Gradient estimates for a class of higher-order elliptic equations of p -growth over a nonsmooth domain

: This article is devoted to a global Calderón-Zygmund estimate in the framework of Lorentz spaces for the m -order gradients of weak solution to a higher-order elliptic equation with p -growth. We prove the main result based on a proper power decay estimation of the upper-level set by the principle of layer cake representation for the L γ q , -estimate of D u m , while the coe ﬃ cient satis ﬁ es a small BMO semi-norm and the boundary of underlying domain is ﬂ at in the sense of Reifenberg. In particular, a tricky ingredient is to establish the normal component of higher derivatives controlled by the horizontal component of higher derivatives of solutions in the neighborhood at any boundary point, which is achieved by comparing the solution under consideration with that for some reference problems.

Ω d for ≥ d 3 be a given bounded domain with its nonsmooth boundary ∂Ω determined later.We consider the following zero Dirichlet problem of a higher-order elliptic equation of p-growth for < < ∞ p 1 : where C is a constant depending only on λ p , Λ, , and Ω, see [28, Part II, Page 94].The main aim of this article is to prove a global regularity in the framework of Lorentz spaces for the m-order gradients of weak solution to problem (1.1) under weaker regular assumptions imposed on the given datum.In other words, we are to impose some minimal regular assumptions on the coefficient ( ) A x and on the rough geometric boundary ∂Ω to ensure the following nonlinear Calderón-Zygmund estimate of the Dirichlet problem (1.1): where the Lorentz spaces ( ) , and > ε 0 as a small constant.In parti- cular, for < = < + , we see that equation (1.5) leads to the L γ -estimate for the Dirichlet problem (1.1), which was just done by Habermann for the elliptic equations of ( ) p x -growth in [28] while ( ) p x is a constant.Especially, it is necessary for the range of < < + to be confined even for a special coefficient ( ) ≡ A x Id and ( ) x f with a higher integrability.It is a well-known fact that the uniform ellipticity and boundedness (1.2) is not enough to ensure the validity of the Calderón-Zygmund type estimate (1.5) if one does not impose more regularity assumptions on ( ) A x and Ω.A counterexample introduced by Meyers in [39] shows that the gradients for the solutions of linear elliptic equations involved in highly oscillatory coefficients cannot be expected to have higher integrability, irrespective of the nice regularity of the data ( ) x f .Another counterexample given by Jerison and Kenig's work [33, Theorem A] also demonstrates that the Poisson equation over a Lipschitz domain may not be solvable in the W p 1, -spaces for large p, unless the boundary is sufficiently smooth.Therefore, to ensure a nonlinear Calderón-Zygmund estimate of (1.1), it is necessary to find minimal regular assumptions for the coefficients ( ) A x and the boundary of underlying domain Ω.For this, let us recall some notations and basic facts.For any fixed point ∈ Assumption 1.1.We say that ( ( ) ) there exists a coordinate system depending on x 0 and r, whose variables are still denoted by x, such that in the new coordinate system, x 0 is the origin and for some ∈ ∂ z Ω 0 , there exists a coordinate system depending on x 0 and r, whose variables are still denoted by x, such that in the new coordinate system, z 0 is the origin, to the ball ( ) B z r 3 0 , and the parameters R 0 and > δ 0 will be specified later.
The domain Ω is called to be ( ) δ R , 0 -Reifenberg flat if equation (1.6) holds in the new coordinate system, see [44].This ensures that Ω locally satisfies the so-called A-type one, which means that there exist > R 0 0 and a small constant > δ 0 such that for any ∈ ∂ x Ω 0 .The estimate (1.7) actually implies that a higher integrability of the m-order gradients of the solution holds for the Dirichlet problem (1.1) in a neighborhood of every boundary point.Moreover, we also note that the extension theorem of Sobolev embedding inequalities is available over the ( ) δ R , 0 -Reifenberg flat domains, see [11].To present our main result precisely, we first introduce the definition of Lorentz spaces.with ≤ < ∞ γ 1 and < < ∞ q 0 is the set of all measurable functions → g U : such that .
For = ∞ q , the space ( ) is set to be the usual Marcinkiewicz space with quasi-norm We observe that if ≤ = < ∞ γ q 1 , then the Lorentz space ( ) L U q q , is nothing but the classical Lebesgue space ( ) L U q , which is equivalently endowed with the norm by Let us now state our research motivation under consideration.It is an important observation that for the linear case of = p 2, the problem (1.1) leads to the following higher-order linear elliptic form: Some recent studies of problem (1.8) are recalled in order.Byun and Ryu [12] established a global Orlicz estimate of higher-order elliptic and parabolic equations on nonsmooth domains, and they got that 2 by a geometric method provided that the principle coefficients satisfy small BMO and the boundary of underlying domain is a Reifenberg flatness.Furthermore, Dong and Kim [17][18][19] Dong and Gallarati [20] proved the L p -solvability for various complex boundary conditions of higher-order elliptic and parabolic systems by a different argument based on an estimate of the Fefferman-Stein sharp function theorem under main assumptions on the coefficients with small partially BMO quasinorm, and the boundary of underlying domain being Reifenberg flat.Wang and Yao [51] verified the L p -regularity for a higher-order elliptic and parabolic equation in the whole space by way of the so-called large-M-inequality principle under assumption that the leading coefficients have small BMO quasi-norm.On the other hand, for a special case of = m 1, this leads to a degenerate elliptic problem of p-growth as follows: (1.9) As we know, Iwaniec [32] first proved a local nonlinear Calderón-Zygmund estimate for the equation of p-Laplacian, and he got that loc for all ≥ q p.It was extended by DiBendetto and Manfredi in [21] to the setting of the elliptic systems of p-Laplacian.For the case of discontinuous coefficients, Caffarelli and Peral [15] used the modified Vitali covering originated from Safonov's "crawling of ink spots" argument [45] to attain the W p loc 1, -estimates for a class of general elliptic equations of p-growth with a small deviation of "variable coefficients" from "constant coefficients."Kinnunen and Zhou proved the interior and boundary L q estimates of problem (1.9) in [34,35] based on the Perturbation approach, respectively, where the coefficients ( ) A x are of VMO class and the domain is C α 1, for any < ≤ α 0 1.Later, Byun et al. [11] reduced the C α 1, boundary regularity to the so-called Reifenberg flat boundary by the geometric method, and they also showed a global W q 1, estimate of the gradients to the above problem (1.9).Recently, we also observed that a global nonlinear Calderón-Zygmund estimate is generalized to the weighted Lorentz spaces by Mengesha and Phuc [40] under the same regular assumptions on the coefficients and the Reifenberg domain.Here, we would like to mention that Acerbi and Mingione [1] first made use of the so-called large-M-inequality technique to prove a local W q 1, -estimate of the weak solution for parabolic systems of p-growth with small BMO coefficients in all ( ) t x , variables.By way of Acerbi and Mingione's idea, Baroni [6,7] established local Lorentz estimates of the gradients to the weak solution of parabolic equations and parabolic obstacle problems of p-growths, respec- tively, where the leading coefficients are allowed to be small BMO seminorms in all the spatial variables and measurable in time variable.In addition, it would be remarked that Eleuteri and Habermann [23] got the Calderón-Zygmund-type estimates for a class of obstacle problems with ( ) p x -growth.Tian and Zheng [50] also proved a global weighted Lorentz estimate for a variable power of the gradients to the weak solution of nonlinear parabolic equations with BMO nonlinearity over the quasiconvex domain by using the so-called large-M-inequality principle.For more study involving the existence and regularity of solutions to p-Laplacian elliptic and parabolic equations, we also refer to the recent articles [31,36,43,47,49].
Note that the research on regularity results of higher-order elliptic and parabolic problems is substantially rare in comparison to the second-order cases.Giaquinta and Modica in [25] showed some regularity results for higher-order nonlinear elliptic systems.Campanato and Cannarsa in [13] established a local ( ) + m 1 -order differentiability of the solutions of nonlinear elliptic systems with quadratic growth.The above result had been expanded by Campanato in [14] to nonlinear elliptic systems with the standard p-growths.Maz'ya [38] dealt with the Wiener test for the Dirichlet problem of strongly elliptic differential operators of an arbitrary even order m 2 with constant real coefficients, which is concerned with an equivalent relation between the capacitary Wiener's-type criterion and the regularity of a boundary point.The partial Hölder regularity of the gradients of the solutions for such a higher-order homogeneous nonlinear elliptic system in divergence form had been completed by Duzaar et al. [22], while Habermann [30] also showed partial regularity for minima of higher-order functionals with ( ) p x -growth.Recently, Miyazaki [42] obtained a global ( ) L Ω p estimate when Ω is a C 1 -smooth domain and the coefficients satisfy uniform continuity.Bögelein [10] gave a higher integrability of the gradients of the solutions by way of proving a reverse Hölder inequality of the m-order gradients for quasilinear parabolic systems of higher-order.Habermann [28] showed a nonlinear Calderón-Zygmund estimate of D u m in ( ) L Ω q loc for any < < − q 1 d d 2 for a higher-order nonlinear elliptic system of ( ) p x -growth, which the nonlinearities are assumed to be continuous in all spatial variables.Furthermore, Habermann [29] improved the integrable exponent of m-order gradients to with a positive constant > ε 0 by proving the validity of the reverse Hölder inequality of the m-order gradients under the same regularity assumption on the nonlinearities.Choi and Kim [16] proved the weighted L p q , -estimates for divergence-type higher-order elliptic and parabolic systems with irregular coefficients on Reifenberg flat domains, where the coefficients do not have any regularity assumptions in the time-variable for the parabolic case, and the leading coefficients are permitted to have small mean oscillations in the spatial-variables for the elliptic case.On the other hand, the Lorentz regularity of gradients of solutions for elliptic and parabolic equations have been substantially studied since Talenti's seminal work [48], for more references see [3][4][5][6]41,50].It is a well-known fact that the Lorentz space is a refined version of the Lebesgue spaces.Therefore, the regularity under our consideration in the scale of Lorentz spaces for the Dirichlet problem (1.1) is inspired by the aforementioned works, which is supposed to be under weaker regular assumptions on the coefficient ( ) A x and the nonsmooth domain Ω.
We are now in a position to summarize the main result of this article.
be the weak solution of the Dirichlet problem (1.1) with the ellipticity and boundedness (1.2).Then, there exists a small constant with the following estimate: where the constant , , , , Λ, , , Ω 0 0 while < ∞ q ; the constant is not dependent on q while = ∞ q .
It would be remarked that the restriction for < + does not appear in the order-two case = m 1 because the solutions of second-order homogeneous elliptic problems with constant coefficients have local Lipschitz regularity, which implies that | | ( ) in this setting.We refer the readers to [6,11,34] for more details.Indeed, a key point for such a higher-order problem of p-growth in comparison to the second-order case is the absence of a priori estimate in the ∞ W m, -spaces for the higher-order elliptic equations with constant coefficients.To this end, our scheme is arranged as follows.We are the first to control the normal component of higher derivatives by the horizontal component of higher derivatives of solutions on the boundary point, which is based on making comparisons among the solutions under consideration with some reference problems.Thanks to a validity of the ( ) + m 1 -order derivatives of the solution for such a higher-order elliptic equation with constant coefficients based on the difference quotient argument, then we attain the + W m p 1, ˜-estimate of the solution of problem (1.1) with a suitable exponent p ˜depending only on d and > p 1.This estimate combined with the Sobolev embed- ding theorem yields a higher integrability of the m-order gradients of the solution, see equation (3.14) in Section 3. Furthermore, an integrability of the m-order gradients is improved by the reverse Hölder inequality based on the definition of the weak solution of the Dirichlet problem (1.1).Here, we would also like to point out that it is infeasible to show an estimate of the ( ) + m 2 -order gradients of the solution by the same way as the difference quotient technique even for the higher-order equation of p-Laplacian.Finally, we are ascribed to study the upper-level set for sufficiently large κ by way of the modified Vitali covering and the large-M-inequality technique.It is essential to show a proper power decay estimate of such upper-level set so that we ensure the validity of the principle of layer cake representation for the L γ q , -estimate of D u m .
A brief outline of this article is given as follows: in Section 2, we introduce some preliminary lemmas; Section 3 is mainly devoted to local interior and boundary estimates of the m-order gradients of the weak solution to the Dirichlet problems (1.1), respectively; finally, the proof of main Theorem 1.3 is given in Section 4.
varying from line to line.We begin this section with a few standard embedding relations for the Lorentz spaces.It is worth noting that the quantity ‖ ‖ ( ) is only a quasi-norm due to the lack of subadditivity.However, the mapping is still a lower semi-continuous, see [8,Section 3] or [40,Proposition 3.9].
Lemma 2.1.For any given ⊂ U d with a finite measure, we have the following conclusions: , then it holds a continuous embedding with , then there holds a continuous embedding with while < ∞ q 2 ; the constant is not dependent on q 2 while = ∞ q 2 .(iii) There holds a continuous embedding for any > γ p and all < ≤ ∞ q 0 .
Note that the boundary of Reifenberg flat domain is locally the graph of a Lipschitz continuous function with a small Lipschitz constant, which ensures the Sobolev embedding theorem and interpolation due to the validity of the Sobolev extension theorem.Let for > r 0. The following interpolation inequality involved in the higher gradients on an annulus of the form with < < < ∞ r R 0 will be frequently used in our main proof, see [10,Lemma 3] , there exists a positive constant = In the next context, we will intensively use the difference quotients to approximate the weak derivative of the solution.Therefore, it is necessary to recall the following relationship between the difference quotient and the weak derivative, see [24, Theorem 3 in Chapter 5].For any locally summable function ( ) g x in the bounded domain U and ⊂ ⊂ V U , we first introduce the notation of the s-th difference quotient with size h denoted by , there holds Next, it is useful to recall the following two basic inequalities regarding general elliptic operator of p-growth, see [52,Lemma 2] or [34,Eq. (3.16)].

Lemma 2.4. Let the coefficients ( )
A x satisfy the uniform elliptic condition (1.2).Then, there exists a positive constant for any ∈ ξ md and ∈ η md .In particular, if ≥ p 2, then Lemma 2.5.Let the coefficients ( ) A x and ( ) B x satisfy the uniform ellipticity and boundedness (1.2), then for any , and there hold where It is necessary for the proof of main theorem to recall the following two useful technical lemmas.The first inequality is actually a variant of the classical Hardy's inequality, whose proof can be found in [6,Lemma 3.4].
Lemma 2.6.Let g be a nonnegative measurable function defined on [ ) +∞ 0, satisfying then for any ≥ α 1 and > τ 0, we have The second lemma is a useful inequality to our main proof, which really encodes the inclusion property with respect to the second exponent of the Lorentz spaces, see [6,Lemma 3.5].
, and , for any ( ] ∈ ε 0, 1 and ≥ β 0, we have . If = ∞ α 2 , then we have with the constant Finally, the following iterating lemma is mainly used in our main proof, which is referred to [26].
Gradient estimates for a class of higher-order elliptic equations of p-growth  7 Lemma 2.8.Let ϕ be a bounded nonnegative function on ( ) r R , .Suppose that for any r r , there holds where the constants ≥ A B , 0, ≤ < ε 0 1, and > θ 0.Then, there exists a positive constant

Local interior and boundary estimates
This section is devoted to some auxiliary estimates by comparing the solution with some reference problems.First, we prove a boundary a priori estimate, which is a crucial ingredient in the proof for a higher integrability of problem (1.1).Note that a similar result can be found in [10, Lemma 6] for problem (1.1) where Proof.Without loss of generality, we let = ∈ x 0 Ω 0 .For any r and R: , we take ( ) as a cutting-off function with Now, we define ≔ η ηχ ¯Ω, where χ Ω is the characteristic function on Ω. Choosing the test function = φ uη ¯in equation (1.3), we obtain (3.4) Now we are to estimate I I I , , 1 2 , and I 3 , respectively.Estimate of I: using the uniform ellipticity equations (1.2) and (3.2), we obtain Estimates of I I , 1 2 , and I 3 : employing uniform boundedness (1.2) and Young's inequality, we obtain that for any > ε 0 1 and some constant for any > ε 0 2 and some constant for some constant Putting the estimates I , I I , 1 2 , and I 3 together into equation (3.4), we obtain where , , Λ is a positive constant.To estimate the term LOT m , we note that . Therefore, we apply the interpolation Lemma 2.2 on Ω \Ω r r for any > ε 0

3
, where , , is a positive constant.Therefore, this in combination with equation (3.5) yields for some > ν p, then for every such that for all < ≤ σ σ 0 0 , we have where , we recall the following Sobolev imbedding inequality , , , , 0 is a positive constant.Here, we would like to point out that the above Sobolev imbedding inequality also holds in the neighborhood of a Reifenberg boundary point.Now, putting it into equation (3.7) deduces Then, we attain the desired inequality (3.6) by using the Gehring-Giaquinta-Modica lemma, see Proposition 1.1 in [26, p. 122].
and consider the localized problem under the assumption of with u as the solution of equation (3.9).We also consider 4 as a weak solution of the following limiting problem: (3.12) with h as the weak solution of equation (3.11) and A ¯4 being an average of ( ) 12) is such a problem with constant coefficients satisfying the uniform ellipticity (1.2).By the standard L p -estimates for problems (3.11) and (3.12), we conclude that for some constant , , , , Λ 0, there hold Here, let us recall an interior higher-order differential of D v m for higher-order p-Laplacian equations.
is the weak solution of problem (3.12), then there exists with the estimate where In the following lemmas, we set as the Sobolev conjugate of an arbitrary > p 1.By the Gagliardo-Nirenberg-Sobolev inequality in [24, Theorem 6] and a higher integrability of D v m shown in Lemma 3.3, it yields the following conclusion.
is the weak solution of problem (3.12), then there exists moreover, there holds the following estimate: We are next to give the following perturbation estimate by comparing the solutions between the original problem and the limiting problem (3.12).More precisely, we have  In the following, we part the estimates of equation (3.16)   (3.17) , , , , Λ is a positive constant.If ≥ p 2, using a similar argument, we readily conclude for any > δ 0 and > ε 0 i with = i 1, 2, 3. On the other hand, we take 4 as a test function, and by making a difference to equations (3.11) and (3.12), we have  , and > σ 0 mentioned in Lemma 3.2, where ( ) = C C p is a positive constant.Therefore, using the boundedness of ( ) A x in equation (1.2), the local BMO seminorm with small > δ 0 for equation (3.8), the standard L p estimate (3.13), as well as a higher integrability of the gradients for equation (3.11) shown in Lemma 3.2, we have

A D h D h A D h A D v D v A D v D h D v x A D h D h A D h A x D h D h A x D h D h D v x
For the estimate of LHS of equation (3.20), we use a similar argument as in the proofs of (3.17) and (3.18) to obtain for any > ε 0

5
, where , , , , Λ is a positive constant.Let us put the above estimates into equation (3.20), and using the standard L p estimate (3.13) and assump- tion (3.10), we obtain  In what follows, we are devoted to the relevant estimates in the neighborhood at a point on the flat boundary.Here, we adapt an argument from Skrypnik's monograph [46] to estimate the normal derivatives controlled by the horizontal derivatives, which is realized by the difference quotient.To this end, we choose a suitable test function consisting of the difference quotient involved the higher derivatives of the solution u to show the relevant estimates for the limiting problem in a half ball.Let for any > r 0. We consider the following limiting problem Gradient estimates for a class of higher-order elliptic equations of p-growth  13 with the following estimate: where Proof.We divide the proof into two aspects: one is first to estimate the horizontal derivatives , with = s d 2,…, .Then, we are to estimate the normal derivatives in accordance with the horizontal derivatives.Estimate of the horizontal direction: we select a smooth cut-off function Let us first confine a small h in the range We use equation (3.24) as a test function, together with the integration by part for the difference quotients to obtain We set and then obtain , and (3.29) Estimate of K 1 : by using equation (3.28), we obtain Estimate of K 2 : we distinguish it into the following two cases:  for any > ε 0, where , , , Λ is a positive constant.Subsequently, we put the above two estimates K K , 1 2 into equation (3.29) and choose = > ε 0  where on Γ 4 , and by using the standard estimates for the difference quotients as in Lemma 2.3 and the Poincaré's inequality of ( ) − k 1 -order, we have Gradient estimates for a class of higher-order elliptic equations of p-growth  15 , , is a positive constant.A combination of the above three estimates yields , we obtain that To estimate the second term in the last inequality above, we use an estimate of the difference quotient shown in Lemma 2.3, Poincaré's inequality, and Young's inequality with two exponents for any > ε 0, which implies Putting the two estimates of K 1 and K 2 into equation (3.29) and using the iterating lemma with > ε 0 small enough, we also give the same estimate as shown equation (3.31) for the case of < ≤ p 1 2. Finally, using Lemma 2.5, we see that .
This means that the sequence is uniformly bounded in ( ) . By Lemma 2.3, we obtain (3.33) Estimate of the normal direction: it only suffices to prove (3.34) For this, we first need a zero-extension of v from + B 4 to B 4 , i.e., Recalling the cut-off function η defined by equation (3.23), we conclude as follows: where + χ d is the characteristic function of the half space + d , and + ± ϒ 1 is a convolution operator with constant symbol in d , defined by the equality Using the integration by part for the difference quotients and Leibniz's rule, it yields that Gradient estimates for a class of higher-order elliptic equations of p-growth  17 and then, we obtain Using the uniform ellipticity and boundedness (1.2), we obtain that for ( ) ∈ g L p d (cf.[27, Chapter 2, Page 104]), we take to conclude that which yields that for any ∈ + x d , there holds Putting it into equation (3.37) and again using Leibniz's rule, we write Estimate of J 1 : using equation (3.40), we have  , and the Poincaré's inequality of ( ) − k 1 -order, we obtain that Putting the above two estimates together, we conclude 2, then we see that for ≥ k 1, the function ( ) It implies that we can use equation (3.38) and the integration by parts for the difference quotients to obtain , we obtain that for any > ε 0, To estimate the last two terms for the above inequality, we use the standard estimate of difference quotient as in Lemma 2.3, the Poincaré's inequality of ( ) − k 1 -order, and Young's inequality with two exponents ≡ > p ¯1 p to obtain that Gradient estimates for a class of higher-order elliptic equations of p-growth  19 Putting the above estimate into the estimate of J 2 , we also give the estimate (3.43) for the case of < ≤ p 1 2. Let us now insert the estimates of J 1 and J 2 as shown in equations (3.42) and (3.43) into equation (3.41), then we use the iterating lemma with > ε 0 small enough to obtain It is easy to check that there exists a constant Thus, this estimate together with equation (3.44) yields We see that the sequence is uniformly bounded in ( ) has the same bound as above in ( ) with the estimate where Proof.With Lemma 3.6 in hand, it suffices to establish a reverse Hölder's inequality of (| | ) , which leads to the desired higher integrability result via the famous Gehring's lemma.
To this end, we are first to estimate the horizontal direction, and test equation (3.22) by the following test function: with a suitable cutting-off function η and the polynomial ( ) P x of degree m, which will be specified later.Here, we confine , it follows from the estimate (3.30) and Young's inequality that To estimate the second term of the right-hand side above, we use Lemma 2.3 and Poincaré's inequality of (3.46) Now we choose the highest-order coefficient of polynomial ( ) Here, we note that the existence of such polynomial P is a well-known fact, see [29].Therefore, putting the above four estimates together, we obtain (3.48) , by the estimates of K 1 and K 2 with equation (3.29) and the Young inequality, we obtain To estimate the second term of the right-hand side above, we also use a similar argument as equations (3.46) and (3.47)

:
Gradient estimates for a class of higher-order elliptic equations of p-growth  21 Gehring-Giaquinta-Modica lemma (cf.[26, Proposition 1.1]) ensures a higher integrability of . This combining with Lemma 3.6 leads to the desired estimate, which completes the proof.□ It is easy to check that we make use of the Gagliardo-Nirenberg-Sobolev inequality (cf.[24,Theorem 6]) and Lemma 3.7 to obtain the following conclusion.
is the weak solution of problem (3.22), then there exists such that for any < ≤ σ σ 0 2 , we have with the following estimate: In what follows, we note that the boundary of Reifenberg domain Ω is rough, which is not flattened by a smooth mapping.So, it is difficult to use the odd and even extension techniques to handle the boundary estimates.We consider a localized Dirichlet problem under the assumption of 5 be the weak solution of the following homogeneous equation: (3.56) Recalling that ( ) is the weak solution of problem (3.22), then the following approximation lemma assures that the solutions between w and v are close enough in the L p -sense.Lemma 3.9.For any > ε 0, there exists a small > δ 0 such that if is the weak solution of equation (3.55) with boundary assumption (3.50), then there exists a corresponding weak solution ( ) Proof.If = m 1, then the proof of lemma can be found in [11,Lemma 4].Hence, we focus on the case of ≥ m 2. We argue it by contradiction.If not, then there would exist > ε 0 is the weak solution of for any weak solution ( )

B m p 4
We note that the Reifenberg domain is locally A-type one, which means that there exists a positive constant , we make use of the generalized Poincaré's inequality of m-order (cf. [37,Theorem 4]) and the assumption (3.58) to obtain ‖ ‖ ( ) . Therefore, this estimate implies that there exists a subsequence, which still denotes by { } w k , and Gradient estimates for a class of higher-order elliptic equations of p-growth  23 Recalling that ( ) A x satisfies the uniform ellipticity and boundedness (1.2), we obtain that A ˆk is bounded in Ω.Then, there exists a subsequence, which we denote by { } A ˆk , such that (3.63)Indeed, by the trace theorem a.e. on Ω 4 .We use the boundary conditions (3.59) and (3.57) to obtain that on Γ 4 immediately.On the other hand, we note that A k satisfies the uniform boundedness as in equation ( 1 . This leads to that there exists a vector-valued function and a subsequence of , which we still denote by the same form, such that Therefore, to attain our claim (3.63), it suffices only to show that a .e i n .
and every ( ) which implies that Therefore, this combined with equation (3.66) yields Putting the above equality into equation (3.67), we have Therefore, it yields Taking → τ 0 and using the Lebesgue-dominated convergence theorem, we obtain for every ( ) and every ( ) . The above inequality also holds for Therefore, we have for every ( ) . Therefore, we obtain the desired formula (3.65) to prove the claim (3.63).
Gradient estimates for a class of higher-order elliptic equations of p-growth  25 Finally, it follows from the weak lower semi-continuity of the semi-norm for D w m k that (3.69) The claim (3.63) combined with equation (3.69) leads to a contradiction to equation (3.60) by taking and k large enough.This completes the proof.□ To make a comparison of the solution u with the limiting problem (3.22), we need a zero-extension of v from + B 4 to B 4 .For this, we let Then, for any > ε 0, there exists a small > δ 0 such that the assumptions (3.50), (3.51), and (3.53) hold, then there exists a corresponding weak solution ( ) where v ¯is the extension of v defined by (3.70).
Proof.First, we select a smooth cutting-off function χ defined on such that Recall that is the weak solution of limiting problem (3.55).Let = w χw ˆ, then w ˆtogether with all its derivative vanish on , and w ˆsatisfies (3.72) In what follows, we let ( )   We are now in a position to estimate the terms LHS and RHS of equation (3.74), respectively.Using Leibniz's rule, it yields To estimate LHS: we first note that Using a similar argument to the proofs of equations (3.17) and (3.18), we obtain for any > ε 0 1 . Hence, it follows from Lemma 2.5 and Young's inequality that .By Leibniz's rule and the Hardy-type inequality (cf.[18, Lemma 7.9]), we obtain Putting the above two estimates together, we obtain that In this section, we are to prove the global Lorentz estimate of the m-order gradients to the Dirichlet problem (1.1) of higher-order elliptic equations with p-growth.For < < ∞ p 1 , we assume that ( ( ) ) A x , Ω satisfies , 0 -vanishing for some > R 0 0 shown as Assumption 1.1, where < ≤ δ 0 1 8 is temporarily confined which is determined later.In the following context, we set where > η 1 will be specified later.We also take sufficiently large κ satisfying and consider the upper-level set Fix any point ( ) ∈ y E κ , we consider a continuous function ( ) y Φ r in r defined by  Consequently, we conclude that for almost every ( ) ∈ y E κ , there exists a maximum ( ) r r y y Then, we infer the following lemma from the well-known Vitali covering lemma due to the property of r y .
Lemma 4.1.Let κ satisfy equation (4.2).Then, there exists a disjoint family { We are now in a position to show a suitable decay estimate to each member ( ) Under the same hypotheses as Lemma 4.1, we have Proof.By Lemma 4.1, we immediately conclude one of the following alternatives: For the first case, we split it into two parts and use the Hölder's inequality to obtain that Here, the constant < ≤ σ σ 0 0 is determined by Lemma 3.2.Therefore, using Lemma 3.2, we obtain Putting it into equation (4.6) and dividing by κ, we derive For the second case, by the classical measure theory, we estimate the upper level set as follows: and Let us make the scaling functions on Ω 35 again as follows: Then, u ˜i is the weak solution of  In similar way, we also check that the Dirichlet problem (4.23) still satisfies the hypotheses of Lemmas 3.8 and 3.10, which implies that there exist small > δ 0 and function v ˜i defined in + B 32 such that 1 is a position constant, and < ≤ σ σ 0 2 as in Lemma 3.8.Here, we made an extension of v ˜i by zero from + B 32 to B 32 and denote it by v ˜i.Scaling these functions back and denoting v i by the translated function of Furthermore, using the expression of domain relation (4.19), we obtain , which is independent of the index i.In the following, we write Gradient estimates for a class of higher-order elliptic equations of p-growth  33 where the parameter  For the interior setting: we first note that for some < ≤ σ σ 0 1 mentioned as in Lemma 3.4.Using the facts that | , and the interior estimates (4.16), we find that   q γ q γ q γ q γ q γ d σ d σ q γ q γ q γ q γ q γ d σ d σ   q γ q γ q γ γ q , Putting above two estimates I 21 and I 22 into equation (4.35), we obtain that ) Combining the estimates I 1 and I 2 with equation (4.33) yields ) Furthermore, we take > ϑ 1 sufficiently large and > ε 0 sufficiently small such that Therefore, it follows from equation (4.36) that    Gradient estimates for a class of higher-order elliptic equations of p-growth  37

1 Introduction
Throughout the article, we assume that ≥ m 1 is a positive integer,

.
β α denotes a standard inner product of ∈ η ξ , md in the following context.Here, the variable coefficient tensor ( ) A x satisfies the uniform ellipticity and boundedness, i.e., there exist two positive constants < classical Lebesgue spaces, we are concerned with the weak solution of the Dirichlet problem (1.1) in the classical Sobolev space ( ) By using a suitable Galerkin-approximation in the L p -norm, there exists a unique weak solution

8
B r is the d-dimensional Lebesgue measure of ( ) by a scale way.

Definition 1 . 2 .
Let U be an open subset of d .The Lorentz space ( ) , , Λ is a positive constant.Finally, to obtain the estimate (3.1), we only need to use the iterating Lemma 2.8 and take > ε 0 i sufficiently small for each = i 1, 2, 3. □ In what follows, we are to show a higher integrability of the D u m to problem (1.1).It merely suffices to prove a generalized reverse Hölder inequality in the neighborhood of boundary point by using the type-A condition of ( ) δ R , 0 -Reifenberg flat domain as in equation (1.7).Lemma 3.2.(Reverse Hölder inequality) Assume that

6 ( 3 . 10 )
We are now to give various interior comparison estimates by comparing the local original problem with a few of reference problems.Let ( ) weak solution of the following boundary problem:

13 )
Unlike the p-Laplacian equation for = m 1, one cannot obtain the ∞ L -bound for the derivative D v m to the higher-order limiting equation of p-growth.Therefore, it is necessary to recall the following interior estimates involved in the equations of p-growth with the constant coefficients as in [29, Lemma 4.1].

2 ,
, and | | > h 0 is a small constant.If < ≤ p 1 then using equation(3.26), the integration by parts for the difference quotients, the ellipticity (1.2) and Young's inequality with exponents ≡ > put the estimates (3.32) and (3.31) together to yield

2 1B 5 2 .
transform of the function ( ) g x .In such way, we employ the same argument as in the proof of Theorem 3.4 from [46, Page 197, line-9], we deduce that the function ( ) ω x vanishes in − d .By considering that the limit of h is small enough that there is still a gap between ( ) moves the boundary Γ 4 downward h-step.This leads to the fact that the support of φ is still in ∕ + Therefore, we can make use of equation(3.35)  as a test function in the limiting problem(3.22)to obtain that

1 on Γ 4 .
Hence, using the standard estimate of difference quotient as in Lemma 2.3, Young's inequality with two exponents ≡ Thanks to the ellipticity (1.2), the definition of cut-off function(3.23), and Young's inequality with exponents ≡

2 2
to make equation (3.34) true.Finally let us put equations (3.33) and (3.34) together, then we obtain the desired estimate as shown in Lemma 3.6, which completes the proof.□ With Lemma 3.6 in hand, we can achieve a higher integrability of D v m by the standard technique of the reverse Hölder inequality.weak solution of problem (3.22), then there exists the Sobolev-Poincaré inequality that

( 3 . 49 )
For the normal direction, we again use the estimate(3.34).Therefore, we put the horizontal estimate(3.49)    and the normal estimate(3.34)together to deduce the following reverse Hölder inequality of | | To consider the boundary setting, we set ∂

4
be the weak solution of a limiting problem L p -estimate for the Dirichlet problems (3.54) and (3.55), we have

i 3 , 4 . 1 1 .
Consequently, we put the estimates (3.75) and (3.76) with equation (3Gradient estimates for a class of higher-order elliptic equations of p-growth  27 Now, we extend v to zero in + B B \ 3 3 as presented in equation (3.70) and recall the fact that ( ) low-order term involving LOT m , we make use of the interpolation of Lemma 2.2 to obtain in equation (3.78) and (3.79), we note that − = χ Therefore, the weak solution w, as a function of y ˆ1, vanishes along with its derivatives up to (

3 σ 0 as in Lemma 3 . 2 . 2 1
This inequality in combination with the Hölder inequality and the reverse Hölder inequality for the Dirichlet problem (Inserting the estimates (3.79) and (3.80) into (3.78),we obtain …, 5 and > δ 0. Subsequently, we use the standard L p estimate (3.56) for the boundary setting and Lemma 3.9 with = ε ε 6 to obtain On the other hand, for the Dirichlet problems (3.52), (3.54), and (3.55), we use the same argument as the proof of Lemma 3 desired result (3.71) by taking enough small constants > ε 0 i , = i 1,…, 7, and > δ 0. It com- pletes the proof.□ 4 Proof of main theorem using equation (4.2), the measure density condition (1.7) near the Reifenberg boundary and the fact of > η 1, then we obtain that

32 ) 1 i
Inserting equations (4.31) and (4.32) into equation (4.30), we obtain are nonoverlapping, which leads to the required result.□ Now we are ready to give the proof of Theorem 1.3.Proof of Theorem 1.3.Case 1:

(γ 1 1 , 34 )
Recalling the definition of the upper-level set ( ) E κ as equation (4.3) and using the change of variables with > ϑ 1 and > T 1 as in equation (4.28), we obtainTo estimate I 1 : we first recall the definition of κ 1 in equation (4.2) and obtain that This combined with the standard L p estimate (1.4) and the Hölder inequality yields the proof of Lemma 4.2 and taking σ sufficiently small such that < = + < η σ by Theorem 2.1, we conclude that there exists a positive constant ( To estimate I 2 : using the power decay estimate of upper-level set in Lemma 4.3, we obtain that
For the estimate I 22 : we consider following two cases:If ≥ > q γ 1, then we use Hardy's inequality shown as Lemma 2.6 to obtain use the reverse-Hölder inequality shown as Lemma 2.Again using a simple change of variable and Hardy's inequality as Lemma 2.6, we obtain the definition of the upper-level set ( ) E κ in equation (4.3) and the change of variables with > ϑ 1 and > T 1 in equation (4.28), we obtain To estimate I 3 , by the definition of κ 1 in equation (4.2), we obtain that

1γ 1 1 . 39 )
This in combination with the standard L p estimate (1.4) and the Hölder inequality yields the proof of Lemma 4.2 and taking σ sufficiently small such that < = + < η σ Thanks to Theorem 2.1, there exists a positive constant( | | ) = C C d m p γ q λ δ R , ,, , , , Λ, , ,To estimate I 4 : using the power decay estimate of upper-level set as in 4.3, we deduce that For the estimate I 41 : a simple change of variable shows For the estimate I 42 : we use the relation < < Combining the estimates I 3 and I 4 with equation (4.38), we obtain and [9, Chapter 1].
in the interior setting.Here, we only focus our L p -estimate on the boundary neighborhood.
In short, combining the interior estimates (4.16) with the boundary estimates (4.27), we write {