Maximal regularity for local minimizers of non-autonomous functionals

We establish local $C^{1,\alpha}$-regularity for some $\alpha\in(0,1)$ and $C^{\alpha}$-regularity for any $\alpha\in(0,1)$ of local minimizers of the functional \[ v\ \mapsto\ \int_\Omega \phi(x,|Dv|)\,dx, \] where $\phi$ satisfies a $(p,q)$-growth condition. Establishing such a regularity theory with sharp, general conditions has been an open problem since the 1980s. In contrast to previous results, we formulate the continuity requirement on $\phi$ in terms of a single condition for the map $(x,t)\mapsto \phi(x,t)$, rather than separately in the $x$- and $t$-directions. Thus we can obtain regularity results for functionals without assuming that the gap $\frac qp$ between the upper and lower growth bounds is close to $1$. Moreover, for $\phi(x,t)$ with particular structure, including $p$-, Orlicz-, $p(x)$- and double phase-growth, our single condition implies known, essentially optimal, regularity conditions. Hence, we handle regularity theory for the above functional in a universal way.


Introduction
The calculus of variations is a classical and still active topic in mathematics which is connected not only to other mathematical fields (partial differential equations, geometry, . . . ) and but also to applications (physics, engineering, economy, . . .).Research on regularity of minimizers of the functional v → F (v, Ω) := ˆΩ F (x, Dv) dx has been a major topic in calculus of variations and PDEs.If F depends only on the gradient, i.e.F (x, z) ≡ F (z), F is called an autonomous functional.The simplest nonlinear model case is the p-power function The corresponding Euler-Lagrange equation is the p-Laplace equation div(|Du| p−2 Du) = 0, and the maximal regularity of weak solutions of p-Laplace equations is C 1,α for some α ∈ (0, 1) depending only on p and the dimension n.We refer to [1,32,40,59,63,78,79,80,82,83] for classical results on C 1,α -regularity for equations and systems of p-Laplacian type.
On the other hand, if F depends on both the space variable and the gradient, F is called a non-autonomous functional, and this has been a central topic in contemporary regularity theory.The main approach to such minimization problems is due to Giaquinta and Giusti [47,48].It is based on the following p-type growth conditions: This essentially corresponds to the perturbed case a(x)|z| p with the same p-type growth assumed at all points.Lieberman [61] extended this to the case where |z| p is replaced by ϕ(|z|).However, such structure conditions fail to accommodate many kinds of energy functionals since the variability in the x-and z-directions are treated separately.
The need to treat the x-and z-directions separately leads Mingione to conclude in his influential survey that "regularity results should be chased [in more general cases] by looking at special classes of functionals and thinking of relevant model examples, thereby limiting the degree of generality one wants to achieve" [71, p. 405].In this spirit, the most significant non-autonomous functionals in the literature have so-called Uhlenbeck structure, i.e.F depends on t := |z| instead of z, F (x, z) = ϕ(x, |z|) = ϕ(x, t), and are the following: I. Perturbed Orlicz: a(x)ψ(t), where 0 < ν a(•) L and ψ ′ (t) ≈ tψ ′′ (t).II.Variable exponent: t p(x) , where 1 < p − p(•) p + < ∞.III.Double phase: t p + a(x)t q , where 1 < p q and a(•) 0. These models were first studied by Zhikov [85,86] in the 1980's in relation to Lavrentiev's phenomenon and have been considered in hundreds of papers since [71,75].In keeping with Mingione's thesis, regularity results for these cases have been established in independent, idiosyncratic ways (cf.Section 2).Moreover, various variants and borderline cases have been investigated, such as: IV. Perturbed variable exponent: t p(x) log(e + t), e.g.[44,60,72,74].
Remark 1.3.In this paper, we consider ϕ(x, t) continuous in x.It is clear that we cannot remove the assumption lim r→0 ω(r) = 0 from (VA1) and still obtain C α -regularity for all α ∈ (0, 1).However, continuity is not strictly speaking necessary, as it is known for ϕ(x, t) = a(x)ψ(t) with a locally VMO (vanishing mean oscillation), that the corresponding minimizer is in C α loc for any α ∈ (0, 1), in fact, in W 1,p loc for any p > 1.It seems that for this result the special multiplicative structure is important.
Remark 1.4.If we consider solutions of the general linear elliptic equation div(A(x)Du) = 0, where A(x) is a bounded and uniformly elliptic n × n matrix, then the continuity of A does not imply that the function is Lipschitz or its derivative is continuous [56,Propositions 1.5 and 1.6].Therefore, we cannot expect to remove the assumption ω(r) cr β from (VA1) and still obtain C 1,α -regularity.
We shall introduce notation, assumptions and properties of generalized Φ-functions and related spaces later in Section 3. Recall that local minimizer means that u satisfies ˆΩ′ ϕ(x, |∇u|) dx and Ω ′ ⋐ Ω.In fact, we will generalize (VA1) to a weaker version, (wVA1), which covers not only (VA1) but its borderline cases (see Remark 4.2) as well as the PDE case (see Remark 4.3), and under this condition we will prove C α -and C 1,α -regularity, see Theorems 7.2 and 7.4.As far as we know, these theorems cover all previously known results (and several new ones) of C α -or C 1,α -regularity for the functionals I-X (see Section 8) with the exception of VMO coefficients (Remark 1.3).
Let us conclude the introduction by outlining the approach of the paper and pointing out the main difficulties and innovations.
The first difficulty for a reasonable regularity theory is to find a well-designed condition for general ϕ.The regularity conditions on ϕ for the types I-III seem unconnected to one another, since in these cases, the behaviors of ϕ with respect to x and t can be investigated separately.Recently, on the other hand, the C α -continuity with some small α > 0 for (quasi-)minimizers of the general non-autonomous functional has been established under the so-called (A1) condition [13,54,55]: From this, it is natural to require L → 1 as r → 0 for higher regularity.Additionally, small values t 1 were previously lumped into an additive constant using decay at infinity.A more precise estimate, on the other hand, requires the previous condition to be extended from [ The main difficulty is to find a suitably regular auxiliary autonomous function φ(t) for the perturbation technique in which one approximates the minimizer with the solution to a related but simpler minimization problem.In order for the perturbation argument to work under the assumption (VA1), the autonomous function φ(t) should satisfy the following requirements: ( (2) For a given B r with small r ∈ (0, 1), φ(t) is sufficiently close in some sense to ϕ(x, t) for all (x, t) where A0), (aInc) 1 , (aDec) q/p and (A1).The construction of such φ is quite nontrivial, since the property (3) is not satisfied in general for either φ(t) = ϕ(y, t) with any choice of y ∈ B r or φ(t) = ϕ − Br (t) (the expected choices based on previous research).Note that for type II (variable exponent) or type III (double phase), one can simply take φ(t) = t pr or φ(t) = t p + a r t q , where p r := inf Br p(•) and a r := inf Br a(•), so this provides no guidance for the general case: in these special cases t → ϕ(x, φ−1 (t)) satisfies (aInc) 1 since a single point captures the slowest growth for all values of t, whereas in general the slowest growth may occur at different locations for different t.
The requirements (1)-( 3) above are crucially used in our comparison step.Let v be a minimizer of an autonomous functional with φ-energy in B r satisfying v = u on ∂B r .Then by (1) and known regularity results for Orlicz growth, we obtain that v is locally C 1,α 0 for some α 0 ∈ (0, 1) (Lemma 4.12).Moreover, from (3) we can deduce a global nonlinear Calderón-Zygmund type estimate in the generalized Orlicz space L θ with θ = θ 1+σ 0 0 for some σ 0 > 0 (Lemma 4.15), which implies that Dv ∈ L ϕ (B r ) and so, with this v, we can use the minimizing property of u.Note that this approach is new even for the double phase problem, type III.
The Calderón-Zygmund type estimates (Lemma 4.15) in generalized Orlicz space L θ for the norm will be obtained by an extrapolation argument [29] and in this process (A1) of θ suffices.However, we need a mean integral version of Calderón-Zygmund type estimate that is stable under the size of underlying domain and here (A1) of θ is not enough.We overcome this problem by replacing θ(x, t) with θ(x, t) + t p 1 for suitable p 1 > 1 along with delicate analysis.Note that θ(x, t) + t p 1 satisfies a stronger assumption than (A1).As a consequence, there is "+1" in the mean integral version of estimate (4.17).
We construct our approximation φ and derive the comparison estimate for ϕ and φ in Section 5.In Proposition 5.12 we show that our approximation satisfies the assumptions in (3), above, and in this step a new framework for generalized Orlicz spaces from [51] is rather crucial.Then a comparison argument along with (2) and a higher integrability result for Du yield that Du is sufficiently close to Dv in the mean oscillation sense (Corollary 6.3).
We present proofs of some regularity results for autonomous problems in Appendices A and B. We start this article with an overview of regularity theory in the (p, q)-growth case (Section 2) and with notation and background (Section 3).Remark 1.6.Constructing a suitable φ is the main problem also in extending this approach to the case without Uhlenbeck structure, i.e. energy functionals depending on the derivative Du, not just its norm.Namely, an approximation φ : Ω × R n → R affords us much less room to operate in than φ : Ω × [0, ∞) → R. Indeed, it is not even clear how to state the appropriate assumptions in this case.In addition, the main tools from [51] concern only the isotropic case ϕ(x, |Du|).Therefore, the regularity of the anisotropic minimization problem ´ϕ(x, Du) dx remains a question for future research.
Remark 1.7.The vectorial case, i.e. u : Ω → R N with N > 1, is also an interesting issue.The main difficulty in this case is the following: in order that the local minimizer of the regular autonomous functional with Orlicz function φ = φ(t) have C 1,α -regularity φ should apparently satisfy not only t φ′′ (t) ≈ φ′ (t) but also a Hölder type vanishing condition on φ′′ , see [35,Assumption 2.2].It is unclear whether (VA1) or some modification implies the additional condition of φ.This is also a future research topic.
2. Overview of regularity for (p, q)-growth and special cases An alternative extension to the approach of Giaquinta and Giusti is to consider different upper and lower growth rates, and replace the exponent on the right-hand side by q > p.This leads to so-called (p, q)-growth functionals, for instance with assumptions This case was introduced and systematically studied by Marcellini [64,65,66,67,68].Several other researchers also contributed to the theory, cf.[11,38,71].For instance, Marcellini [65] started by showing that that every minimizer in W 1,q loc (Ω) has locally bounded gradient provided 2 p q and q p 1 + 2 n − 2 , when n > 2; (the proof uses PDE techniques and entails several additional assumptions, which are not presented here; see also a recent improvement in [12]).Note, however, that W 1,q loc (Ω) is already higher integrability, so this is not a natural assumption in this context and was addressed in [65, Section 3].Later, Esposito, Leonetti and Mingione [39] showed that every minimizer in W 1,p loc (Ω) also belongs to W 1,q loc (Ω), but only when Furthermore, they provide an example showing that if the latter condition does not hold, then a minimizer in W 1,p loc (Ω) need not belong to W 1,q loc (Ω) so the Lavrentiev phenomenon occurs.
It seems that (p, q)-growth is the most general class of non-autonomous functionals in the calculus of variations.Regularity theory, including C α -and C 1,α -regularity, in this general class is not easily obtained from classical regularity theory for functionals with standard p-growth, see for instance [71].Furthermore, there are no general results in the (p, q)-case which cover the special cases I-X, so in that sense the theory is incomplete.We note that some recent papers [13,23,24,54,55,81] deal with calculus of variation in generalized Orlicz spaces, but these papers do not cover higher regularity.
Indeed, the C α -and C 1,α -regularity theories for type I-III functionals have been proved in independent ways.For I, ϕ is nothing but an autonomous functional with coefficient, and so regularity results can be obtained by using a standard perturbation argument.On the other hand, II and III are quite different from I, since they are potentially nonuniformly elliptic problems.Formally, we can rewrite the energy functions as II: |Du| p(x)−p − |Du| p − and III: (1 + a(x)|Du| q−p )|Du| p .
Here, |Du| p(x)−p − and 1 + a(x)|Du| q−p blow up or vanish when |Du| does.Therefore, by identifying a(x) in I with |Du| p(x)−p − or 1 + a(x)|Du| q−p , we see that a is neither bounded nor far away from the zero.Let us briefly introduce regularity results for the above types.Let u be a minimizer of the ϕ-energy (1.2) with ϕ being one of I-III.Then the following is known: For type I, i.e. ϕ(x, t) = a(x)ψ(t), suppose a is continuous with modulus of continuity ω a .Then lim r→0 + ω a (r) = 0 =⇒ u ∈ C α for any α ∈ (0, 1), ω a (r) r β for some β > 0 =⇒ u ∈ C 1,α for some α ∈ (0, 1), (2.1) see for instance [71] and references therein.
For this result, we refer to the series of papers of Baroni, Colombo and Mingione [10,25], see also [8,15,26,27,73].Note that no independent condition implies C α -regularity.In other words, we cannot ensure even C α -regularity for u if q p > 1 + β n .We also mention that the C 1,α -regularity for type III was first proved under the following condition instead of (2.3): (2.4) for some α ∈ (0, 1), see [25], and later it was extended to the borderline case q p = 1 + β n in [10], see also [31].As mentioned in the introduction, our general results cover all of these special cases.Specifically, Theorem 1.1(1) implies (2.1) 1 and (2.2) 1 and Theorem 1.1(2) implies (2.1) 2 , (2.2) 2 and (2.4).We notice that Theorem 1.1(2) does not imply (2.3).In fact, (VA1) holds when ϕ(x, t) = t p + a(x)t q with a(•) ∈ C 0,β if and only if the strict inequality q p < 1 + β n holds.This gap will be filled by Theorem 7.4; this is one main reason why we consider the slightly weaker assumption (wVA1).
Furthermore, many other, previously unstudied cases can also be covered, cf., e.g.Corollary 8.3, and Section 8 more generally.Originally, the double phase model was introduced to model the situation when two phases (the p and the q-growth phases) mix.Since only the larger exponent affects the nature of the problem, this was simplified in the form t p + a(x)t q that we have seen.However, we can also consider a variant which is more closely related to the original motivation: (2.5) ϕ(x, t) = (1 − a(x))t p + a(x)t q , where 1 < p q, a(•) : Ω → [0, 1].Now a indicates the relative amount of material at a point from the q-phase.Such functionals have been treated by Eleuteri-Marcellini-Mascolo [36,37,38].More generally, we can also deal with general double phase problems of the type where L and ψ ′ , ξ ′ satisfy (A0), (Inc) p−1 and (Dec) q−1 , which includes the following examples: t p + a(x)t q , a(x)t p + t q , a(x)t p + b(x)t q , and ψ(t) + a(x)ψ(t) ln(e + t).
We present conditions for above functions to satisfy (wVA1) or (VA1) in Corollaries 8.4 and 8.6, so that C α -and C 1,α -regularity results for (2.5) are obtained as special cases.We note that the second example a(x)t p + t q can be understood as a functional with standard q-growth and hence q/p has no upper bound to obtain the regularity results.Here, we explain the regularity results for this functional as a special case of double phase problems.In addition, in the same spirit, one could consider functionals with infinitely many phases such that which satisfies the fundamental assumption of Theorem 1.1.

Generalized Orlicz spaces
Notation and assumptions.For x 0 ∈ R n and r > 0, B r (x 0 ) is the ball in R n with radius r and center x 0 .We write B r = B r (x 0 ) when the center is clear or unimportant.For an integrable function f in U ⊂ R n , we define (f ) U by the average of f in U in the integral sense, that is, (f Lf (s) or f (s) Lf (t), respectively.In particular, if L = 1 we say f is nondecreasing or non-increasing.
We refer to [51] for more details about basics of Φ-functions and generalized Orlicz spaces.For If the map t → ϕ(x, t) is non-decreasing for every x ∈ Ω, then the (left-continuous) inverse function with respect to t is defined by If ϕ is strictly increasing and continuous in t, then this is just the normal inverse function.
We define some conditions related to regularity with respect to the t-variable.
Note that this version of (A0) is slightly stronger than the one used in [51], but they are equivalent under the doubling assumption (aDec).Let 0 On the other hand, if ϕ satisfies (aDec) γ with the constant L 1, then Remark 3.2.If ϕ satisfies (aInc) γ or (aDec) γ for some γ > 0, then so do ϕ − Br and ϕ + Br for any B r ⊂ Ω.
Cf (y), respectively, for all y ∈ U.In particular, in this paper we shall use these symbols when the relevant constants C depend only on n and constants from the fundamental conditions (aInc) γ , (aDec) γ , (Inc) γ , (Dec) γ and (A0).By following this, for instance, (A0) can be written as ϕ(•, 1) ≈ 1 in Ω.We use some results from papers with a weaker notion of equivalence: f ≃ g (in U) which means that there exists C 1 such that f (C −1 y) g(y) f (Cy) for all y ∈ U.However, if (aDec) holds, then ≃ and ≈ are equivalent and furthermore constants can be moved inside and outside of ϕ as observed above.
Basic properties of generalized ϕ-functions and related functions spaces.We next introduce classes of Φ-functions.Let L 0 (Ω) be the set of the measurable functions on Ω.In the sequel we omit the words "generalized" and "weak" from the parentheses.
For ϕ : [0, ∞) → [0, ∞), we define the conjugate function by By definition, we have the following Young inequality: We state some properties of Φ-functions, for which we refer to [51, Chapter 2].
Proof.We start with (1) and suppose that ϕ ′ satisfies (aInc) γ .Fix 0 < t < s < ∞ and set a := s t > 1.Then (aInc) γ of ϕ ′ implies that ϕ(x, t) which means ϕ satisfies (aInc) γ+1 .In the same way we can also prove that (aDec) γ of ϕ ′ implies (aDec) γ+1 of ϕ.The claims regarding (Inc) and (Dec) follow when L = 1.We next prove (2).Since ϕ ′ is non-decreasing, it follows that By the (aDec) γ condition of ϕ ′ , we have Then, we prove (3).By ( 2) and (A0) of ϕ ′ it follows that ϕ(•, 1) Finally, we prove (4).Since ϕ is convex, ϕ(x, s) ϕ(x, t) + k(s − t), where k := ϕ ′ (x, t) is the slope.Then from the definition of the conjugate function we have We end this subsection with some properties for C 1 -regular Φ-functions.Note that Proposition 3.8(2) below is proved for C 2 -functions in [33, Lemma 3] -here we provide a more elementary proof which is based on a reduction to the same claim for the function t p , that is While versions of this claim are commonly known, we have not found this precise formulation in the literature.Rather than providing a proof of (3.7), we just invoke [33, Lemma 3], since t p is certainly a C 2 -function.

Preliminary regularity results
Assumptions for higher regularity.Here we introduce the new assumptions that are used to obtain C α -regularity for any α ∈ (0, 1) or C 1,α -regularity for some α ∈ (0, 1) of local minimizers of (1.2).We also restate the definition of (VA1) from the introduction, so that it can be more easily compared with its weaker variant, (wVA1).
In the next definition, we have several conditions which are assumed to hold "for any small ball"; this means that it holds for all r < r 0 for some r 0 > 0. Definition 4.1.Let ϕ ∈ Φ w (Ω).We define some conditions related to regularity with respect to the x-variable.
Remark 4.3.Finally, we would like to explain why we adapt the methodology of calculus of variations, instead of one of partial differential equations, since indeed u is a minimizer of (1.2) if and only if it is a weak solution to see [53].In the comparison step in our approach, we take advantage of the minimizing property of u.If we would instead use the PDE approach, to the best of our understanding, the main assumption (VA1) would be replaced by the assumption . Compared with (VA1), ϕ is replaced by ϕ ′ in the inequality.Since small values are not covered in this assumption or (VA1), these two assumptions are not comparable, i.e. one may hold but not the other, in either direction.However, if ϕ satisfies the basic assumption in Theorem 1.1 (this is always assumed in our main theorems), we show that (wVA1) is implied by this assumption: for any ε > 0, any small B r ⋐ Ω, any t > 0 satisfying ϕ Thus (wVA1) holds with function cω(r) p/q .Furthermore, we could also consider a (wVA1)-type assumption with ϕ ′ instead of ϕ, but the same argument shows that this also implies (wVA1).
We note that such difference between regularity assumptions for the minimizer and the PDE problem does not appear in types I-III.This also shows that regularity theory for general ϕ(x, t) cannot be understood easily by just mixing the ones for types I-III.
Higher integrability and reverse Hölder type inequality.We prove higher integrability of minimizers of (1.2) and, as a corollary, a reverse Hölder type inequality.In this subsection we assume (A1).
The next lemma contains reverse Hölder type estimates for Du.
We move on the the second claim.The first inequality directly follows from Hölder's inequality, hence we prove the second inequality.Taking t = 1 q in (4.8), we see that We notice that the map t → [ϕ + B 2r (t)] 1 q satisfies (aDec) 1 , since ϕ + B 2r satisfies (aDec) q .Therefore, by Jensen's inequality with Proposition 3.5(2), we have (4.9) for some c = c(c 1 , q, L) 1.In addition, since 1, (A0) gives an upper bound of c for the right-hand side of (4.9).
Regularity results for the autonomous case.In this subsection, we consider ϕ ∈ Φ c ∩ C 1 ([0, ∞)) ∩ C 2 ((0, ∞)) with ϕ ′ satisfying (Inc) p−1 and (Dec) q−1 for some 1 < p q. Fix v 0 ∈ W 1,ϕ (B r ) and let v ∈ v 0 + W 1,ϕ 0 (B r ) be a solution of the minimization problem or equivalently a weak solution to We start with the C 1,α -regularity in the autonomous case, with appropriate estimates.
The previous lemma is expected from [61].In particular, we refer to [7] for the case p 2. However, we cannot find any result treating the case p < 2 with the above estimates in the literature.Hence, we give a proof of the above lemma in Appendix A. We also note that (Inc) p−1 and (Dec) q−1 of ϕ ′ are equivalent to tϕ ′′ (t) ≈ ϕ ′ (t) by Remark 3.3, since we assume ϕ ∈ C 2 ((0, ∞)).
We next state Calderón-Zygmund type estimates in B r with non-zero boundary data.
Construction of a regularized Orlicz function.We construct a regularized function φ ∈ C 1 ([0, ∞)) ∩ C 2 ((0, ∞)) with t φ′′ (t) ≈ φ′ (t), which is independent of the x variable and sufficiently close to ϕ(x 0 , t) in a suitable range of t.This procedure is quite delicate since we want improved differentiability and, moreover, want to find φ satisfying in particular the assumptions of Proposition 5.12, below.The challenge lies in ensuring that ϕ(x, φ−1 (t)) satisfies (aInc) 1 and (aDec) γ with some γ > 1 for small and large values of t, as we only have the comparison property when t is in some range [t 1 , t 2 ].We approach this problem by requiring p-growth for small and large values of t.This is counter-intuitive, because it means that the resulting function is neither a lower nor an upper bound of the original function, in contrast to estimates used in previous articles.
For the functions defined above, we have the following properties.
Here, the constants c > 0 depend only on n, p, q and L.
Comparison estimates.Let φ : [0, ∞) → [0, ∞) be the function constructed in the previous subsection.We then consider the minimizer v ∈ W 1, φ(B r ) of (5.13) where u ∈ W 1,ϕ loc (Ω) is a minimizer of (1.2), and derive a comparison estimate between the gradients of u and v.We note from Proposition 5.10(3) that u ∈ W 1, φ(B r ), so it is an appropriate boundary-value function and thus there exists a unique minimizer of (5.13).The minimizer v is also a weak solution to Before stating the main comparison result, we observe the following reverse Hölder type estimate for Du and Calderón-Zygmund type estimate for the problem (5.13).
in the last step we used that ffl Br [ϕ(x, |Du|) − ϕ(x, |Dv|)] dx 0 since u is a minimizer of (1.2).We shall estimate I 2 .We split B r into three regions E 1 , E 2 and E 3 defined by In the set E 1 , (Dec) q and (A0) of ϕ imply that |Dv| ω(2r) In the set E 3 , Proposition 5.10(3) and the fact that 1 Integrating this inequality over E 3 and using (6.1), we find that  Recall that t 1 and t 2 are defined in (5.5).In the set E 2 , we observe that Hence it follows from (wVA1) and Proposition 5. Therefore, applying (5.17We have shown that |Du| dx + 1 .
The estimate for I 1 is analogous, with sets E i defined with Du instead of Dv.
The following corollary is the key to the regularity results in the next section.Indeed, once we have the estimate from the corollary, the main results follow using standard methods.for some γ 1 = γ 1 (n, p, q, L) ∈ (0, 1) and c = c(n, p, q, L) 1.
Suppose next that ϕ satisfies (VA1) with function ω.Then, for r Rȃdulescu and colleagues [22,76,84] have considered a functional with model case ϕ(x, t) = t p(x) + t q(x) , which they call "double phase" (it is different from the double phase functional of Zhikov, considered below).To the best of our knowledge, this is the first regularity result this functional.Corollary 8.3 (Rȃdulescu's double phase).Let p, q : Ω → [p 1 , p 2 ] for some 1 < p 1 p 2 and ϕ(x, t) = t p(x) + t q(x) .Then ϕ satisfies (VA1) if there exist ω m and ω M with This result can be proved with the same methods as Corollary 8.2; the details are left to the interested reader.Note that the regularity required of the minimum is lower than the regularity required of the maximum.This is due to the fact that we only require the inequality of (VA1) in the range [ω(r), 1] where the minimum determines ϕ, whereas the maximum is used in the range [1, |B r |].
We now consider double phase problems in the sense of Zhikov and Mingione.
We also notice that the weak solution w to (B.8) has value zero on ∂Ω 5 ∩ B 5 .We next compare (B.8), which assumes zero boundary values on ∂Ω 5 ∩ B 5 , with an equation defined in B + 2 with zero boundary values on B 2 ∩ {x n = 0}.A similar result can be found in [14,Lemma 3.6].The proof of that lemma employs a compactness argument.Here we give a more direct approach which clearly shows the dependence on δ.
Lemma B.11.Let η = η(x n ) ∈ C ∞ (R) with η = 0 if x n 0, η = 1 if x n δ and |η ′ | 2 δ .For any ε > 0 there exists a small δ > 0 depending on n, p, q and ε, such that, under the assumptions of the above lemma, if w 0 is the weak solution to Here constants c depend on n, p and q, but are independent of ε.
), if and only if a ∈ C ωa .Proof.For any B r ⊂ Ω, Corollary 8.2 (Variable exponent case).Let p : Ω → [p 1 , p 2 ] for some 1 < p 1 p 2 .Define ϕ(x, t) := t p(x).Then ϕ satisfies (VA1) if and only if there exists ω p with Let us derive an equivalent form of the inequality in condition (VA1).We may consider the range [|B r |, |B r | −1 ] in the condition, since it turns out that this choice of lower bound entails no additional restrictions in the variable exponent case.When t 1, we have When t 1, the exponents p + and p − are interchanged.Since we consider the range t r = 0 and p ∈ C ωp .Moreover, ϕ satisfies (VA1) with ω(r) r β for some β > 0 if and only ifω (r) r β for some β > 0. Proof.Fix B r ⊂ Ω with |B r |1 and set p ± = p ± Br .Then we have ϕ − Br (t) = t p − and ϕ + Br (t) = t p + for t 1 as well as ϕ − Br (t) = t p − and ϕ + Br (t) = t p + for t < 1. + −p − − 1)t p − = (t p + −p − − 1)ϕ − Br (t).