is small enough, the minimizers of (1.1) are regular. This result is not very surprising but, moreover, an upper bound (although probably not optimal) of the above mentioned quantity is designed. In a case when the mentioned condition EJDE-2020/69 H ¨ OLDER CONTINUITY 3 is not fulfilled a sufficient condition for regularity of minimizers of functional (1.1) is stated as well. A basic advantage of the second condition in the paper [3] is, that it admits (for sufficiently big ellipticity constant ν) an arbitrary growth of the continuity modulus ω=ω(t) when tis near by zero. Here it is needful to note that the second condition works likewise when νis small but, in this case, the modulus of continuity ωhas to grow slowly enough. A disadvantage of the condition is its ”local character”, analogous to the regularity conditions in partial regularity theory. The present paper essentially extends results of [1] and [3]. Here we study the regularity for variational integrals, coefficients of which satisfy (iii) with modulus of continuity given by (1.5). Together with more delicate estimates and careful designing of some parameters in proof, it allows us to state the regularity condition preserving all the advantages of the previous mentioned conditions from [1, 3] and, moreover, the condition is formulated much simpler and more exactly than the previous ones in [1, 3]. Consequently, it improves the possibility of immediate application (it is well visible mainly in the case of the Dirichlet problem - see Remark 1.4 below). It is worth to mention that the regularity condition (expressed by (1.6), (1.7), (1.8)) has, compared to that one from [3, Thm. 2], global features. The methods of proving the main results are based on those that were developed in the classic partial regularity theory ( see for example [5, 10]), but they are essentially modified. In Remark 4.2 it is shown that, in a case of split coefficients, joining the results of this paper with those from [12], we are able to guarantee the regularity of minimizers of (1.1) in Ω. Now we can formulate the main result. Theorem 1.1. Let Ω0⊂⊂ Ω,n−2≤ϑ < n be given and the coefficients Aαβ ij of the functional (1.1) satisfy (i), (ii), (iii) and (iv). There exists a positive constant Msuch that if the minimizer uof the functional (1.1) satisfies the condition 1 |Ω|1−2/n ZΩ|Du|2dy ≤1 M2(1.6) then ubelongs to C0,(ϑ−n+2)/2(Ω0,RnN )when ϑ>n−2and to BMO(Ω0,RnN ) when ϑ=n−2. Here M= sup t0<t<∞e Ψω(t) ε−e Ψω(t0) ε t−t0 and e Ψω(t0) ε≤2n+2pC2.(1.7) Remark 1.2. In the foregoing formula the function e Ψ(u) = ue(u/2√µ)2/(2µ−1) (for further properties of e Ψ see (2.1) below), t0≥0 (t0is the parameter from the definition of ω, see (1.5)), ε=ω∞/Cρ µ,Cµ= (µ/((p−1)e))µ, the constants µ≥6 and ρ > 1/p are such that 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 the case when the coefficients Aαβ ij depend only on u. Here p > 1 is from Lemma 2.9, K= 2(n+11+(n+3)ϑ/(n−ϑ))p−(2n+5) κ1−p n,Lis the constant from Lemma 2.7 below, C1,C2are the constants from Lemma 2.9 and 2.10 respectively, d= dist(Ω0, ∂Ω)/2>0 and the symbol |·|stands for the n-dimensional Lebesgue measure (κnis the Lebesgue measure of the unit ball in Rn). If Aαβ ij =Aαβ ij (x, u) then, formally, the constant Kon the right-hand side of (1.8) is substituted by 2K(here, as it is visible at the end of the proof of Theorem 1.1, 4 J. DANˇ Eˇ CEK, E. VISZUS EJDE-2020/69 the multiplier 2 could be substituted by another one, bigger than 1). It is important to release that the dependence of the coefficients Aαβ ij on variable xtends to the choice d= min{R0,dist(Ω0, ∂Ω)/2}(for definition of R0see (3.25) below) and so d and, consequently, the value of the constant Cρp−1 µfrom (1.8) depend on ”VMOquality” of x-dependence of coefficients Aαβ ij as well. Broadly speaking, the bigger R0is, the better regularity result one can obtain. Remark 1.3. It is easily seen that instead of the assumption (iv) in the foregoing Theorem 1.1 one can suppose the coefficients Aαβ ij of the functional (1.1) to be of BMO-class with suitable small BMO semi-norms (see (3.25) below). Remark 1.4. It is a consequence of the estimate (1.4) that if u∈W1,2(Ω,RN), mentioned in the foregoing theorem, is such that u−g∈W1,2 0(Ω,RN) for some g∈W1,2(Ω,RN) (the Dirichlet problem for functional (1.1)), then the left-hand side of (1.6) can be replaced by the term M ν|Ω|1−2/n ZΩ|Dg|2dy . The regularity theorem, we formulated above, can be illustrated with two samples of the function ω, defined by (1.5), for which we give estimates of the parameter M. Broadly speaking, if the coefficients of the functional satisfy (iii) with some ω given below and (1.8) is fulfilled, we have the regularity. Example 1.5. Let ω(t) =        ω0(t) for 0 ≤t<t0, ω∞ln 1 + eε/ω∞−1 tγ 0tγfor t0≤t≤t1,0< γ ≤1, ω∞for t>t1 (1.9) where ω0is an arbitrary continuous, concave, nondecreasing function such that ω0(0) = 0 and the points t0,t1are chosen so that ωis continuous and concave on [0,∞). If we put ε=ω∞/Cρ µin (1.9) then the right-hand side of (1.6) can be chosen in the form (see Appendix for more information) 1 M2=t0 10C 2 2µ−1ρ µ min n1,3C 2 2µ−1ρ µ eC 2 2µ−1ρ µo2.(1.10) Here µ≥6, ρ > 1/p and t0>0. Example 1.6. Let ω(t) = 2ω∞ πarctan t Cτ µfor 0 ≤t < ∞(1.11) then the constant from (1.6) can have the form (in this case t0= 0, see Appendix as well) 1 M2=Cτ−ρ µ eCρ µ 2√µ2 2µ−12 .(1.12) Here τ > ρ > 1/p,µ≥6 satisfy (1.8) and e Ψ(ω(t0)/ε) = 0. EJDE-2020/69 H ¨ OLDER CONTINUITY 5 2. Preliminaries If x∈Rnand ris a positive real number, we set Br(x) = {y∈Rn:|y−x|< r}, Ωr(x) = Ω ∩Br(x). Denote by ux,r =1 |Ωr(x)|ZΩr(x) u(y)dy =− ZΩr(x) u(y)dy the mean value of the function u∈L1(Ω,RN) over the set Ωr(x) where the symbol |·|denotes the n-dimensional Lebesgue measure. Moreover, we set φ(r) = φ(x, r) = RBr(x)|Du(y)|2dy,Ur=Ur(x) = r2−nφ(x, r) for Br(x)⊂Ω. Beside the standard space C∞ 0(Ω,RN), H¨older space C0,α(Ω,RN) and Sobolev spaces Wk,p(Ω,RN), Wk,p 0(Ω,RN) we use Morrey spaces Lq,λ(Ω,RN) (see, e.g. [5, 14]). We will denote by Xloc(Ω,RN) the space of all functions which belong to X(e Ω,RN) for any bounded subdomain e Ω with smooth boundary which is compactly embedded in Ω. We recall a definition of V MO - spaces and a few properties of Morrey spaces. We set for f∈L1(Ω), 0 <a<∞ Na(f, Ω) := sup x∈Ω,r<a − ZΩr(x)|f(y)−fx,r|dy. Definition 2.1 (see [20]).A function f∈L1(Ω) is said to belong to BMO(Ω) if Ndiam Ω(f, Ω) <∞. A function f∈L1(Ω) is said to belong to V MO(Ω) if lim a→0Na(f, Ω) = 0. Proposition 2.2. For a bounded domain Ω⊂Rnwith the Lipschitz boundary, for q∈(1,∞)and 0<λ<µ<∞we have the following: (a) Lq,µ(Ω,RN)⊂Lq,λ(Ω,RN). (b) If u∈W1,2 loc (Ω,RN)and Du ∈L2,λ loc (Ω,RnN ),n−2< λ < n then u∈ C0,(λ−n+2)/2(Ω,RN). (c) If u∈W1,2 loc (Ω,RN)and Du ∈L2,n−2 loc (Ω,RnN )then u∈BMOloc(Ω,RN). (d) Lq,n(Ω,RN)is isomorphic to the L∞(Ω,RN). (e) L∞(Ω,RN)$BMO(Ω,RN). Let now Φ, Ψ be a pair of complementary Young functions Φ(u) = ulnµ +(au) for u≥0, Ψ(u)≤Ψ(u) = 1 aue(u 2√µ)2/(2µ−1) =1 ae Ψ(u) for u≥0(2.1) where a > 0, µ≥2 are constants, and ln+(au) = (0 for 0 ≤u < 1/a, ln(au) for u≥1/a. (2.2) Then the Young inequality for Φ and Ψ reads uv ≤Φ(u) + Ψ(v), u, v ≥0.(2.3) 6 J. DANˇ Eˇ CEK, E. VISZUS EJDE-2020/69 Lemma 2.3 ([21, p.37]).Let φ: [0,∞)→[0,∞)be a non decreasing function which is absolutely continuous on every closed interval of finite length, φ(0) = 0. If w≥0is measurable and l(t) = {y∈Rn:w(y)> t}then ZRn φ◦w dy =Z∞ 0|l(t)|φ0(t)dt. Lemma 2.4. Let v≥0,b > 0,µ > 0and q > 1be arbitrary. Then vlnµ +(bv)≤Cµbq−1vq(2.4) where Cµ=µ (q−1)e µ. For a proof of the above lemma, calculate sup lnµ +(bv) vq−1;v∈(0,∞). The next Lemma is taken from [1, Lemma 6]. Lemma 2.5. Let A,R0≤R1be positive numbers, n−2≤ϑ<n,ηa nonnegative and nondecreasing function on (0,∞). Then there exist 0,cpositive so that for any nonnegative, nondecreasing function φdefined on [0,2R1]and satisfying with (B1+B2η(U2R0)) ∈[0, 0]the inequality φ(σ)≤Aσ Rn+1 21 + Aσ Rn[B1+B2η(U2R)]φ(2R) (2.5) for all σ,Rsuch that 0< σ < R ≤R0, it holds φ(σ)≤cσϑφ(2R0),∀σ: 0 < σ ≤R0.(2.6) Remark 2.6. Note that we can take 0=1 2(2n+1A)ϑ n−ϑ , c =(2n+1A)1 n−ϑ 2R0ϑ. Lemma 2.7 ([5, p.78]).Given the system −DαAαβ ij Dβuj= 0, i = 1, . . . , N where Aαβ ij are constants satisfying (i) and (ii). There exists a constant L= L(n, N, M/ν)≥1such that for every weak solution u∈W1,2(Ω,RN), for every x∈Ωand 0< σ ≤R≤dist(x, ∂Ω) the following estimate holds, ZBσ(x)|Du(y)|2dy ≤Lσ RnZBR(x)|Du(y)|2dy . Remark 2.8. Note that L=c(n, N)M ν2k, k = 1 + n 2 and for n= 3 and N= 2 it holds L < 104M ν4.(2.7) One of the tools for the proof of our main result is the following reverse H¨older inequality that is standard in our setting . EJDE-2020/69 H ¨ OLDER CONTINUITY 7 Lemma 2.9 (see [5, 10]).Let u∈W1,2(Ω,RN)be a minimum of the functional (1.1) under the assumptions (i) and (ii). Then Du ∈L2p loc(Ω,RnN )for some p > 1 and there exists a constant C1=C1(n, N, M/ν)such that for all balls B2R(x)⊂Ω, − ZBR(x)|Du|2pdy1/2p≤C1− ZB2R(x)|Du|2dy1/2. Let x0be any fixed point of Ω, 0 < R ≤dist(x0, ∂Ω). We set Aαβ ij (ux0,R)x0,R =− ZBR(x0) Aαβ ij (y, ux0,R)dy . Asolution to the system DαAαβ ij (ux0,R)x0,RDβvj= 0 in BR(x0), v−u∈W1,2 0(BR(x0),RN) (2.8) posses the following property. Lemma 2.10 (see [5, 6, 10]).Let v∈W1,2(BR(x0),RN)be a solution to (2.8) with u∈W1,2p(BR(x0),RN),p≥1. Then ZBR(x0)|Dv|2pdy ≤C2ZBR(x0)|Du|2pdy. Here C2:= C2(M/ν). Remark 2.11. Revising proofs of Lemmas 2.9 and 2.10 one can see that the constants from the foregoing estimates depend increasingly on M/ν. Moreover, in a case p= 1, the constant C2from Lemma 2.10 can be computed as C2= 21+(M/ν)2. In the proof of Theorem 1.1 we use an inequality which is a consequence of the Natanson’s Lemma (see e.g. [17, pg. 262]). It reads as follows. Lemma 2.12 (see [2, Lemma 3.7]).Let f: [a, ∞)→Rbe a nonnegative function which is integrable on [a, b]for all a < b < ∞and N= sup 0<h<∞ 1 hZa+h a f(t)dt < ∞. Let g: [a, ∞)→Rbe an arbitrary nonnegative, non-increasing and integrable function. Then R∞ af(t)g(t)dt exists and Z∞ a f(t)g(t)dt ≤ N Z∞ a g(t)dt. The next two propositions will be used in the proof of Theorem 1.1. Proposition 2.13. Let u∈W1,2(Ω,RN)be a minimizer of the functional (1.1) under the assumptions (i) and (ii). Then for every ball B2R(x)⊂Ω, arbitrary constants b > 0,µ≥2and the constant p > 1from Lemma 2.9 we have ZBR(x)|Du|2lnµ +(b|Du|2)dy ≤2−nC2p 1Cµb− ZB2R(x)|Du|2dyp−1ZB2R(x)|Du|2dy where C1is the constant from Lemma 2.9. The above proposition is a straightforward consequence of Lemmas 2.4 and 2.9. 8 J. DANˇ Eˇ CEK, E. VISZUS EJDE-2020/69 Proposition 2.14. Let v∈W1,2(BR(x0),RN)be a weak solution to (2.8) where u∈W1,2(Ω,RN)be a minimizer of the functional (1.1) under the assumptions (i) and (ii). Then for ball B2R(x0)⊂Ω, arbitrary constants b > 0,µ≥2and the constant p > 1from Lemma 2.9 we have ZBR(x0)|Dv|2lnµ +b|Dv|2dx ≤2−nC2p 1C2Cµb− ZB2R(x0)|Du|2dxp−1ZB2R(x0)|Du|2dx (2.9) where C2is the constant from Lemma 2.10. The proof of the above proposition is a consequence of Lemmas 2.4, 2.10 and 2.9. 3. Proof of Theorem 1.1 We divide the proof into two parts. In the first part of the proof we assume that the coefficients Aαβ ij of the functional (1.1) depend only on u, and the second part we consider the proof of the theorem in its full generality. Case Aαβ ij =Aαβ ij (u).We set φ(r) = φ(x, r) = RBr(x)|Du|2dy and Ur=Ur(x) = r2−nφ(x, r) for Br(x)⊂Ω. Now let xbe any fixed point of Ω0⊂Ω, dist(Ω0, ∂Ω) = 2d > 0, B2R(x)⊂Ω, 0 < R ≤dand vbe a minimizer of the frozen functional A0(v;BR(x)) = ZBR(x) Aαβ ij (uR)DαviDβvjdy among all the functions in W1,2(BR(x),RN) taking the values uon ∂BR(x). From the Euler equation for vand from Lemma 2.7 we have ZBσ(x)|Dv|2dy ≤Lσ RnZBR(x)|Dv|2dy, for 0 < σ ≤R. (3.1) Put w=u−v. It is clear that w∈W1,2 0(BR(x),RN). Using (3.1) by standard arguments we obtain 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 estimate the first integral on the right-hand side of (3.2). From [7, Lemma 2.1] we have ZBR(x)|Dw|2dy ≤2 νA0(u;BR(x)) −A0(v;BR(x)) ≤2 νnZBR(x0)Aαβ ij (uR)−Aαβ ij (u)DαuiDβujdx +ZBR(x0)Aαβ ij (v)−Aαβ ij (uR)DαviDβvjdx +A(u;BR(x0)) −A(v;BR(x0)) o =2 ν{I+II +A(u;BR(x)) −A(v;BR(x))} ≤2 ν(I+II). (3.3) Note that A(u;BR(x))−A(v;BR(x)) ≤0, since uis a minimizer. Now we estimate terms Iand II from (3.3). Assumption (iii) and the Young inequality (2.3) give |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 Proposition 2.13 we have I1=εZBR(x)|Du|2lnµ +aε|Du|2dy ≤ε2−nC2p 1Cµaε − ZB2R(x)|Du|2dyp−1φ(2R). (3.5) According to Lemma 2.3 (see (2.1) as well) we have I2=ZBR(x) Ψ1 εω(|u−uR|)dy =1 aZ∞ 0 d dt e Ψω(t) εmR(t)dt =1 ae I2(3.6) where mR(t) = |{y∈BR(x) : |u(y)−uR|> t}|. 16 J. DANˇ Eˇ CEK, E. VISZUS EJDE-2020/69 and the modulus of continuity ωis given in Example 1.6. This is a sample of functional, regularity properties of which could be well understood through Theorem 1.1. Example 4.4. To complete reader’s notion of practical consequences of the results formulated in Theorem 1.1, we give two charts of possible values of the basic parameters appearing in the theorem. The first chart corresponds to the function ωdefined by (1.11) and the second one corresponds to (1.9). For the simplicity, we put Ω = BR(0) ⊂R3and Ω0=BR/2(0) (we use the same denotation as in Example 4.3). Choosing in the previous example a= 16π b we have M/ν = 10.6, P=ω∞/ν = 0.1, C1= 104,C2= 102, from (2.7) we obtain L= 1.2·108, by means of Remark 2.8 we have 0= 2.3·10−6. In this case the function ωis defined by (1.11) and choosing p= 1.5, ϑ= 1.05 we can present the following chart. ν= 1030 1040 1050 1060 1070 ω∞= 1029 1039 1049 1059 1069 ω(ω∞)≈1081027 1044 1059 1069 t1≈1051 1051 1055 1059 1065 real value 1 M2≈1041061091014 1022 estimate 1 M2by means of (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 where t1is the point for which ω(t1)=0.95 ·ω∞. In the case when the function ωis defined by (1.9), for the foregoing parameters we obtain the following chart. ω∞= 1030 1040 1050 1060 1070 t0= 1071010 1013 1016 1019 ω(t0)≈1 1010 1019 1030 1040 ω(ω∞)≈1015 1028 1042 1057 1070 t1≈1056 1060 1062 1066 1068 real value 1 M2≈1011 1017 1022 1028 1035 estimate 1 M2by means of (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 where t1is the point for which ω(t1) = ω∞. We note that for above mentioned parameters the second condition from (1.7) is satisfied. 5. Appendix We give estimates of the constant Mfrom (1.7) where ωis defined by Examples 1.5 and 1.6. e Ψω(t) ε−e Ψω(t0) ε t−t0 =d dt e Ψω(t) ε|t=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, for t0< ξ < t ≤t1. EJDE-2020/69 H ¨ OLDER CONTINUITY 17 (a) Estimate of Mrelated to the function ωfrom Example 1.5. Here we consider µ≥6, ρ > 1/p, 0 < γ < 1, t0>0, Cµ>1. M= sup t0<t<t1e Ψω(t) ε−e Ψω(t0) ε t−t0 = sup t0<t<t1ω0(t) εe(1 2√µ ω(t) ε)2 2µ−1h1 + 2 2µ−11 2√µ ω(t) ε2 2µ−1i = sup t0<t<t1γCρ µ e1/Cρ µ−1 tγ 0+e1/Cρ µ−1tγtγ−1eCρ µ 2√µln 1+ e1/Cρ µ−1 tγ 0 tγ 2 2µ−1 ×h1 + 2 2µ−1Cρ µ 2√µln 1 + e1/Cρ µ−1 tγ 0 tγ 2 2µ−1i ≤sup t0<t<t1γCρ µ e1/Cρ µ−1 tγ 0+e1/Cρ µ−1tγsup t0<t<t1tγ−1eCρ µ 2√µln 1+ e1/Cρ µ−1 tγ 0 tγ 2 2µ−1 ×h1 + 2 2µ−1sup t0<t<t1Cρ µ 2√µln 1 + e1/Cρ µ−1 tγ 0 tγ 2 2µ−1i =S1S21 + 2 2µ−1S3. (5.1) The estimates of S1,S2and S3are as follows. S1≤γCρ µ e1/Cρ µ−1 tγ 0≤γ(e −1) tγ 0 ,∀t0≤t≤t1; S2≤sup t0<t<t1 e1 √µ(t t0)γ2 2µ−1 t1−γ. If we define f(t) = e1 √µ(t t0)γ2 2µ−1 t1−γ, t ∈(0,∞), then the standard method of differential calculus gives us the estimate S2≤max{f(t0), f(t1)} ≤ max ne1 √µ2 2µ−1 t1−γ 0 ,e2Cρ µ √µ2 2µ−1 C 1−γ γρ µt1−γ 0o ≤1 t1−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,∀t0≤t≤t1. 18 J. DANˇ Eˇ CEK, E. VISZUS EJDE-2020/69 Inserting the above estimates into (5.1), we obtain M ≤ γ(e −1) t0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 t0 max n1,eC 2 2µ−1ρ µ 3C 1−γ γρ µo. (5.2) The term e Ψ(ω(t0) ε) from the definition of Mwe can estimate as e Ψω(t0) ε=ω(t0) εeω(t0) 2√µ ε 2/(2µ−1) ≤e1 2√µ2 2µ−1≤3,∀t0>0. (b) Estimate of Mfor ωfrom 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(t0)/ε) = 0. Acknowledgements. E. Viszus was supported by the research project Slovak Grant Agency No. 1/0078/17 and No. 1/0358/20. References [1] J. Danˇeˇcek, E. Viszus; Interior C0,γ - regularity for vector-valued minimizers of quasilinear functionals. Nonlinear Anal., 74 (2011), 5274–5285. [2] J. Danˇeˇcek, E. Viszus; Regularity on the interior for the gradient of weak solutions to nonlinear second-order elliptic systems. Electron. J. Diff. Equations, 2013, 121 (2013), 1–17. [3] J. Danˇeˇcek, E. 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Josef Danˇ eˇ cek Vˇ SB - Technical University of Ostrava, FEECS, Department of Applied Mathematics, 17. listopadu 15/2172, 70833 Ostrava-Poruba, Czech Republic Email address:[email protected] Eugen Viszus Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics Comenius University, Mlynsk´ a dolina, 84248 Bratislava, Slovak Republic Email address:[email protected]