Besov regularity for the elliptic p-harmonic equations in the non-quadratic case

Fengping Yao

doi:10.1515/anona-2024-0044

Article Open Access

Besov regularity for the elliptic p-harmonic equations in the non-quadratic case

Fengping Yao

Published/Copyright: November 8, 2024

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From the journal Advances in Nonlinear Analysis Volume 13 Issue 1

Abstract

In this article, we mainly establish the local extra fractional differentiability (Besov regularity) of weak solutions for the following divergence nonlinear elliptic equations of p -Laplacian type:

div A ( D u , x ) = div F ,

where A is a Carathéodory function with p − 1 growth for 1 < p < 2 . The standard example for the aforementioned equations is the classical elliptic p -Laplacian equation

div ( ∣ D u ∣ p − 2 D u ) = div F , for 1 < p < 2 .

Remarkably, the case 1 < p < 2 is very different from the case p ≥ 2 since the modulus of ellipticity in the elliptic p -Laplacian equation tends to infinity when ∣ D u ∣ → 0 .

Keywords: Besov; fractional; regularity; p-Laplacian; elliptic; weak solutions

MSC 2010: 35J60; 35J70

1 Introduction

In this article, we mainly establish the local extra fractional differentiability (Besov regularity) of weak solutions for the following divergence nonlinear elliptic equations of p -Laplacian type with 1 < p < 2 :

(1.1) div A ( D u , x ) = div F , in Ω ,

where Ω in R n for n ≥ 2 is an open domain and the vector-valued function F = ( f 1 , ⋅ ⋅ ⋅ , f n ) . Here, A ( ξ , x ) : R n × Ω → R n satisfies the following Carathéodory conditions:

(1.2) [ A ( ξ 1 , x ) − A ( ξ 2 , x ) ] ⋅ ( ξ 1 − ξ 2 ) ≥ C ( μ 2 + ∣ ξ 1 ∣ 2 + ∣ ξ 2 ∣ 2 ) p − 2 2 ∣ ξ 1 − ξ 2 ∣ 2 ,

(1.3) ∣ A ( ξ 1 , x ) − A ( ξ 2 , x ) ∣ ≤ C ( μ 2 + ∣ ξ 1 ∣ 2 + ∣ ξ 2 ∣ 2 ) p − 2 2 ∣ ξ 1 − ξ 2 ∣ ,

(1.4) ∣ A ( ξ , x ) ∣ ≤ C ( μ 2 + ∣ ξ ∣ 2 ) p − 1 2 ,

for any ξ , ξ 1 , ξ 2 ∈ R n , some μ ∈ [ 0 , 1 ] , and a.e. x ∈ Ω .

The elliptic nonlinear p -Laplacian equation

div ( ∣ D u ∣ p − 2 D u ) = div F

and the general nonlinear equations come from many important practical problems of natural science: non-Newtonian fluid theory, nonlinear elasticity and glaciology, turbulent flows of a gas in porous media, radiation of heat, and so on. Actually, the nonlinear partial differential equations (PDEs) can also be derived from some financial and economic problems, and simultaneously, the solutions of the nonlinear PDEs and their properties illustrate the features of these problems. Since the structure models in some real financial products and the option price can be reduced to some nonlinear PDE boundary problems, it is useful to adopt the existing theory and methods of PDEs as a fundamental approach to the study of the financial and economic theory [37]. So far, there has been a rapid scientific development in the theory of L p estimates [8,18,24,25,38,43], Hölder estimates [1,7,17,23,26,27,36,44], and Lipschitz regularity [6,11–13] for weak solutions of the elliptic p -Laplace equation (1.1) and the general case with different coefficient and domain assumptions.

Since no extra differentiability for solutions of problem (1.1) can be expected without a few more assumptions on the x -dependence of A , some experts try to find conditions on A under which fractional differentiability assumptions on F can transfer to D u with no loss in the order of differentiation. First of all, Baisón et al. [2] made an in-depth study of the local higher fractional differentiability of weak solutions to the following linear elliptic homogeneous and non-homogeneous equation (1.1) in divergence form:

div A ( D u , x ) = div F , in Ω ,

where A has linear growth in D u and satisfies the following Carathéodory conditions:

[ A ( ξ 1 , x ) − A ( ξ 2 , x ) ] ⋅ ( ξ 1 − ξ 2 ) ≥ C ∣ ξ 1 − ξ 2 ∣ 2 , ∣ A ( ξ 1 , x ) ∣ ≤ C ( μ 2 + ∣ ξ 1 ∣ 2 ) 1 2 , ∣ A ( ξ 1 , x ) − A ( ξ 2 , x ) ∣ ≤ C ∣ ξ 1 − ξ 2 ∣ .

More precisely, they showed that a local higher fractional differentiability assumption for the operator A with respect to the x -variable yields a fractional differentiability for weak solutions. Moreover, it is worth mentioning that the authors [2,16] used the following two conditions:

(1.5) ∣ A ( ξ , x 1 ) − A ( ξ , x 2 ) ∣ ≤ ( μ 2 + ∣ ξ ∣ 2 ) p − 1 2 ( g ( x 1 ) + g ( x 2 ) ) ∣ x 1 − x 2 ∣ β ,

for the homogeneous case F ≡ 0 with p ≥ 2 and

(1.6) ∣ A ( ξ , x 1 ) − A ( ξ , x 2 ) ∣ ≤ ( μ 2 + ∣ ξ ∣ 2 ) p − 1 2 ( g i ( x 1 ) + g i ( x 2 ) ) ∣ x 1 − x 2 ∣ β ,

for the inhomogeneous case F ≢ 0 , where β ∈ ( 0 , 1 ) and the nonnegative measurable functions g ( x ) and g i ( x ) for i ∈ N satisfy some proper conditions. As a matter of fact, various higher integrability results for nonlinear equations are also discussed from the aforementioned similar pointwise assumptions on the partial map A ( ξ , x ) (see, for instance, [29,42,45,47]). In addition, we would like to point out therefore that assumptions (1.5) and (1.6) on the coefficients A are more adaptable to the Triebel-Lizorkin spaces F n β , ∞ α and Besov space B n β , q α (see §2), respectively. More explanations about those can be found in Theorem 1.2 and Remark 3.3 of [40]. Then, Clop et al. [16] undertook further research of the extra fractional differentiability of weak solutions to the nonlinear elliptic equations (1.1) in divergence form, when A is a Carathéodory function with p − 1 growth for p ≥ 2 satisfying (1.2)–(1.4). It is worth mentioning that many authors [4,21,35] also studied the higher differentiability in Besov spaces for the elliptic equations of p -Laplacian type. Moreover, Eleuteri and Passarelli di Napoli [28] investigated the regularity theory in the context of Besov spaces for weak solutions to the following variational obstacle problems:

∫ Ω A ( D u , x ) ⋅ D ( φ − u ) d x ≥ 0 ,

where the p -harmonic-type operator A ( ξ , x ) satisfies the aforementioned conditions (1.2)–(1.4). Recently, Giova [30] proved the regularity estimates in Besov spaces of the gradient of weak solutions for the elliptic p ( x ) -Laplacian equations with p ( x ) ≥ 2 . More to the point, Dahlke et al. [21,35] also investigated the regularity theory for the elliptic p -Laplacian equation in the adaptivity scale B τ σ ( L τ ( Ω ) ) , 1 τ = σ n + 1 p of Besov spaces, where the smoothness σ determines the order of approximation, which can be gained by adaptive and other nonlinear approximation methods. And more importantly, many other scholars [14,15,19,20,22,32,33,39,41,46] have made a deep study of regularity estimates in Besov spaces for PDEs of various types.

Now, we state the main results of this work. Remarkably, the present case 1 < p < 2 is very different from the case p ≥ 2 since the modulus of ellipticity in the elliptic p -Laplacian equation tends to infinity when ∣ D u ∣ → 0 . In the beginning, we shall give the following conclusion for weak solutions of the homogeneous equation (1.1) for F ≡ 0 .

Theorem 1.1

If u ∈ W loc 1 , p ( Ω ) is a local weak solution (Definition 2.1) of the following homogeneous equation:

div A ( D u , x ) = 0 i n Ω ,

where A ( ξ , x ) satisfies assumptions (1.2)–(1.4) and

(1.7) ∣ A ( ξ , x 1 ) − A ( ξ , x 2 ) ∣ ≤ ∣ x 1 − x 2 ∣ β ( g ( x 1 ) + g ( x 2 ) ) ( μ 2 + ∣ ξ ∣ 2 ) p − 1 2 ,

for a.e. x 1 , x 2 ∈ Ω , 0 < β < p 2 with the measurable function g ( x ) ≥ 0 belonging to the Morrey spaces L loc n β , ϑ ( Ω ) (Definition 2.2) for some 0 < ϑ < n , then we have

(1.8) V p ( D u ) ∈ B 2 , ∞ β ( Ω ′ ) , f o r Ω ′ ⊂ ⊂ Ω ,

where

(1.9) V p ( ξ ) = ( μ 2 + ∣ ξ ∣ 2 ) p − 2 4 ξ , f o r ξ ∈ R n .

The following is the second conclusion of this article for weak solutions of the inhomogeneous equation (1.1) for F ≢ 0 . Since the embedding theorem B p , q β ⊂ L p β * = L n p n − p β in Lemma 2.3 only holds for 1 < q ≤ p β * ≔ n p n − p β and fails otherwise, there are several hurdles to be obtained over in this situation. Properly speaking, we are able to obtain the higher fractional differentiability listed below.

Theorem 1.2

If u ∈ W loc 1 , p ( Ω ) is a local weak solution for the following nonhomogeneous quasilinear elliptic equation:

div A ( D u , x ) = div F , i n Ω ,

where A ( ξ , x ) satisfies assumptions (1.2)–(1.4) and

(1.10) ∣ A ( ξ , x 1 ) − A ( ξ , x 2 ) ∣ ≤ ∣ x 1 − x 2 ∣ β ( g i ( x 1 ) + g i ( x 2 ) ) ( μ 2 + ∣ ξ ∣ 2 ) p − 1 2 ,

for a.e. x 1 , x 2 ∈ Ω , 0 < β < 1 , 2 − i diam ( Ω ) ≤ ∣ x 1 − x 2 ∣ ≤ 2 − i + 1 diam ( Ω ) , i ∈ N and a sequence of measurable functions g i ( x ) ≥ 0 satisfying

(1.11) ∑ i ∈ N ‖ g i ‖ L loc n β , ϑ ( Ω ) q < ∞ , f o r s o m e ϑ ∈ ( 0 , n ) ,

then we have

V p ( D u ) ∈ B 2 , q α ( Ω ′ ) , for Ω ′ ⊂ ⊂ Ω ,

provided that F ∈ B p p − 1 , q β ( Ω ) , 0 < α < β 2 , 2 < p p − 1 < n β , and 1 < q ≤ n p n ( p − 1 ) − p β .

2 Notations and preliminaries

Let 0 < α < 1 , h ∈ R n and Δ h v ( x ) = [ v ( x + h ) − v ( x ) ] ⋅ χ Ω ( x + h ) . As in Section 2.5.12 of [48], we say that a measurable functions v : R n → R belongs to the Besov space B p , q α ( Ω ) for 1 < p < ∞ if v ∈ L p ( Ω ) and

‖ v ‖ B p , q α ( Ω ) ≔ ‖ v ‖ L p ( Ω ) + [ v ] 64 p , q α ( Ω ) < ∞ ,

where

‖ v ‖ L p ( Ω ) ≔ ∫ Ω ∣ v ( x ) ∣ p d x 1 p , [ v ] 64 p , q α ( Ω ) ≔ ∫ R n ∫ Ω ∣ Δ h v ( x ) ∣ p ∣ h ∣ α p d x q p d h ∣ h ∣ n 1 q , for 1 < q < ∞

and

[ v ] 64 p , ∞ α ( Ω ) ≔ sup h ∈ R n ∫ Ω ∣ Δ h v ( x ) ∣ p ∣ h ∣ α p d x 1 p < ∞ .

Since

∫ { ∣ h ∣ ≥ δ } ∫ Ω ∣ Δ h v ( x ) ∣ p ∣ h ∣ α p d x q p d h ∣ h ∣ n 1 q ≤ C ∫ { ∣ h ∣ ≥ δ } d h ∣ h ∣ n + α q 1 q ‖ v ‖ L p ( Ω ) ≤ c ( n , α , p , q , δ ) ‖ v ‖ L p ( Ω )

and

sup { ∣ h ∣ ≥ δ } ∫ Ω ∣ Δ h v ( x ) ∣ p ∣ h ∣ α p d x 1 p ≤ 1 δ α sup { ∣ h ∣ ≥ δ } ∫ Ω ∣ Δ h v ( x ) ∣ p d x 1 p ≤ c ( n , α , p , δ ) ‖ v ‖ L p ( Ω ) ,

we can obtain the equivalent norms of ‖ v ‖ B p , q α ( Ω ) for 1 < q ≤ ∞ by only considering the ball B δ for a fixed δ > 0 . In particular, when p = q , the Besov space B p , q α ( Ω ) is equal to the fractional Sobolev space W α , q ( Ω ) (Definition 3 in [5]).

Now, we shall give the definition of the solutions of (1.1) in a weak sense.

Definition 2.1

Assume that F ∈ L loc p p − 1 ( Ω ) . A function u ∈ W loc 1 , p ( Ω ) is a local weak solution of (1.1) in Ω if for any function φ ∈ W 0 1 , p ( Ω ) , we have

∫ Ω A ( D u , x ) ⋅ D φ d x = ∫ Ω F ⋅ D φ d x .

Additionally, we recall the definition of the following Morrey spaces L q , ϑ ( Ω ) .

Definition 2.2

We say that a function u ∈ L q ( Ω ) for q ∈ [ 1 , + ∞ ) and a bounded domain Ω of R n belong to the Morrey space L q , ϑ ( Ω ) for ϑ ∈ [ 0 , n ] if

‖ u ‖ L q , ϑ ( Ω ) ≔ sup y ∈ Ω r ∈ ( 0 , diam Ω ) 1 r ϑ ∫ B r ( y ) ∩ Ω ∣ u ( x ) ∣ q d x 1 q < + ∞ .

Now, we would like to remark that L q , ϑ ( Ω ) is a Banach space, L q , n ( Ω ) = L ∞ ( Ω ) , L q , 0 ( Ω ) = L q ( Ω ) , and L q , n + ν ( Ω ) = { 0 } for any ν > 0 . Moreover, in this work, we shall use the current version of Sobolev embedding theorem (see Lemma 2.2 in [28] and Proposition 7.12 of [34]).

Lemma 2.3

Assume that 0 < α < 1 . If 1 < p < n α and 1 < q ≤ p α * ≔ n p n − α p , then there is a continuous embedding B p , q α ( Ω ) ⊂ L p α * ( Ω ) .

The subsequent two lemmas are necessary for the proofs of our main results in the next section. First, we give some results described below [16,28,31].

Lemma 2.4

If F ∈ W 1 , p ( Ω ) and G ∈ L q ( Ω ) for p , q > 1 and 1 p + 1 q = 1 , and

Ω ∣ h ∣ ≔ { x ∈ Ω : dist ( x , ∂ Ω ) > ∣ h ∣ } ,

then for 0 < ρ < R and ∣ h ∣ < R − ρ 2 with B R ⊂ Ω , we have

∫ Ω F Δ h G d x = ∫ Ω G Δ − h F d x
when either of the following cases supp F ⊂ Ω ∣ h ∣ or supp G ⊂ Ω ∣ h ∣ is true.
D i ( Δ h F ) = Δ h ( D i F ) , Δ h F ∈ W 1 , p ( Ω ∣ h ∣ ) , and
Δ h ( F G ) ( x ) = G ( x ) Δ h F ( x ) + F ( x + h ) Δ h G ( x ) ,
∫ B ρ ∣ F ( x + h ) ∣ p d x ≤ ∫ B R ∣ F ( x ) ∣ p d x ,
and
∫ B ρ ∣ Δ h F ∣ p d x ≤ C ( n , p ) ∣ h ∣ p ∫ B R ∣ D F ∣ p d x .

Next, we shall give the following important results on the function V p ( ξ ) in the singular case 1 < p < 2 .

Lemma 2.5

Let A ( ξ , x ) satisfy (1.2)–(1.4) for 1 < p < 2 and

V p ( ξ ) = ( μ 2 + ∣ ξ ∣ 2 ) p − 2 4 ξ , for ξ ∈ R n .

Then, for every ξ , η ∈ R n , there exist two positive constants C 1 and C 2 such that

C 1 ( μ 2 + ∣ ξ ∣ 2 + ∣ η ∣ 2 ) p − 2 2 ∣ ξ − η ∣ 2 ≤ ∣ V p ( ξ ) − V p ( η ) ∣ 2 ≤ C 2 ( μ 2 + ∣ ξ ∣ 2 + ∣ η ∣ 2 ) p − 2 2 ∣ ξ − η ∣ 2 .

Proof

Without loss of generality, we may as well assume that μ = 0 . In the first instance, we find that

∣ V p ( ξ ) − V p ( η ) ∣ = ∣ ξ ∣ p − 2 2 ξ − ∣ η ∣ p − 2 2 η = ∫ 0 1 d ∣ s ξ + ( 1 − s ) η ∣ p − 2 2 ( s ξ + ( 1 − s ) η ) d s d s ≤ ∣ ξ − η ∣ ∫ 0 1 ∣ s ξ + ( 1 − s ) η ∣ p − 2 2 d s + ∣ p − 2 ∣ 2 ∫ 0 1 ∣ s ξ + ( 1 − s ) η ∣ p − 2 2 − 2 ( s ξ + ( 1 − s ) η ) [ s ξ + ( 1 − s ) η ] ⋅ ( ξ − η ) d s ≤ 1 + ∣ p − 2 ∣ 2 ∣ ξ − η ∣ ∫ 0 1 ∣ s ξ + ( 1 − s ) η ∣ p − 2 2 d s

and

∣ V p ( ξ ) − V p ( η ) ∣ ≥ ( ξ − η ) ∫ 0 1 ∣ s ξ + ( 1 − s ) η ∣ p − 2 2 d s − ( p − 2 ) 2 ∫ 0 1 ∣ s ξ + ( 1 − s ) η ∣ p − 2 2 − 2 ( s ξ + ( 1 − s ) η ) [ s ξ + ( 1 − s ) η ] ⋅ ( ξ − η ) d s ≥ ∣ ξ − η ∣ ∫ 0 1 ∣ s ξ + ( 1 − s ) η ∣ p − 2 2 d s − ∣ p − 2 ∣ 2 ∣ ξ − η ∣ ∫ 0 1 ∣ s ξ + ( 1 − s ) η ∣ p − 2 2 d s = 1 − ∣ p − 2 ∣ 2 ∣ ξ − η ∣ ∫ 0 1 ∣ s ξ + ( 1 − s ) η ∣ p − 2 2 d s .

On the one hand, in view of the fact that

∣ s ξ + ( 1 − s ) η ∣ ≤ ( ∣ ξ ∣ + ∣ η ∣ ) , for any s ∈ [ 0 , 1 ] ,

we find that

∫ 0 1 ∣ s ξ + ( 1 − s ) η ∣ p − 2 2 d s ≥ ( ∣ ξ ∣ + ∣ η ∣ ) p − 2 2 ∫ 0 1 d s = ( ∣ ξ ∣ + ∣ η ∣ ) p − 2 2 , for 1 < p < 2 ,

which implies that

∣ V p ( ξ ) − V p ( η ) ∣ 2 ≥ 1 − ∣ p − 2 ∣ 2 2 ( ∣ ξ ∣ + ∣ η ∣ ) p − 2 ∣ ξ − η ∣ 2 .

Furthermore, we may as well assume that 0 < ∣ ξ ∣ ≤ ∣ η ∣ and ξ ≠ η , and then define

s 0 ≔ ∣ η 0 − η ∣ ∣ ξ − η ∣ ,

where η 0 is the minimum norm point on the line through ξ and η . Obviously, it is easy to verify that s 0 ≥ 1 2 . Now, we divide into two cases.

Case 1: s 0 ≥ 1 . Then, ∣ s ξ + ( 1 − s ) η ∣ ≥ ∣ s η 0 + ( 1 − s ) η ∣ and ∣ s η 0 + ( 1 − s ) η ∣ ≥ ∣ s 0 + ( 1 − s ) η ∣ = ( 1 − s ) ∣ η ∣ ≥ ( 1 − s ) 2 ( ∣ ξ ∣ + ∣ η ∣ ) for any s ∈ [ 0 , 1 ] and ∣ η ∣ ≥ ∣ ξ ∣ > 0 . Since 1 < p < 2 , we deduce that

∣ s ξ + ( 1 − s ) η ∣ p − 2 2 ≤ ( 1 − s ) 2 p − 2 2 ( ∣ ξ ∣ + ∣ η ∣ ) p − 2 2

and then

∫ 0 1 ∣ s ξ + ( 1 − s ) η ∣ p − 2 2 d s ≤ C ( ∣ ξ ∣ + ∣ η ∣ ) p − 2 2 ∫ 0 1 ( 1 − s ) p − 2 2 d s ≤ C ( ∣ ξ ∣ + ∣ η ∣ ) p − 2 2 .

Case 2: 1 2 ≤ s 0 < 1 . Then, ∣ θ η 0 + ( 1 − θ ) η ∣ ≥ ∣ θ 0 + ( 1 − θ ) η ∣ = ( 1 − θ ) ∣ η ∣ ≥ ( 1 − θ ) 2 ( ∣ ξ ∣ + ∣ η ∣ ) for any θ ∈ [ 0 , 1 ] . Therefore, by recalling the definition of η 0 and choosing s = θ s 0 , we conclude that

∫ 0 1 ∣ s ξ + ( 1 − s ) η ∣ p − 2 2 d s ≤ 2 ∫ 0 s 0 ∣ s ξ + ( 1 − s ) η ∣ p − 2 2 d s ≤ C ∫ 0 1 ∣ θ η 0 + ( 1 − θ ) η ∣ p − 2 2 d θ ≤ C ( ∣ ξ ∣ + ∣ η ∣ ) p − 2 2 ∫ 0 1 ( 1 − θ ) p − 2 2 d θ ≤ C ( ∣ ξ ∣ + ∣ η ∣ ) p − 2 2 .

Based on an overall consideration of the aforementioned two cases, we conclude that

∣ V p ( ξ ) − V p ( η ) ∣ 2 ≤ C ( ∣ ξ ∣ + ∣ η ∣ ) p − 2 ∣ ξ − η ∣ 2 ,

which completes the final proof of this lemma.□

3 Proofs of Theorems 1.1 and 1.2

This final section is largely devoted to the proofs of the two main results in this article. Generally speaking, we shall need a control on the oscillations of the matrix A ( ξ , x ) of coefficients when we study regularity estimates for various kinds of linear/nonlinear elliptic and parabolic PDEs. Now, we shall give the definition of the vanishing mean oscillation (VMO) condition on the matrix A ( ξ , x ) of coefficients.

Definition 3.1

(VMO coefficients). We say that the matrix A ( ξ , x ) of coefficients satisfies a VMO condition if

(3.1) lim ρ → 0 sup B ρ ( y ) ⊂ Ω ∫ B ρ ( y ) ∣ Θ ( A , B ρ ( y ) ) ( x ) ∣ d x = 0 ,

where

(3.2) Θ ( A , B ρ ( y ) ) ( x ) ≔ sup ξ ∈ R n ∣ A ( ξ , x ) − A ¯ B ρ ( y ) ( ξ ) ∣ ( μ 2 + ∣ ξ ∣ 2 ) p − 1 2

and A ¯ B ρ ( y ) ( ξ ) is the integral average of A ( ξ , x ′ ) in the variable x ′ over B ρ ( y ) for a fixed ξ ∈ R n , i.e.,

A ¯ B ρ ( y ) ( ξ ) ≔ ∫ B ρ ( y ) A ( ξ , x ′ ) d x ′ = 1 ∣ B ρ ( y ) ∣ ∫ B ρ ( y ) A ( ξ , x ′ ) d x ′ .

Here, we want to point out that if a function satisfies the VMO condition, then it satisfies the small bounded mean oscillation (BMO)/ ( δ , R ) -vanishing condition (Definition 2.1 in [9]). More precisely, we say that the matrix A ( ξ , x ) of coefficients is ( δ , R ) -vanishing for small δ > 0 if there exists a small positive constant R ∈ ( 0 , 1 ) such that

sup 0 < ρ ≤ R sup B ρ ( y ) ⊂ Ω ∫ B ρ ( y ) ∣ Θ ( A , B ρ ( y ) ) ( x ) ∣ d x ≤ δ .

As a matter of fact, the matrix A ( ξ , x ) of coefficients can satisfy a VMO condition under the assumptions on A ( ξ , x ) in the main results: Theorems 1.1 and 1.2.

Lemma 3.2

If the matrix A ( ξ , x ) of coefficients satisfies assumptions (1.2)–(1.4) and (1.7)/(1.10)–(1.11), then A ( ξ , x ) satisfies a VMO condition.

Proof

(1) Let A ( ξ , x ) satisfy (1.2)–(1.4) and B ρ ( y ) ⊂ Ω . From (1.7), (3.2), and Hölder’s inequality, we conclude that

∫ B ρ ( y ) ∣ Θ ( A , B ρ ( y ) ) ( x ) ∣ d x = ∫ B ρ ( y ) sup ξ ∈ R n ∣ A ( ξ , x ) − A ¯ B ρ ( y ) ( ξ ) ∣ ( μ 2 + ∣ ξ ∣ 2 ) p − 1 2 d x ≤ ∫ B ρ ( y ) sup ξ ∈ R n ∫ B ρ ( y ) ∣ A ( ξ , x ) − A ( ξ , x ′ ) ∣ d x ′ ( μ 2 + ∣ ξ ∣ 2 ) p − 1 2 d x ≤ ∫ B ρ ( y ) ∫ B ρ ( y ) ∣ x − x ′ ∣ β [ g ( x ) + g ( x ′ ) ] d x ′ d x ≤ C ρ β ∫ B ρ ( y ) g ( x ) d x ≤ C ∫ B ρ ( y ) g ( x ) n β d x β n = C ρ β ϑ n 1 ρ ϑ ∫ B ρ ( y ) g ( x ) n β d x β n ≤ C ρ β ϑ n .

Therefore, we know that A ( ξ , x ) satisfies a VMO condition.

(2) Additionally, let A ( ξ , x ) satisfy the conditions (1.2)–(1.4), x , y ∈ Ω , and A i ( x ) ≔ { y ∈ Ω : 2 − i diam ( Ω ) ≤ ∣ x − y ∣ ≤ 2 − i + 1 diam ( Ω ) } . Then, by virtue of (3.2) and the given assumption (1.10), we have

∫ B ρ ( y ) ∣ Θ ( A , B ρ ( y ) ) ( x ) ∣ d x = ∫ B ρ ( y ) sup ξ ∈ R n ∣ A ( ξ , x ) − A ¯ B ρ ( y ) ( ξ ) ∣ ( μ 2 + ∣ ξ ∣ 2 ) p − 1 2 d x ≤ ∫ B ρ ( y ) sup ξ ∈ R n ∫ B ρ ( y ) ∣ A ( ξ , x ) − A ( ξ , x ′ ) ∣ d x ′ ( μ 2 + ∣ ξ ∣ 2 ) p − 1 2 d x ≤ 1 ∣ B ρ ( y ) ∣ 2 ∫ B ρ ( y ) sup ξ ∈ R n ∫ B ρ ( y ) ∣ A ( ξ , x ) − A ( ξ , x ′ ) ∣ d x ′ ( μ 2 + ∣ ξ ∣ 2 ) p − 1 2 d x ≤ 1 ∣ B ρ ( y ) ∣ 2 ∫ B ρ ( y ) sup ξ ∈ R n ∑ i ∈ N ∫ B ρ ( y ) ∩ A i ( x ) ∣ A ( ξ , x ) − A ( ξ , x ′ ) ∣ d x ′ ( μ 2 + ∣ ξ ∣ 2 ) p − 1 2 d x ≤ C ρ β ∣ B ρ ( y ) ∣ 2 ∑ i ∈ N ∫ B ρ ( y ) ∫ B ρ ( y ) ∩ A i ( x ) ( g i ( x ) + g i ( x ′ ) ) d x ′ d x ≤ C ρ β ∣ B ρ ( y ) ∣ 2 ∑ i ∈ N ∣ B ρ ( y ) ∩ A i ( x ) ∣ ∫ B ρ ( y ) g i ( x ) d x + C ρ β ∣ B ρ ( y ) ∣ ∑ i ∈ N ∫ B ρ ( y ) ∩ A i ( x ) g i ( x ′ ) d x ′ ≕ I 1 + I 2 .

Estimate of I 1 . From Hölder’s inequality and (1.11), we have

I 1 ≤ C ρ − n ∑ i ∈ N ∣ B ρ ( y ) ∩ A i ( x ) ∣ ∫ B ρ ( y ) g i n β d x β n ≤ C ρ − n ∑ i ∈ N ∫ B ρ ( y ) g i β n d x q β n 1 q ∑ i ∈ N ∣ B ρ ( y ) ∩ A i ( x ) ∣ q q − 1 q − 1 q ≤ C ρ β ϑ n ∑ i ∈ N ‖ g i ‖ L n β , ϑ ( B ρ ( y ) ) q 1 q ≤ C ρ β ϑ n .

Estimate of I 2 . Now, we apply Hölder’s inequality again and (1.11) to deduce that

I 2 ≤ C ρ β ∣ B ρ ( y ) ∣ ∑ i ∈ N ∫ B ρ ( y ) ∩ A i ( x ) g i n β d x ′ β n ∣ B ρ ( y ) ∩ A i ( x ) ∣ 1 − β n ≤ C ρ β ∣ B ρ ( y ) ∣ ∑ i ∈ N ∫ B ρ ( y ) g i n β d x β q n 1 q ∑ i ∈ N ∣ B ρ ( y ) ∩ A i ( x ) ∣ ( n − β ) q n ( q − 1 ) q − 1 q ≤ C ρ β ϑ n ∑ i ∈ N ‖ g i ‖ L n β , ϑ ( B ρ ( y ) ) q 1 q ≤ C ρ β ϑ n .

Combining the estimates of I 1 and I 2 , we have

lim ρ → 0 sup B ρ ( y ) ⊂ Ω ∫ B ρ ( y ) ∣ Θ ( A , B ρ ( y ) ) ( x ) ∣ d x = 0 ,

which can imply that A ( ξ , x ) satisfies a VMO condition. Thus, we complete the proof.□

Several years ago, Byun et al. [10] have proved the local/global L p -type estimates for the weak solutions of (1.1) for the special case A ( ξ , x ) = ( A D u ⋅ D u ) p − 2 2 A D u with p > 1 and the small BMO seminorm assumption on the matrix A ( x ) of coefficients. Subsequently, Byun and Wang [9] made a deep study of the local/global L p -type estimates for the weak solutions of the general case (1.1) with the small BMO seminorm assumption on the coefficients matrix A ( ξ , x ) . Recently, Balci et al. [3] also obtained new local Calderón-Zygmund estimates for linear and nonlinear elliptic equations, whose model case is A ( ξ , x ) = ( A D u ⋅ D u ) p − 2 2 A D u , with a novel log-BMO condition on the coefficient matrix. Here, we shall use an appropriate modified version for our purposes.

Lemma 3.3

Suppose that the matrix A ( ξ , x ) of coefficients satisfies assumptions (1.2)–(1.4) and (1.7)/(1.10)–(1.11). If u ∈ W 1 , p ( Ω ) is a local weak solution of (1.1) and ∣ F ∣ p p − 1 ∈ L loc q ( Ω ) for any q ∈ ( 1 , ∞ ) , then ∣ D u ∣ p ∈ L loc q ( Ω ) with the following estimate:

∫ B R ∣ D u ( x ) ∣ p q d x ≤ C ∫ B 2 R ∣ D u ( x ) ∣ p d x q + ∫ B 2 R ∣ F ( x ) ∣ p q p − 1 d x + 1 ,

where the constant C is independent of u and F and B 2 R ⊂ Ω .

Now, let us prove the first main result in Theorem 1.1.

Proof of Theorem 1.1

Let B 2 R ⊂ Ω and η ( x ) ∈ C 0 ∞ ( Ω ) be a cut-off function satisfying

(3.3) 0 ≤ η ≤ 1 , η ≡ 1 , in B R ⁄ 2 , η ≡ 0 , in Ω \ B R and ∣ D η ∣ ≤ C R .

Now, we choose a test function φ = Δ − h ( η 2 Δ h u ) for h ∈ B δ and δ < R and then use Definition 2.1 to deduce that

∫ B R A ( D u , x ) ⋅ D ( Δ − h ( η 2 Δ h u ) ) d x = ∫ B R Δ h A ( D u , x ) ⋅ D ( η 2 Δ h u ) d x = 0 .

After a direct calculation, the resulting expression is shown as follows:

I 1 = I 2 + I 3 ,

where

I 1 ≔ ∫ B R [ A ( D u ( x + h ) , x + h ) − A ( D u ( x ) , x + h ) ] ⋅ η 2 Δ h D u d x , I 2 ≔ − ∫ B R [ A ( D u ( x ) , x + h ) − A ( D u ( x ) , x ) ] ⋅ η 2 Δ h D u d x , I 3 ≔ − ∫ B R [ A ( D u ( x + h ) , x + h ) − A ( D u ( x ) , x ) ] ⋅ 2 η D η Δ h u d x .

Estimate of I 1 . We first apply (1.2) to obtain

I 1 ≥ C ∫ B R η 2 ( μ 2 + ∣ D u ( x + h ) ∣ 2 + ∣ D u ( x ) ∣ 2 ) p − 2 2 ∣ Δ h D u ∣ 2 d x .

Estimate of I 2 . By virtue of (1.7), (3.3), Cauchy’s inequality, and Lemma 2.4, we find that

∣ I 2 ∣ ≤ ∫ B R ∣ A ( D u ( x ) , x + h ) − A ( D u ( x ) , x ) ∣ η 2 ∣ Δ h D u ∣ d x ≤ C ∣ h ∣ β ∫ B R η 2 [ g ( x ) + g ( x + h ) ] ( μ 2 + ∣ D u ( x ) ∣ 2 ) p − 1 2 ∣ Δ h D u ∣ d x ≤ C ∣ h ∣ β ∫ B R η 2 [ g ( x ) + g ( x + h ) ] ( μ 2 + ∣ D u ( x + h ) ∣ 2 + ∣ D u ( x ) ∣ 2 ) p − 1 2 ∣ Δ h D u ∣ d x ≤ ε ∫ B R η 2 ( μ 2 + ∣ D u ( x + h ) ∣ 2 + ∣ D u ( x ) ∣ 2 ) p − 2 2 ∣ Δ h D u ∣ 2 d x + C ( ε ) ∣ h ∣ 2 β ∫ B R ( μ 2 + ∣ D u ( x + h ) ∣ 2 + ∣ D u ( x ) ∣ 2 ) p 2 [ g ( x ) + g ( x + h ) ] 2 d x ≤ C ε I 1 + C ( ε ) ∣ h ∣ 2 β ∫ B R [ g ( x + h ) + g ( x ) ] n β d x 2 β n ∫ B R ( 1 + ∣ D u ( x + h ) ∣ + ∣ D u ( x ) ∣ ) n p n − 2 β d x n − 2 β n ≤ C ε I 1 + C ( ε ) ∣ h ∣ 2 β ∥ ∣ D u ∣ p ∥ L n n − 2 β ( B 2 R ) + 1 ≤ C ε I 1 + C ( ε ) ∣ h ∣ 2 β ,

since

(3.4) ∫ B R [ g ( x ) + g ( x + h ) ] n β d x ≤ C R ϑ ‖ g ‖ L n β , ϑ ( B 2 R ) n β ≤ C

in view of the facts that g ∈ L loc n β , ϑ ( Ω ) and ∣ D u ∣ p ∈ L loc n n − 2 β ( Ω ) by Lemma 3.3 for F ≡ 0 .

Estimate of I 3 . Just like I 1 and I 2 , we divide I 3 into two terms

∣ I 3 ∣ ≤ ∫ B R ∣ A ( D u ( x + h ) , x + h ) − A ( D u ( x ) , x + h ) ∣ 2 η ∣ D η ∣ ∣ Δ h u ∣ d x + ∫ B R ∣ A ( D u ( x ) , x + h ) − A ( D u ( x ) , x ) ∣ 2 η ∣ D η ∣ ∣ Δ h u ∣ d x ≕ I 31 + I 32 .

Estimate of I 31 . Since C ∞ ( B 2 R ¯ ) is dense in W 1 , p ( B 2 R ) , we may as well assume that u ( x ) ∈ C ∞ ( B 2 R ¯ ) in the following proof. By virtue of (1.3), Cauchy’s inequality, and Lagrange’s mean-value theorem, we obtain

∣ I 31 ∣ ≤ C ∫ B R η ( μ 2 + ∣ D u ( x + h ) ∣ 2 + ∣ D u ( x ) ∣ 2 ) p − 2 2 ∣ Δ h D u ∣ ∣ Δ h u ∣ d x ≤ C ∫ { x ∈ B R : μ 2 + ∣ D u ( x + h ) ∣ 2 + ∣ D u ( x ) ∣ 2 ≥ ∣ Δ h u ∣ } η ( μ 2 + ∣ D u ( x + h ) ∣ 2 + ∣ D u ( x ) ∣ 2 ) p − 2 2 ∣ Δ h D u ∣ ∣ Δ h u ∣ d x + C ∫ { x ∈ B R : μ 2 + ∣ D u ( x + h ) ∣ 2 + ∣ D u ( x ) ∣ 2 < ∣ Δ h u ∣ } η ( μ 2 + ∣ D u ( x + h ) ∣ 2 + ∣ D u ( x ) ∣ 2 ) p − 2 2 ∣ Δ h D u ∣ ∣ Δ h u ∣ d x ≤ C ∫ { x ∈ B R : μ 2 + ∣ D u ( x + h ) ∣ 2 + ∣ D u ( x ) ∣ 2 ≥ ∣ Δ h u ∣ } η ( μ 2 + ∣ D u ( x + h ) ∣ 2 + ∣ D u ( x ) ∣ 2 ) p − 2 2 ∣ Δ h D u ∣ ∣ Δ h u ∣ d x + C ∫ { x ∈ B R : μ 2 + ∣ D u ( x + h ) ∣ 2 + ∣ D u ( x ) ∣ 2 < ∣ Δ h u ∣ } ∣ Δ h u ∣ p d x ≤ C ε I 1 + C ( ε ) ∫ { x ∈ B R : μ 2 + ∣ D u ( x + h ) ∣ 2 + ∣ D u ( x ) ∣ 2 ≥ ∣ Δ h u ∣ } ( μ 2 + ∣ D u ( x + h ) ∣ 2 + ∣ D u ( x ) ∣ 2 ) p − 2 2 ∣ Δ h u ∣ 2 d x + C ∫ B R ∣ Δ h u ∣ p d x ≤ C ε I 1 + C ( ε ) ∫ B R ∣ Δ h u ∣ p d x ≤ C ε I 1 + C ( ε ) ∣ h ∣ p ∫ B R ∣ D u ( x + θ h ) ∣ p d x ≤ C ε I 1 + C ( ε ) ∣ h ∣ p ,

for some θ ∈ ( 0 , 1 ) , in view of the fact that u ∈ W loc 1 , p ( Ω ) .

Estimate of I 32 . Similar to the previous estimate of I 31 , we use (1.7), Lagrange’s mean value theorem, and Young’s inequality to obtain that

∣ I 32 ∣ ≤ C ∣ h ∣ β ∫ B R [ g ( x ) + g ( x + h ) ] ( μ 2 + ∣ D u ( x ) ∣ 2 ) p − 1 2 ∣ Δ h u ∣ d x ≤ C ∣ h ∣ 1 + β ∫ B R [ g ( x ) + g ( x + h ) ] ( μ 2 + ∣ D u ( x ) ∣ 2 ) p − 1 2 ∣ D u ( x + θ h ) ∣ d x ≤ C ∣ h ∣ 1 + β ∫ B R [ g ( x ) + g ( x + h ) ] [ 1 + ∣ D u ( x ) ∣ p + ∣ D u ( x + θ h ) ∣ p ] d x ≤ C ∣ h ∣ 1 + β ∫ B R [ g ( x ) + g ( x + h ) ] n β d x β n ∫ B R ( 1 + ∣ D u ( x ) ∣ p + ∣ D u ( x + θ h ) ∣ p ) n n − β d x n − β n ≤ C ∣ h ∣ 1 + β ,

for some θ ∈ ( 0 , 1 ) , in view of (3.4), and the fact that ∣ D u ∣ p ∈ L loc n n − β ( Ω ) by Lemma 3.3 for F ≡ 0 . Therefore, from the aforementioned estimates of I 31 and I 32 , we obtain

(3.5) ∣ I 3 ∣ ≤ C ε I 1 + C ( ∣ h ∣ p + ∣ h ∣ 1 + β ) ≤ C ε I 1 + C ( ∣ h ∣ p + ∣ h ∣ 2 β ) .

Finally, by combining all the estimates of I i ( 1 ≤ i ≤ 3 ) and choosing ε small enough, from Lemma 2.5, we conclude that

∫ B R ⁄ 2 ∣ Δ h V p ( D u ) ∣ 2 d x = ∫ B R ⁄ 2 ∣ V p ( D u ( x + h ) ) − V p ( D u ( x ) ) ∣ 2 d x ≤ C ∫ B R ⁄ 2 ( μ 2 + ∣ D u ( x + h ) ∣ 2 + ∣ D u ( x ) ∣ 2 ) p − 2 2 ∣ Δ h D u ∣ 2 d x ≤ C ∫ B R η 2 ( μ 2 + ∣ D u ( x + h ) ∣ 2 + ∣ D u ( x ) ∣ 2 ) p − 2 2 ∣ Δ h D u ∣ 2 d x ≤ C ( ∣ h ∣ p + ∣ h ∣ 2 β ) .

Ultimately, we divide the aforementioned inequality by a small number ∣ h ∣ 2 β , and then by Lemma 2.4, find that

∫ B R ⁄ 2 Δ h V p ( D u ) ∣ h ∣ β 2 d x ≤ C ∣ h ∣ p − 2 β + C ≤ C ,

for β < p 2 , which completes the proof by taking supremum over h ∈ B δ for some δ < R .□

We now come to give the proof of the second result, Theorem 1.2.

Proof of Theorem 1.2

Let B 2 R be a ball in Ω and η ( x ) ∈ C 0 ∞ ( Ω ) be a cut-off function satisfying

(3.6) 0 ≤ η ≤ 1 , η ≡ 1 , in B R ⁄ 2 , η ≡ 0 , in Ω \ B R , and ∣ D η ∣ ≤ C R .

Choosing a test function φ = Δ − h ( η 2 Δ h u ) in Definition 2.1, where h ∈ B δ for some small constant δ < R , we have

∫ B R A ( D u , x ) ⋅ D ( Δ − h ( η 2 Δ h u ) ) d x = ∫ B R F ⋅ D ( Δ − h ( η 2 Δ h u ) ) d x .

Through a simple operation, we can obtain the following equality:

I 1 = I 2 + I 3 + I 4 + I 5 ,

where I i for 1 ≤ i ≤ 3 are defined in the proof of Theorem 1.1 and

I 4 ≔ ∫ B R η 2 Δ h F ⋅ Δ h D u d x , I 5 ≔ 2 ∫ B R η Δ h F ⋅ Δ h u D η d x .

Estimates of I 1 – I 3 . Since F ∈ B p p − 1 , q β ( Ω ) locally for 0 < β < 1 , 2 < p p − 1 < n β and 1 < q ≤ p p − 1 β * ≔ n p p − 1 n − β p p − 1 , from Lemma 2.3, we find that F ∈ L loc n p p − 1 n − β p p − 1 ( Ω ) and then ∣ F ∣ p p − 1 ∈ L loc n n − β p p − 1 ( Ω ) ⊂ L loc n n − 2 β ( Ω ) in virtue of the fact that p p − 1 > 2 . Therefore, Lemma 3.3 implies that ∣ D u ∣ p ∈ L loc n n − 2 β ( Ω ) ⊂ L loc n n − β ( Ω ) . Without loss of generality, we might just as well suppose that diam ( Ω ) = I 0 R for some positive constant I 0 > 1 and 2 − i I 0 R ≤ ∣ h ∣ ≤ 2 − i + 1 I 0 R ≤ δ for some positive integer i > i 0 , where i 0 = 1 + log 2 I 0 R δ ∈ N and [ θ ] is the integer part of θ . Similar to the proof of Theorem 1.1, we deduce that

I 1 ≥ C ∫ B R η 2 ( μ 2 + ∣ D u ( x + h ) ∣ 2 + ∣ D u ( x ) ∣ 2 ) p − 2 2 ∣ Δ h D u ∣ 2 d x , ∣ I 2 ∣ ≤ C ε I 1 + C ( ε ) ∣ h ∣ 2 β ∫ B R + ∣ h ∣ g i n β ( x ) d x 2 β n ∫ B R + ∣ h ∣ ∣ D u ∣ n p n − 2 β d x + 1 n − 2 β n ≤ C ε I 1 + C ( ε ) ∣ h ∣ 2 β ∫ B R + ∣ h ∣ g i n β ( x ) d x 2 β n , ∣ I 3 ∣ ≤ C ε I 1 + C ( ε ) ∣ h ∣ p + C ∣ h ∣ 1 + β ∫ B R + ∣ h ∣ g i n β ( x ) d x β n ≤ C ε I 1 + C ( ε ) ( ∣ h ∣ p + ∣ h ∣ 1 + β ) + C ∣ h ∣ 1 + β ∫ B R + ∣ h ∣ g i n β ( x ) d x 2 β n .

Estimates of I 4 – I 5 . Using Hölder’s inequality and the fact that u ∈ W loc 1 , p ( Ω ) , we find that

∣ I 4 ∣ ≤ C ∫ B R ∣ Δ h f ∣ p p − 1 d x p − 1 p ∫ B R ∣ Δ h D u ∣ p d x 1 p ≤ C ∫ B R ∣ Δ h f ∣ p p − 1 d x p − 1 p ∫ B R + ∣ h ∣ ∣ D u ∣ p d x 1 p ≤ C ∫ B R ∣ Δ h f ∣ p p − 1 d x p − 1 p ≤ C ∣ h ∣ β ∫ B R ∣ Δ h f ∣ ∣ h ∣ β p p − 1 d x p − 1 p

and

∣ I 5 ∣ ≤ C ∫ B R ∣ Δ h f ∣ p p − 1 d x p − 1 p ∫ B R ∣ Δ h u ∣ p d x 1 p ≤ C ∣ h ∣ ∫ B R ∣ Δ h f ∣ p p − 1 d x p − 1 p ∫ B R + ∣ h ∣ ∣ D u ∣ p d x 1 p ≤ C ∣ h ∣ ∫ B R ∣ Δ h f ∣ p p − 1 d x p − 1 p ≤ C ∣ h ∣ 1 + β ∫ B R ∣ Δ h f ∣ ∣ h ∣ β p p − 1 d x p − 1 p .

Combining the aforementioned estimates of I 1 – I 5 and selecting a sufficiently small number ε > 0 , we deduce that

∫ B R ⁄ 2 ∣ Δ h V p ( D u ) ∣ 2 d x ≤ C ∫ B R η 2 ( μ 2 + ∣ D u ( x + h ) ∣ 2 + ∣ D u ( x ) ∣ 2 ) p − 2 2 ∣ Δ h D u ∣ 2 d x ≤ C ( ∣ h ∣ 1 + β + ∣ h ∣ p ) + C ∣ h ∣ β ∫ B R ∣ Δ h f ∣ ∣ h ∣ β p p − 1 d x p − 1 p + C ∣ h ∣ 2 β ∫ B R + ∣ h ∣ g i n β ( x ) d x 2 β n .

Furthermore, we divide the aforementioned inequality by a small number ∣ h ∣ α and then take the L q norm in B δ for δ < R to obtain that

∫ B δ ∫ B R ⁄ 2 Δ h V p ( D u ) ∣ h ∣ α 2 d x q 2 d h ∣ h ∣ n 1 q ≤ C + C ∫ B δ ∣ h ∣ β 2 − α q ∫ B R Δ h F ∣ h ∣ β p p − 1 d x q ( p − 1 ) 2 p d h ∣ h ∣ n 1 q + ∑ i = i 0 ∞ ∫ C i ∣ h ∣ ( β − α ) q ∫ B R + ∣ h ∣ ∣ g i ∣ n β d x β q n d h ∣ h ∣ n 1 q ≤ C + J 1 + J 2 ,

for 0 < α < β 2 < p 2 < 1 , where C i ≔ B 2 − i + 1 I 0 R \ B 2 − i I 0 R .

Estimate of J 1 . Applying Hölder’s inequality, and the fact that F ∈ B p p − 1 , q β ( Ω ) locally, we obtain

J 1 ≤ C ∫ B δ ∣ h ∣ β 2 − α q ∫ B R Δ h F ∣ h ∣ β p p − 1 d x q ( p − 1 ) 2 p d h ∣ h ∣ n 1 q ≤ C ∫ B δ ∫ B R Δ h F ∣ h ∣ β p p − 1 d x q ( p − 1 ) p d h ∣ h ∣ n 1 2 q ∫ B δ ∣ h ∣ ( β − 2 α ) q − n d h 1 2 q ≤ C ∫ B δ ∫ B R Δ h F ∣ h ∣ β p p − 1 d x q ( p − 1 ) p d h ∣ h ∣ n 1 2 q ∫ 0 δ ρ ( β − 2 α ) q − 1 d ρ 1 2 q ≤ C ∥ F ∥ B p p − 1 , q β ( B 2 R ) 1 2 < + ∞ .

Estimate of J 2 . Using assumptions (1.10)–(1.11) and Hölder’s inequality, we obtain

J 2 ≤ C ∑ i = i 0 ∞ ∫ C i ∣ h ∣ ( β − α ) q ∫ B R + ∣ h ∣ ∣ g i ∣ n β d x β q n d h ∣ h ∣ n 1 q ≤ C ∑ i = i 0 ∞ ‖ g i ‖ L loc n β , ϑ ( Ω ) q ∫ C i ∣ h ∣ ( β − α ) q d h ∣ h ∣ n 1 q ≤ C ∑ i = i 0 ∞ ‖ g i ‖ L loc n β , ϑ ( Ω ) q 1 q < + ∞ .

Consequently, we can prove that V p ( D u ) ∈ B 2 , q α ( Ω ) locally for 2 < p p − 1 < n β , 0 < α < β 2 and 1 < q ≤ p p − 1 β * ≔ n p p − 1 n − β p p − 1 . So we can complete the proof of the result of Theorem 1.2.□

Acknowledgement

The author wishes to thank the anonymous reviewers for many valuable comments and suggestions to improve the expressions.

Funding information: The author would like to thank the Newtouch Center for Mathematics of Shanghai University for the support of this work.
Author contribution: The author independently completed the writing and revision of the paper.
Conflict of interest: The author states no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2024-01-02

Revised: 2024-05-27

Accepted: 2024-08-22

Published Online: 2024-11-08

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/anona-2024-0044

Keywords for this article

Besov; fractional; regularity; p-Laplacian; elliptic; weak solutions

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