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 HÖLDER CONTINUITY FOR VECTOR-VALUED MINIMIZERS OF QUADRATIC FUNCTIONALS JOSEF DANĚČEK, EUGEN VISZUS Abstract. In this article we give a sufficient condition for interior everywhere Hölder continuity of weak minimizers of a class of quadratic functionals with coefficients Aαβij (·, u) belonging to the VMO-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) = ∫ Ω Aαβij (x, u)Dαu iDβu j dx (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 = {Dαu i}, 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 ∈ W 1,2(Ω,RN ) is a minimizer of the functional A(u; Ω) if and only if A(u; Ω) ≤ A(v; Ω) for every v ∈ W 1,2(Ω,RN ) such that u− v ∈W 1,2 0 (Ω,RN ). For more information see [5, 10]. On the functional A we assume: (i) Aαβij = Aβαji , Aαβij are continuous functions in u ∈ RN for every x ∈ Ω and there exists M > 0 such that ∑ i,j,α,β |A αβ ij (x, u)| ≤ M , for all x ∈ Ω, and all u ∈ RN . (ii) (ellipticity) There exists ν > 0 such that Aαβij (x, u)ξiαξ j β ≥ ν|ξ| 2, ∀x ∈ Ω, ∀u ∈ RN , ∀ξ ∈ RnN . (1.2) (iii) (oscillation of coefficients) There exists a real function ω continuous 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. DANĚČEK, E. VISZUS EJDE-2020/69 for all x ∈ Ω and u, v ∈ RN∑ i,j,α,β |Aαβij (x, u)−Aαβij (x, v)| ≤ ω (|u− v|) . (1.3) We set ω∞ = limt→∞ ω(t) ≤ 2M . (iv) For all u ∈ RN , Aαβij (·, u) ∈ VMO(Ω) (uniformly with respect to u ∈ RN ). Assumptions (i) and (ii) allow us to conclude that if u ∈W 1,2(Ω,RN ) is a minimizer of (1.1) then for any admissible function v ∈W 1,2(Ω,RN )∫ Ω |Du|2 dx ≤ M ν ∫ Ω |Dv|2 dx . (1.4) Concerning the assumption (iii) it is worth to point out (see [5, p.169]) that for uniformly continuous coefficients Aαβij there exists a real function ω satisfying the assumption (iii) and, viceversa, (iii) implies the uniform continuity of coefficients and absolute continuity of ω on [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 ω0 is an arbitrary continuous, concave, nondecreasing function, increasing on a neighbourhood of zero such that ω0(0) = 0 and the point t0 and the function ω1 are chosen in such a way that ω preserves 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 VMO, the continuity of coefficients Aαβij = Aαβij (x, u) with respect to x is not supposed. In the linear case, when the coefficients Aαβij = Aαβij (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 VMO, hence possibly discontinuous (see [4, 19] and references therein). It is well known that even in the continuous case the dependence of coefficients Aαβij on u leads to weaker regularity results for minimizers. In dimension n ≥ 3 there are examples of vectorial quadratic functionals (N > 1) with analytic coef- ficients Aαβij = Aαβij (u) whose minimizers are discontinuous (see [10, p. 317], [11]). For the analytic coefficients Aαβij = Aαβij (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 Hölder 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 HÖLDER 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 t is near by zero. Here it is needful to note that the second condition works likewise when ν is small but, in this case, the modulus of continuity ω has 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 Aαβij of the functional (1.1) satisfy (i), (ii), (iii) and (iv). There exists a positive constant M such that if the minimizer u of the functional (1.1) satisfies the condition 1 |Ω|1−2/n ∫ Ω |Du|2 dy ≤ 1 M2 (1.6) then u belongs to C0,(ϑ−n+2)/2(Ω0,RnN ) when ϑ > n − 2 and to BMO(Ω0,RnN ) when ϑ = n− 2. Here M = sup t0 1/p are such that Cρp−1 µ ≥ K C2p 1 C (p+1)/2 2 Lpϑ/(n−ϑ) (ω∞ ν )p( |Ω|1−2/n (2d)n−2 )(p−1)/2 (1.8) in the case when the coefficients Aαβij 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 , L is the constant from Lemma 2.7 below, C1, C2 are 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 (κn is the Lebesgue measure of the unit ball in Rn). If Aαβij = Aαβij (x, u) then, formally, the constant K on 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. DANĚČEK, 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 Aαβij on variable x tends to the choice d = min{R0,dist(Ω0, ∂Ω)/2} (for definition of R0 see (3.25) below) and so d and, consequently, the value of the constant Cρp−1 µ from (1.8) depend on ”VMO- quality” of x-dependence of coefficients Aαβij as well. Broadly speaking, the bigger R0 is, 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 Aαβij 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 ∈ W 1,2(Ω,RN ), mentioned in the foregoing theorem, is such that u − g ∈ W 1,2 0 (Ω,RN ) for some g ∈ W 1,2(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 ∫ Ω |Dg|2 dy . 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 tγ0 tγ ) for t0 ≤ t ≤ t1, 0 < γ ≤ 1, ω∞ for t > t1 (1.9) where ω0 is an arbitrary continuous, concave, nondecreasing function such that ω0(0) = 0 and the points t0, t1 are chosen so that ω is continuous and concave on [0,∞). If we put ε = ω∞/C ρ µ in (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 { 1, 3C 2 2µ−1ρ µ eC 2 2µ−1 ρ µ })2 . (1.10) Here µ ≥ 6, ρ > 1/p and t0 > 0. Example 1.6. Let ω(t) = 2ω∞ π arctan ( t Cτµ ) 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−ρµ e ( C ρ µ 2 √ µ ) 2 2µ−1 )2 . (1.12) Here τ > ρ > 1/p, µ ≥ 6 satisfy (1.8) and Ψ̃(ω(t0)/ε) = 0. EJDE-2020/69 HÖLDER CONTINUITY 5 2. Preliminaries If x ∈ Rn and r is 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)| ∫ Ωr(x) u(y) dy = − ∫ Ωr(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) =∫ Br(x) |Du(y)|2 dy, Ur = Ur(x) = r2−nφ(x, r) for Br(x) ⊂ Ω. Beside the standard space C∞0 (Ω,RN ), Hölder space C0,α(Ω,RN ) and Sobolev spaces W k,p(Ω,RN ), W k,p 0 (Ω,RN ) we use Morrey spaces Lq,λ(Ω,RN ) (see, e.g. [5, 14]). We will denote byXloc(Ω,RN ) the space of all functions which belong toX(Ω̃,RN ) for any bounded subdomain Ω̃ with smooth boundary which is compactly embedded in Ω. We recall a definition of VMO - spaces and a few properties of Morrey spaces. We set for f ∈ L1(Ω), 0 < a <∞ Na(f,Ω) := sup x∈Ω,r 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. DANĚČEK, 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 ≥ 0 is measurable and l(t) = {y ∈ Rn : w(y) > t} then∫ Rn φ ◦ w dy = ∫ ∞ 0 |l(t)|φ′(t) dt. Lemma 2.4. Let v ≥ 0, b > 0, µ > 0 and q > 1 be arbitrary. Then v lnµ+(bv) ≤ Cµ bq−1 vq (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 ≤ R1 be positive numbers, n− 2 ≤ ϑ < n, η a nonnegative and nondecreasing function on (0,∞). Then there exist ε0, c positive so that for any nonnegative, nondecreasing function φ defined on [0, 2R1] and satisfying with (B1 +B2η(U2R0 )) ∈ [0, ε0] 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, R such 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 ε0 = 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 ( Aαβij Dβu j ) = 0, i = 1, . . . , N where Aαβij are constants satisfying (i) and (ii). There exists a constant L = L(n,N,M/ν) ≥ 1 such that for every weak solution u ∈ W 1,2(Ω,RN ), for every x ∈ Ω and 0 < σ ≤ R ≤ dist(x, ∂Ω) the following estimate holds,∫ Bσ(x) |Du(y)|2 dy ≤ L ( σ R )n ∫ BR(x) |Du(y)|2 dy . 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 Hölder inequality that is standard in our setting . EJDE-2020/69 HÖLDER CONTINUITY 7 Lemma 2.9 (see [5, 10]). Let u ∈ W 1,2(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,( − ∫ BR(x) |Du|2p dy )1/2p ≤ C1 ( − ∫ B2R(x) |Du|2 dy )1/2 . Let x0 be any fixed point of Ω, 0 < R ≤ dist(x0, ∂Ω). We set Aαβij (ux0,R)x0,R = − ∫ BR(x0) Aαβij (y, ux0,R) dy . Asolution to the system Dα ( Aαβij (ux0,R)x0,RDβv j ) = 0 in BR(x0), v − u ∈W 1,2 0 (BR(x0),RN ) (2.8) posses the following property. Lemma 2.10 (see [5, 6, 10]). Let v ∈W 1,2(BR(x0),RN ) be a solution to (2.8) with u ∈W 1,2p(BR(x0),RN ), p ≥ 1. Then∫ BR(x0) |Dv|2p dy ≤ C2 ∫ BR(x0) |Du|2p dy. 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 C2 from 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,∞)→ R be a nonnegative function which is integrable on [a, b] for all a < b <∞ and N = sup 0 0, µ ≥ 2 and the constant p > 1 from Lemma 2.9 we have∫ BR(x) |Du|2 lnµ+(b|Du|2) dy ≤ 2−nC2p 1 Cµ ( b− ∫ B2R(x) |Du|2 dy )p−1 ∫ B2R(x) |Du|2 dy where C1 is the constant from Lemma 2.9. The above proposition is a straightforward consequence of Lemmas 2.4 and 2.9. 8 J. DANĚČEK, E. VISZUS EJDE-2020/69 Proposition 2.14. Let v ∈ W 1,2(BR(x0),RN ) be a weak solution to (2.8) where u ∈W 1,2(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 ≥ 2 and the constant p > 1 from Lemma 2.9 we have ∫ BR(x0) |Dv|2 lnµ+ ( b|Dv|2 ) dx ≤ 2−nC2p 1 C2Cµ ( b− ∫ B2R(x0) |Du|2 dx )p−1 ∫ B2R(x0) |Du|2 dx (2.9) where C2 is 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 Aαβij 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 Aαβij = Aαβij (u). We set φ(r) = φ(x, r) = ∫ Br(x) |Du|2 dy and Ur = Ur(x) = r2−nφ(x, r) for Br(x) ⊂ Ω. Now let x be any fixed point of Ω0 ⊂ Ω, dist(Ω0, ∂Ω) = 2d > 0, B2R(x) ⊂ Ω, 0 < R ≤ d and v be a minimizer of the frozen functional A0(v;BR(x)) = ∫ BR(x) Aαβij (uR)Dαv iDβv j dy among all the functions in W 1,2(BR(x),RN ) taking the values u on ∂BR(x). From the Euler equation for v and from Lemma 2.7 we have ∫ Bσ(x) |Dv|2 dy ≤ L ( σ R )n ∫ BR(x) |Dv|2 dy, for 0 < σ ≤ R. (3.1) Put w = u− v. It is clear that w ∈ W 1,2 0 (BR(x),RN ). Using (3.1) by standard arguments we obtain ∫ Bσ(x) |Du|2 dy ≤ 2 ( 1 + 2L ( σ R )n)∫ BR(x) |Dw|2 dy + 4L ( σ R )n ∫ BR(x) |Du|2 dy. (3.2) EJDE-2020/69 HÖLDER CONTINUITY 9 Now we estimate the first integral on the right-hand side of (3.2). From [7, Lemma 2.1] we have ∫ BR(x) |Dw|2 dy ≤ 2 ν ( A0 (u;BR(x))−A0 (v;BR(x)) ) ≤ 2 ν {∫ BR(x0) ( Aαβij (uR)−Aαβij (u) ) Dαu iDβu j dx + ∫ BR(x0) ( Aαβij (v)−Aαβij (uR) ) Dαv iDβv j dx +A (u;BR(x0))−A (v;BR(x0)) } = 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 u is a minimizer. Now we estimate terms I and II from (3.3). Assumption (iii) and the Young inequality (2.3) give |I| ≤ ∫ BR(x) ω (|u− uR|) |Du|2 dy ≤ ∫ BR(x) Φ ( ε|Du|2 ) dy + ∫ BR(x) Ψ ( 1 ε ω (|u− uR|) ) dy = I1 + I2. (3.4) By Proposition 2.13 we have I1 = ε ∫ BR(x) |Du|2 lnµ+ ( aε|Du|2 ) dy ≤ ε 2−nC2p 1 Cµ ( aε− ∫ B2R(x) |Du|2 dy )p−1 φ(2R). (3.5) According to Lemma 2.3 (see (2.1) as well) we have I2 = ∫ BR(x) Ψ (1 ε ω (|u− uR|) ) dy = 1 a ∫ ∞ 0 d dt Ψ̃ (ω(t) ε ) mR(t) dt = 1 a Ĩ2 (3.6) where mR(t) = | {y ∈ BR(x) : |u(y)− uR| > t} |. 10 J. DANĚČEK, E. VISZUS EJDE-2020/69 Estimating the term Ĩ2 we use the fact that mR(t) ≤ κnR n and the constant from the Poincaré inequality on the ball equals to 22n. By Lemma 2.12 we obtain Ĩ2 ≤ ∫ t0 0 d dt Ψ̃ (ω(t) ε ) mR(t) dt+ ∫ ∞ t0 d dt Ψ̃ (ω(t) ε ) mR(t) dt ≤ κnRn ∫ t0 0 d dt Ψ̃ (ω(t) ε ) dt + sup t0 0 (3.15) 12 J. DANĚČEK, E. VISZUS EJDE-2020/69 where δ = 1 + ln ε0/ ln ν, ε0 = 1 2(2n+3L)ϑ/(n−ϑ) , Cµ = ( µ (p−1)e )µ and ρ, µ ∈ R are suitable constants. We obtain φ(σ) ≤4L ( σ R )n φ(2R) + 1 2 ( 1 + 2L ( σ R )n) × [2(n+10)p−2(n+3)(PC2 1 )pC (p+1)/2 2 κp−1 n εp−1 0 Cpρ−1 µ ( |Ω|1−2/n (2d)n−2 )(p−1)/2 + ε0 ( (2d)n−2 |Ω|1−2/n )1/2( 1 2n+4 √ C2 Ψ̃ (ω(t0) ε ) + 1 4 M √ U2R )] φ(2R), (3.16) where P = ω∞/ν. The constants ρ > 1/p and µ ≥ 6 can be always chosen in such a way that 2(n+10)p−2(n+3)(PC2 1 )pC (p+1)/2 2 κp−1 n εp−1 0 Cpρ−1 µ ( |Ω|1−2/n (2d)n−2 )(p−1)/2 ≤ 1 2 ε0, which is equivalent to the estimate Cρp−1 µ ≥ 2(n+10)p−(2n+5)(PC2 1 )pC (p+1)/2 2 κp−1 n εp0 ( |Ω|1−2/n (2d)n−2 )(p−1)/2 . Using the second term in (1.7) and taking into account that ((2d)n−2/|Ω|1−2/n)1/2 ≤ 1, and we obtain φ(σ) ≤ 4L ( σ R )n φ(2R) + 1 2 ( 1 + 2L ( σ R )n) × [3 4 + 1 4 ( (2d)n−2 |Ω|1−2/n )1/2 M √ U2R ] ε0 φ(2R), for 0 < σ ≤ R ≤ d. (3.17) For R = d by (1.6) we obtain( (2d)n−2 |Ω|1−2/n )1/2 M √ U2d(x) ≤M ( 1 |Ω|1−2/n ∫ Ω |Du|2 dy )1/2 ≤ 1 . Putting A = 4L, B1 = 3ε0/4, and B2 = ε0/4 in (3.17) and using Lemma 2.5, we can conclude that φ(σ) ≤ cσϑφ(2R), for 0 < σ ≤ R . Now, the result follows from Proposition 2.2. Case Aαβij = Aαβij (x, u). Let x be any fixed point of Ω0 ⊂ Ω, dist(Ω0, ∂Ω) = 2d0 > 0, B2R(x) ⊂ Ω, 0 < R ≤ d0 and v be a minimizer of the functional A0(v;BR(x)) = ∫ BR(x) Aαβij (uR)RDαv iDβv j dy among all the functions in W 1,2(BR(x),RN ) taking the values u on ∂BR(x) where Aαβij (z)R = − ∫ BR(x) Aαβij (y, z) dy. Arguments, analogous to those at the beginning of the proof of Theorem 1.1, give us ∫ Bσ(x) |Du|2 dy ≤ 2 ( 1 + 2L ( σ R )n)∫ BR(x) |Dw|2 dy + 4L ( σ R )n ∫ BR(x) |Du|2 dy (3.18) EJDE-2020/69 HÖLDER CONTINUITY 13 where w = (u − v) ∈ W 1,2 0 (BR(x),RN ). Now we estimate the first integral on the right hand side of (3.2). From [7, Lemma 2.1] we have∫ BR(x) |Dw|2 dy ≤ 2 ν ( A0 (u;BR(x))−A0 (v;BR(x)) ) ≤ 2 ν {∫ BR(x) ( Aαβij (uR)R −Aαβij (y, uR) ) Dαu iDβu j dy + ∫ BR(x) ( Aαβij (y, uR)−Aαβij (y, u) ) Dαu iDβu j dy + ∫ BR(x) ( Aαβij (y, uR)−Aαβij (uR)R ) Dαv iDβv j dy + ∫ BR(x) ( Aαβij (y, v)−Aαβij (y, uR) ) Dαv iDβv j dy +A (u;BR(x))−A (v;BR(x)) } = 2 ν {I + II + III + IV +A (u;BR(x))−A (v;BR(x))} ≤ 2 ν (I + II + III + IV ) . (3.19) Notice that A (u;BR(x)) − A (v;BR(x)) ≤ 0, since u is a minimizer. Now we will estimate the terms I, II, III and IV from (3.19). In the following we will denote A := (Aαβij ). Using Hölder inequality, higher integrability of gradient of minima (Lemma 2.9, p > 1, p′ = p/(p− 1)) we obtain |I| ≤ |BR(x)|1/p (∫ BR(x) |A(uR)R −A(y, uR)|p ′ dy )1/p′( − ∫ BR(x) |Du|2p dy )1/p ≤ C2 1 |BR(x)|1/p (∫ BR(x) |A(uR)R −A(y, uR)|p ′ dy )1/p′ − ∫ B2R(x) |Du|2 dy. Taking into account assumptions (i), (iv) and Definition 2.1 we obtain( − ∫ BR(x) |A(uR)R −A(y, uR)|p ′ dy )1/p′ ≤ (2M)1/p (NR (A(·, uR))) 1−1/p and then, using the above two estimates, we have |I| ≤ 2−nC2 1 (2M)1/p (NR (A(·, uR))) 1−1/p φ(2R). (3.20) A similarity of the terms I and III enables us to write (by Lemma 2.10) the inequality |III| ≤ 2−nC2 1C 1/p 2 (2M)1/p (NR (A(·, uR))) 1−1/p φ(2R). (3.21) Now it remains to estimate the terms II and IV from (3.19). Estimating these two terms is step by step the same as estimating the terms I and II from (3.3) in the previous part of the proof. So we have |II| ≤ ε2−nC2p 1 Cµ ( aε− ∫ B2R(x) |Du|2 dy )p−1 φ(2R) + 4R2 a ( Ψ̃ (ω(t0) ε ) U2R + 2n−1M√ U2R ) φ(2R) (3.22) 14 J. DANĚČEK, E. VISZUS EJDE-2020/69 and |IV | ≤ 2−nC2p 1 C2Cµ ε ( aε− ∫ B2R(x) |Du|2 dy )p−1 φ(2R) + 4R2 a ( Ψ̃ (ω(t0) ε ) U2R + 2n+2 √ C2M√ U2R ) φ(2R). (3.23) Substituting (3.20)–(3.23) into (3.19) and, consequently, (3.19) into (3.18) we obtain φ(σ) ≤ 4L ( σ R )n φ(2R) + 2 ( 1 + 2L ( σ R )n )[K1(R) ν ε0 + 22−nC2p 1 C2Cµ ν ε ( aε− ∫ B2R(x) |Du|2 dy )p−1 + 16R2 aν ( Ψ̃ (ω(t0) ε ) U2R + 2n+2 √ C2√ U2R M )] φ(2R) (3.24) for σ < R ≤ d0, where ε0 = 1 2(2n+3L) ϑ n−ϑ , K1(R) = C2 1 (MC2)1/p (NR (A(·, uR))) 1−1/p 2n−3ε0 , see Remark 2.6 and Lemma 2.7 as well. Assumption (iv) implies that there exists R0 > 0, such that K1(R) ν ≤ 1 16 ⇐⇒ NR (A(·, uR)) ≤M ( 2n−7ε0ν C2 1C 1/p 2 M )p/(p−1) (3.25) for 0 < R ≤ R0 (here we recall that the choice of the constant R0 does not depend on x ∈ Ω0). Let us put d = min{d0, R0}. Then, in the estimate (3.24), we can choose the constants ε and a in the following way: ε = ω∞ Cρµ , a = 2n+10 √ C2R 2 νδ U2R ( |Ω|1−2/n (2d)n−2 )1/2 for U2R > 0, (3.26) where δ = 1 + ln ε0/ ln ν, Cµ = ( µ (p−1)e )µ and ρ, µ ∈ R are suitable constants. We obtain φ(σ) ≤ 4L ( σ R )n φ(2R) + 1 2 ( 1 + 2L ( σ R )n) × [1 4 ε0 + 2(n+10)p−2(n+3)(PC2 1 )pC (p+1)/2 2 κp−1 n εp−1 0 Cpρ−1 µ ( |Ω|1−2/n (2d)n−2 )(p−1)/2 + ε0 ( (2d)n−2 |Ω|1−2/n )1/2( 1 2n+4 √ C2 Ψ̃ (ω(t0) ε ) + 1 4 M √ U2R )] φ(2R), (3.27) where P = ω∞/ν. The constants ρ > 1/p and µ ≥ 6 can be always chosen in such a way that 2(n+10)p−2(n+3)(PC2 1 )pC (p+1)/2 2 κp−1 n εp−1 0 Cpρ−1 µ ( |Ω|1−2/n (2d)n−2 )(p−1)/2 ≤ 1 4 ε0 which is equivalent to the estimate Cρp−1 µ ≥ 2(n+10)p−2(n+2)(PC2 1 )pC (p+1)/2 2 κp−1 n εp0 ( |Ω|1−2/n (2d)n−2 )(p−1)/2 . EJDE-2020/69 HÖLDER CONTINUITY 15 Using the second term in (1.7) and taking into account ((2d)n−2/|Ω|1−2/n)1/2 ≤ 1, we obtain φ(σ) ≤ 4L ( σ R )n φ(2R) + 1 2 ( 1 + 2L ( σ R )n) [3 4 + 1 4 ( (2d)n−2 |Ω|1−2/n )1/2 M √ U2R ] ε0 φ(2R), for 0 < σ ≤ R ≤ d. The above estimate is formally the same as (3.17) in the first part of the proof. So, one can see that the result follows in the same way as it is demonstrated at the end of the previous case. 4. Illustrating examples and comments Here, for simplicity, we consider Aαβij = Aαβij (u). Example 4.1. Let Ω = BR(0) ⊂ Rn in Theorem 1.1, the function g, mentioned in Remark 1.4, belong to W 1,2 loc (Rn,RN ), and, for n−2 ≤ λ ≤ n it satisfy the condition sup0<σ≤R σ −λ ∫ Bσ(0) |Dg(y)|2 dy ≤ cλ, cλ > 0. Then choosing Ω0 = BR/2(0), d = R/4 and using (1.4), condition (1.6) will have the form cλM κ 1−2/n n ν Rλ−n+2 ≤ 1 M2 . (4.1) So, for sufficiently small R we obtain regularity of minimizer u in BR/2(0) by Theorem 1.1 in the case when λ > n− 2. The case λ = n − 2 leads to the regularity condition that depends only on the parameters of functional (1.1) and the function g. For sufficiently big R we obtain regularity of minimizer u in BR/2(0) by Theorem 1.1 in the case when 0 < λ < n− 2. Remark 4.2. Theorem 1.1 with a result from [12] can guarantee the everywhere regularity up to the boundary in a specific case. More precisely, if we consider Ω = BR(0), split coefficients Aαβij (u) = γαβaij(u) in (1.1), and suppose that Aαβij are uniformly continuous on RN with the modulus of continuity (1.5), the function g, introduced in Remark 1.4, belongs to W 1,s(BR(0),RN ), s > n and the minimizer u is bounded, then, according to [12], there exists a constant 0 < R1 < R such that u ∈ C0,1−n/s(BR(0) \ BR1(0),RN ) . Now, choosing in Theorem 1.1 ϑ = n − 2n/s, d = (R − R2)/2, 0 < R1 < R2 < R, if condition (1.6) is fulfilled, then u ∈ C0,1−n/s(BR(0),RN ). Example 4.3. In Ω = BR(0) ⊂ R3 we consider the quasilinear variational integral A(u; Ω) = ∫ Ω Aαβij (u)Dαu iDβu j dx where Aαβij (u) = aδijδαβ + b ( δiα arctan |ui| Cτµ + δjβ arctan |uj | Cτµ ) for α, β = 1, 2, 3, i, j = 1, 2, u−g ∈W 1,2 0 (BR(0),R2), g ∈W 1,2 loc (R3,R2) a > 6πb > 0 (Cµ is from Remark 1.2 and τ > ρ > 0). In this case we have M = 6a+ 10πb, ν = a− 6πb, ω∞ = π b 16 J. DANĚČEK, E. VISZUS EJDE-2020/69 and the modulus of continuity ω is given in Example 1.6. This is a sample of func- tional, 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 re- sults 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 ω defined by (1.11) and the second one corresponds to (1.9). For the simplicity, we put Ω = BR(0) ⊂ R3 and 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 ε0 = 2.3 · 10−6. In this case the function ω is 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∞) ≈ 108 1027 1044 1059 1069 t1 ≈ 1051 1051 1055 1059 1065 real value 1 M2 ≈ 104 106 109 1014 1022 estimate 1 M2 by means of (1.12) ≈ 102 104 107 1013 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 t1 is the point for which ω(t1) = 0.95 · ω∞. In the case when the function ω is defined by (1.9), for the foregoing parameters we obtain the following chart. ω∞ = 1030 1040 1050 1060 1070 t0 = 107 1010 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 M2 by means of (1.10) ≈ 10 107 1011 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 t1 is 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 constantM from (1.7) where ω is defined by Examples 1.5 and 1.6. Ψ̃ (ω(t) ε ) − Ψ̃ (ω(t0) ε ) t− t0 = ( d dt Ψ̃ (ω(t) ε )) |t=ξ = ω′(ξ) ε [ 1 + 2 2µ− 1 ( 1 2 √ µ ω(ξ) ε ) 2 2µ−1 ] e ( 1 2 √ µ ω(ξ) ε ) 2 2µ−1 , for t0 < ξ < t ≤ t1. EJDE-2020/69 HÖLDER CONTINUITY 17 (a) Estimate of M related to the function ω from Example 1.5. Here we consider µ ≥ 6, ρ > 1/p, 0 < γ < 1, t0 > 0, Cµ > 1. M = sup t0 0. (b) Estimate of M for ω from Example 1.6: M≤ e ( C ρ µ 2 √ µ ) 2 2µ−1 Cτ−ρµ , τ > ρ > 1 p (5.3) and Ψ̃(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. Daněček, E. Viszus; Interior C0,γ - regularity for vector-valued minimizers of quasilinear functionals. Nonlinear Anal., 74 (2011), 5274–5285. [2] J. Daněček, E. Viszus; Regularity on the interior for the gradient of weak solutions to non- linear second-order elliptic systems. Electron. J. Diff. Equations, 2013, 121 (2013), 1–17. [3] J. Daněček, E. Viszus; Interior C0,γ - regularity for vector-valued minimizers of quasilinear functionals with VMO-coefficients. Mediterr. J. Math., 12 (2015), 1287–1305. [4] P. Di Gironimo, L. Esposito, L. Sgambati; A remark on L2,λ - regularity for minimizers of quasilinear functionals. Manuscripta Math. 113, (2004), 143–151. 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Tachikawa; Partial regularity of the minimizers of quadratic functionals with VMO coefficients. J. London Math. Soc., (2),72 (2005), 609–620. [20] D. Sarason; Functions of vanishing mean oscillation. Trans. Amer. Math. Soc., 207 (1975), 391–405. [21] W. P. Ziemer, Weakly differentiable functions. Springer-Verlag, Heidelberg, 1989. Josef Daněček VŠB - Technical University of Ostrava, FEECS, Department of Applied Mathematics, 17. listopadu 15/2172, 70833 Ostrava-Poruba, Czech Republic Email address: danecek.j@seznam.cz Eugen Viszus Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathe- matics, Physics and Informatics Comenius University, Mlynská dolina, 84248 Bratislava, Slovak Republic Email address: eugen.viszus@fmph.uniba.sk 1. Introduction 2. Preliminaries 3. Proof of Theorem 1.1 Case Aij=Aij(u). Case Aij=Aij(x,u). 4. Illustrating examples and comments 5. Appendix Acknowledgements References