In this article, we investigate the regularities of solutions to a class of nonlinear elliptic equations with measure data. These equations involve the N-functions, and the solutions belong to the Sobolev-Orlicz spaces. Through the application of comparison arguments, Caccioppoli-type inequality, and maximal estimate, we derive pointwise Riesz potential estimates for both the gradient of the solutions and the solutions themselves. Furthermore, we establish Hölder continuity estimates for the solutions.
Citation: Zhaoyue Sui, Feng Zhou. Pointwise potential estimates for solutions to a class of nonlinear elliptic equations with measure data[J]. AIMS Mathematics, 2025, 10(4): 8066-8094. doi: 10.3934/math.2025370
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In this article, we investigate the regularities of solutions to a class of nonlinear elliptic equations with measure data. These equations involve the N-functions, and the solutions belong to the Sobolev-Orlicz spaces. Through the application of comparison arguments, Caccioppoli-type inequality, and maximal estimate, we derive pointwise Riesz potential estimates for both the gradient of the solutions and the solutions themselves. Furthermore, we establish Hölder continuity estimates for the solutions.
This article deals with the nonlinear elliptic equation with measure data of the type
−divA(x,∇u)=μin Ω. | (1.1) |
Here Ω⊂Rn (n≥2) is a bounded domain, the unknown u∈W1,P(Ω) with an N-function P(⋅) which will be introduced in Section 2, and A(x,∇u)∈L1loc(Ω,Rn). In (1.1), μ is a Radon measure defined on Ω with finite total mass |μ|(Ω)<∞. Moreover, if the measure μ is actually an L1-function, then
|μ|(Q):=∫Q|μ(x)|dx | (1.2) |
for a measurable subset Q⊂Ω. We assume that measure μ satisfies the following condition:
(M) There exists some θ0∈(0,n−1) such that
|μ|(B(x0,2R))≤(2R)θ0 | (1.3) |
holds for all x0∈Ω and every R>0.
In this article, we assume that the Carathéodory vector field A:Ω×Rn→Rn is C1-regular in the gradient variable ξ, and satisfies A(x,0)=0, and the growth, ellipticity, and continuity conditions, i.e., there are constants 0≤s≤1, 0<ν≤L, and K≥1 such that
(A1) {|A(x,ξ)|+|Aξ(x,ξ)|(|ξ|2+s2)12≤LP((|ξ|2+s2)12)(|ξ|2+s2)12,ν−1P(|ξ2−ξ1|)≤⟨A(x,ξ2)−A(x,ξ1),ξ2−ξ1⟩,|A(x,ξ)−A(y,ξ)|≤Kω(|x−y|)P((|ξ|2+s2)12)(|ξ|2+s2)12 |
for any x,y∈Ω and ξ,ξ1,ξ2∈Rn. In (A1), the function ω:[0,∞)→[0,∞) has the following hypothesis.
(A2)ωis a non-decreasing concave function such thatω(0)=limρ↘0ω(ρ)=0andω(⋅)≤1. |
Moreover, ω is assumed to satisfy the Dini-continuous condition:
(A3) d(R):=∫R0ω(ρ)dρρ<∞ |
for every 0<R≤1.
A significant example of (1.1) is the p-Laplacian type equation, for which s=0, p∈(1,∞), P(x)=xp, and A(∇u)=|∇u|p−2∇u. Then Eq (1.1) can be expressed as
−div(|∇u|p−2∇u)=μ. |
The relevant research on the regularities of solutions to elliptic equations starts with Kilpeläinen and Malý [1,2], and extends with a different technique by Trudinger and Wang [3]. Later, Duzaar and Mingione make a further study in [4]. Those results show a standard fact that solutions to non-homogeneous p-Laplacian-type equations with measure data can be pointwise estimated in a natural way by involving the classical nonlinear Wolff potential Wμβ,p(x,R) [5], that is,
Wμβ,p(x,R):=∫R0(|μ|(B(x,ρ))ρn−βp)1p−1dρρ,β∈(0,np]. |
Based on the relationship between Wolff potential and Riesz potential, Mingione et al. [6,7] find out pointwise gradient estimates hold for general quasilinear degenerate equations by applying the Riesz potential
Iμβ(x,R):=Wμβ2,2(x,R)=∫R0|μ|(B(x,ρ))ρn−βdρρ,β>0. | (1.4) |
Baroni proves pointwise gradient bounds for solutions in terms of linear Riesz potentials in [8]. In addition, the caloric Riesz potential serves as a means for pointwise estimation of the spatial gradient of solutions to nonlinear degenerate parabolic equations [9].
Further, pointwise gradient estimates via the nonlinear Wolff potentials for weak solutions to various quasilinear elliptic equations with measure data are obtained by Mingione [10] and Yao [11]. More generally, pointwise potential estimates for elliptic equations and systems with Orlicz growth are studied in [12,13,14], respectively.
In recent years, a great deal of effort has gone into investigating nonlinear elliptic equations and systems involving measure data. Chilebicka et al. [15] study estimates including precise continuity and Hölder continuity criteria by the means of potential of a Wolff type; they also provide regularity estimates of the solutions and their gradients in the generalized Marcinkiewicz scale [16]. The existence of solutions in the framework of renormalized solutions is introduced in [17]. There are many interesting results in [18,19,20].
In this article, a weak solution to (1.1) is a function u∈W1,P(Ω) such that
∫Ω⟨A(x,∇u),∇φ⟩dζ=∫Ωφdμ, | (1.5) |
whenever φ∈C∞0(Ω). Inspired by Mingione et al. [4,10,21], the main objective is to present pointwise potential estimates and interior Hölder continuity of weak solutions to (1.1) by using (1.4) in the Sobolev-Orlicz spaces.
We state our pointwise estimates in Theorems 1.1 and 1.2. It is important to note that θ0 in the following theorems will be introduced in (1.3). The first main result is the gradient pointwise estimates of u as follows.
Theorem 1.1. Let u∈C1(Ω)∩W1,P(Ω) be a weak solution to (1.1) under the assumptions (A1), (A2), (A3), Radon measure μ satisfy μ∈L1(Ω) and (M), and P be an N-function with Δ2(P,˜P)<∞ satisfying Assumption 2.4 and (2.12). There exists a constant C≡C(n,ν,L,K,s,θ0,CΔ2,C1)>0 and a positive radius ˜R<1 such that the pointwise estimate
P(|∇u(x0)|)≤C−∫B(x0,R)[P(|∇u|)+P(s)]dx+CIμn−θ0(x0,4R) | (1.6) |
holds whenever B(x0,R)⊂Ω and 0<R≤˜R. In (1.6), −∫ denotes integral average, Δ2-condition, CΔ2, and C1 will be described in Definition 2.3 and Lemma 2.5, respectively.
On the basis of Theorem 1.1, we demonstrate the pointwise estimate of u.
Theorem 1.2. Let u∈C0(Ω)∩W1,P(Ω) be a weak solution to (1.1) under the assumptions (A1)1, (A1)2, Radon measure μ satisfy μ∈L1(Ω) and (M), and P be an N-function with Δ2(P,˜P)<∞ satisfying Assumption 2.4, and (2.12). There exists a constant C≡C(n,ν,L,θ0,CΔ2,C1,diam(Ω))>0 and a positive radius R<1 such that for every r≤R the pointwise estimate
P(|u(x0)|)≤Cr1+ε0−∫B(x0,r)P(|u|r)dx+CrαIμn−θ0−α(x0,2r) | (1.7) |
holds, where ε0 and α will be introduced in Lemmas 2.5 and 3.3, respectively.
Inspired by Mingione's result [10], the following theorem expounds the Hölder continuity of the solution u.
Theorem 1.3 (Interior Hölder type estimate). Let u∈C0(Ω)∩W1,P(Ω) be a weak solution to (1.1) under the assumptions (A1), (A2), let μ be a Radon measure satisfying μ∈L1(Ω) and (M), and let P be an N-function satisfying Δ2(P,˜P)<∞, Assumption 2.4, and (2.12). Then there exists constants α∈[0,1) and 0<R<1 such that for every x,y∈B(x0,2R)⊂Ω, there holds
P(|u(x)−u(y)|)≤C[Iμn−θ0−α(x,2R)+Iμn−θ0−α(y,2R)+−∫B(x0,2R)P(|u|R)dζ+P(s)]⋅|x−y|α, |
where the constant C depends on n,ν,L,θ0,ω(⋅),CΔ2,C1, and diam(Ω).
Remark 1.4. Drawing upon the Riesz potential, our results present pointwise estimates and Hölder continuity within the more generalized framework of Sobolev-Orlicz spaces. Notably, (1.6) provides an estimate of ∇u, whereas (1.7) estimates u itself. We shall leverage Lemma 2.11 to relate u to ∇u.
In this article, we adopt several technical tools and methods in Sobolev-Orlicz spaces, and explore the properties of solutions of homogeneous equations to those of inhomogeneous equations with measure data. We first establish Proposition 3.2, which reveals the density of the Riesz potential and serves as a crucial conclusion in facilitating the proof of subsequent comparison lemmas. The proof of the comparison estimate is divided into two intricate steps; the first step requires Sobolev-type embedding, while the second primarily employs scaling changes. Our primary objective is to prove pointwise potential estimates and Hölder continuity, which is shown essentially by oscillation estimates of solutions. By utilizing summation methods and involving the Riesz potential, we proceed from deriving the estimates of the gradient of u to those of u itself. Ultimately, we employ the sharp maximal functions, and achieve the interior Hölder estimates of solutions.
This article is organized as follows: In Section 2, we state fundamental tools and definitions such as N-functions and maximal functions. Section 3 is devoted to the proof of Lemmas 3.4 and 3.9, and Section 4 presents supporting results, to gather Caccioppoli-type inequality and maximal estimate towards the proof of the main theorems. In the last section, we present the proofs of Theorems 1.1–1.3, respectively.
In this section, we give the definitions and tools of N-functions, function spaces; classical inequalities, and maximal functions.
The following definitions and results are standard in the context of N-function; see [22].
Definition 2.1. A function P:[0,∞)→[0,∞) is said to be an N-function, if P is convex, differentiable, and its derivative P′ is a right continuous, non-decreasing function satisfying that P′(0)=0 and P′(t)>0 for t>0.
Definition 2.2. The complementary function ˜P:[0,∞)→[0,∞) is defined by
˜P(x)=supt≥0[xt−P(t)]. |
Definition 2.3. We say that P satisfied the Δ2-condition, if there exists CΔ2>0 such that
P(2t)≤CΔ2P(t) |
for all t≥0. By Δ2(P) we denote the smallest constant CΔ2. The function P is said to satisfy the ∇2-condition, if ˜P∈Δ2. Then we define
Δ2(P,˜P):=max{Δ2(P),Δ2(˜P)}. | (2.1) |
By the Δ2-condition of Definition 2.3, we can easily obtain
P(x+y)≤CΔ2[P(x)+P(y)] | (2.2) |
for every x,y≥0.
Assumption 2.4. Let P be a convex function that satisfies Δ2(P,˜P)<∞ as (2.1), and P is C2 on (0,∞). Moreover, let P′(0)=0, limt→∞P′(t)=∞ and uniformly in t≥0
P′(t)∼tP″(t). |
Assumption 2.4 assures that P is an N-function.
We let P−1:[0,∞)→[0,∞) be the right-continuous inverse function of P, and (P′)−1:[0,∞)→[0,∞) the inverse function of P′. Then ~˜P(t)=P(t) and (˜P)′(t)=(P′)−1(t) hold. By [23], one has
P(t)∼tP′(t), and ˜P(P′(t))∼P(t) | (2.3) |
hold uniformly in t≥0. By (2.2) and (2.3), we have
P′(x+y)≤CΔ2[P′(x)+P′(y)]. | (2.4) |
Lemma 2.5. [23] Let P be an N-function with Δ2(P,˜P)<∞. Then there exist ε0>0, C1>0 which only depend on Δ2(P,˜P) such that for all t≥0 and all λ∈[0,1], one has
P(λt)≤C1λ1+ε0P(t). | (2.5) |
From Definition 2.2 of N-function ˜P(⋅), it is easy to obtain that (2.5) holds for ˜P, i.e.,
˜P(λt)≤C1λ1+ε0˜P(t). | (2.6) |
Using (2.3) and (2.5), it is not difficult to obtain
P′(λt)≤C1λε0P′(t). | (2.7) |
In this article, we denote a∼b by Cb≤a≤C′b for two constants C and C′.
Lemma 2.6. [23] Let P be an N-function under Assumption 2.4, and A satisfies the continuity and growth condition of (A1)1, (A1)2. Then
(A(x,ξ1)−A(x,ξ2))⋅(ξ1−ξ2) ∼ P(|ξ1−ξ2|) ∼ |ξ1−ξ2|P′(|ξ1|+|ξ2|). |
Moreover,
A(x,ξ)⋅ξ∼P(|ξ|) |
uniformly in ξ∈Rn and x∈Ω.
Example 2.7. [24] Assume that P is a Young function such that
P(t)≈tp1(logt)p2,t≫1, |
where p1>1 and p2∈R. The derivative P′ of P is as follow:
P′(t)≈tp1−1(log(1+t))p2nearinfinity. |
The complementary function ˜P satisfies
˜P(t)≈tp1p1−1(log(t))−p2p1−1nearinfinity. |
It is not difficult to verify that P(⋅) is an N-function satisfying Definitions 2.1–2.3, and Assumption 2.4.
In this article, we need the following definitions of function spaces. The classical Orlicz spaces LP(Rn) with its norm are given via [22]
LP(Rn)={f∈L1(Rn)|∫RnP(|f(x)|)dx<∞}, |
and
‖f(x)‖LP(Rn)=inf{k>0|∫RnP(f(x)k)dx≤1}. | (2.8) |
If ‖f(x)‖LP(Rn) is finite, then f(x)∈LP(Rn). The Sobolev-Orlicz spaces W1,P(Rn) and its norm are given by [25]
W1,P(Rn)={f∈LP(Rn)|∇f∈LP(Rn)}, |
and
‖f(x)‖W1,P(Rn)=‖f‖LP(Rn)+‖∇f‖LP(Rn). |
If ‖f(x)‖W1,P(Rn) is finite, then f(x)∈W1,P(Rn). Both LP(Rn) and W1,P(Rn) are Banach spaces.
In this subsection, we recall several classical inequalities.
Lemma 2.8. (Young's inequality [26]) For all ε>0, there exist Cε, ˜Cε depending on Δ2(P,˜P), such that for all ζ1,ζ2≥0, there holds
ζ1⋅ζ2≤εP(ζ1)+Cε˜P(ζ2), | (2.9) |
and
ζ1⋅ζ2≤ε˜P(ζ1)+˜CεP(ζ2). | (2.10) |
Lemma 2.9. Let P be an N-function with the Δ2-condition. Then for all ε>0, there exists a constant C such that
|P(x)−P(y)|≤εCΔ2P(y)+CP(|x−y|) |
for x,y>0, where the constant C depends on CΔ2 and C1.
Proof. By the mean value theorem, for x>y>0, there exists λ0∈(0,1) such that
P(x)−P(y)=P′[λ0x+(1−λ0)y](x−y)=P′[y+λ0(x−y)](x−y). |
We use (2.4) and (2.7) to get
P(x)−P(y)≤CΔ2[P′(y)(x−y)+C1λε00P′(x−y)(x−y)]. |
Then applying (2.3) and (2.10), we obtain that
|P(x)−P(y)|≤CΔ2[P′(y)|x−y|+C1λε00P(|x−y|)]≤εCΔ2˜P(P′(y))+˜CεP(|x−y|)+CP(|x−y|)≤εCΔ2P(y)+CP(|x−y|). |
We complete the proof of Lemma 2.9.
Let B be a measurable set with positive measure, and f:B→Rn a measurable function. We denote the integral average of f by
(f)B=−∫Bf(x)dx=1|B|∫Bf(x)dx. |
Lemma 2.10. (Jesen's inequality [27]) Let P be an N-function with Δ2(P,˜P)<∞. If f∈W1,P(B(x,R)), then there exists C≡C(n) for B(x,R)⊂Ω such that
P(|−∫B(x,R)fdζ|)≤C−∫B(x,R)P(|f|)dζ. |
Lemma 2.11. (Sobolev-Poincaré's inequality [27]) Let P be an N-function with Δ2(P,˜P)<∞ and satisfy Assumption 2.4. If f∈W1,P(B(x,R)), then there exist 0<θ1<1 and C>0 such that
−∫B(x,R)P(|f−(f)B(x,R)|R)dζ≤C(−∫B(x,R)Pθ1(|∇f|)dζ)1θ1 | (2.11) |
holds whenever B(x,R)⊂Ω.
The following lemma describes an embedding into a space of continuous functions; see Theorem 8.39 in [28] and (2.22) in [24].
Lemma 2.12. If an N-function P satisfies that
∫∞1P−1(x)xn+1ndx<∞, | (2.12) |
then W1,P0(Ω)↪C0(Ω)∩L∞(Ω), that is, there exists a constant C≡C(n) such that
‖f‖L∞(Ω)≤C‖f‖W1,P0(Ω)∼C‖∇f‖LP(Ω) |
for all f∈W1,P(Ω).
The following iteration lemma plays an essential role in proving the Caccioppoli-type inequality (4.1).
Lemma 2.13. (Iteration lemma [29]) Let f:[γR,R]→[0,∞) be a bounded function such that the inequality
f(ϱ)≤12f(r)+C2(r−ϱ)κ |
holds for fixed constants C2,κ≥0, and γR≤ϱ≤r≤R with 0<γ<1. Then we have
f(γR)≤C[(1−γ)R]κ |
for a constant C depending only on κ.
Throughout this subsection, we provide powerful tools for Hölder estimate of Theorem 1.3. Analogously to the definition of the classical maximal operator in [30], we define the generalized maximal operator and sharp maximal function in Sobolev-Orlicz spaces.
Definition 2.14. [31] Let −1<τ<n, R<dist(x,∂Ω),x∈Ω, where ∂Ω is the boundary of Ω. Let f be a function in Orlicz space LP(Ω) or a measure with finite mass, and P be an N-function with Δ2(P,˜P)<∞. The function defined by
MPτ,R(f)(x):=sup0<r≤Rrτ−∫B(x,r)P(|f|)dζ | (2.13) |
is called the restricted fractional τ generalized maximal function of f.
Definition 2.15. [31] Let β∈(0,1), x∈Ω, and R<dist(x,∂Ω), let f∈LP(Ω) and P be an N-function with Δ2(P,˜P)<∞. The function defined by
M#,Pβ,R(f)(x):=sup0<r≤Rr−β−∫B(x,r)P(|f−(f)B(x,r)|)dζ |
is called the restricted fractional β generalized sharp maximal function of f. For θ>0, we also denote
˜M#,Pθ,R(f)(x):=sup0<r≤Rrθ−∫B(x,r)P(|f−(f)B(x,r)|r)dζ. | (2.14) |
The following note gives us a connection between maximal functions and sharp maximal functions.
Remark 2.16. Combining the generalized sharp maximal functions and Lemma 2.11, we see that if Assumption 2.4 holds, then it follows that
˜M#,Pθ,R(f)(x)≤Csup0<r≤Rrθ(−∫B(x,r)Pθ1(|∇f|)dζ)1θ1. | (2.15) |
In this section, we collect the relevant difference estimates and decay estimates, and consider the density of the Riesz potential Iμβ explicitly. We define v∈u+W1,P0(B(x0,2R)) as the unique solution to the homogeneous Dirichlet problem
{divA(x,∇v)=0 in B(x0,2R),v=u on∂B(x0,2R). | (3.1) |
The existence of v is guaranteed by a standard monotonicity argument; see [13]. One obtains the following control estimate.
Lemma 3.1. Let u∈W1,P(Ω) be as in Theorem 1.1 satisfying the continuity and growth condition of (A1), v∈u+W1,P0(B(x0,2R)) be a solution to (3.13)1, let P be an N-function with Δ2(P,˜P)<∞ and satisfy Assumption 2.4. Then the following estimate
−∫B(x0,R)P(|∇v|)dx≤C−∫B(x0,2R)P(|∇u|+s)dx | (3.2) |
holds, where C depends on n,ν,s,L.
We consider the estimate of the difference of a solution to (1.1), and that of the corresponding solution to the Dirichlet problem (3.1).
Proposition 3.2. Let u∈W1,P(Ω) be as in Theorem 1.1 satisfying (A1)2, v∈u+W1,P0(B(x0,2R)) be a solution to (3.1), let P be an N-function with Δ2(P,˜P)<∞, and satisfy Assumption 2.4, (2.12). Radon measure μ satisfies (M). There exists a constant C≡C(n,ν,C1) such that
−∫B(x0,2R)P(|∇u−∇v|)dx≤C|μ|(B(x0,2R))(2R)θ0. | (3.3) |
Proof. Without loss of generality, we first assume that B(x0,2R)≡B(0,1) and |μ|(B(0,1))=1. Then we shall remove these two conditions for general situations.
Step 1. We assume that B(x0,2R)≡B(0,1) with |μ|(B(0,1))=1. By choosing φ≡u−v as a test function in (1.5), and using (1.1) and (3.1)1, we obtain
∫B(0,1)⟨A(x,∇u)−A(x,∇v),∇u−∇v⟩dx=∫B(0,1)(u−v)dμ. | (3.4) |
By using the ellipticity assumption (A1)2, we deduce that
ν−1∫B(0,1)P(|∇u−∇v|)dx≤∫B(0,1)⟨A(x,∇u)−A(x,∇v),∇u−∇v⟩dx. |
By (3.4) and Lemma 2.12, it follows that
ν−1∫B(0,1)P(|∇u−∇v|)dx≤∫B(0,1)(u−v)dμ≤supB(0,1)|u−v|⋅|μ|(B(0,1))≤C‖∇u−∇v‖LP(B(0,1)). |
That is, there exists a constant C3 such that
∫B(0,1)P(|∇u−∇v|)dx≤C3‖∇u−∇v‖LP(B(0,1)). | (3.5) |
We claim that there exists a positive constant C≡C(n,ν) such that
∫B(0,1)P(|∇u−∇v|)dx≤C | (3.6) |
with 2R=1 and |μ|(B(0,1))=1. According to (2.8), we write ‖∇u−∇v‖LP(B(0,1))=k. We shall prove (3.6) in the following two scenarios. On the one hand, if 0<k≤1, then it follows from (3.5) that
∫B(0,1)P(|∇u−∇v|)dx≤C3k≤C. |
On the other hand, if k>1, then we set k=(C1C3)−(2+ε0), where C1 and C3 are given in (2.5) and (3.5), respectively. It is obvious that 1C1C3>1, and so
∫B(0,1)(C1C3)1+ε0P(|∇u−∇v|)dx≤(C1C3)1+ε0C31(C1C3)2+ε0=1C1. |
By involving λ=C1C3<1 in (2.5), we obtain that
∫B(0,1)P(C1C3|∇u−∇v|)dx≤C1∫B(0,1)(C1C3)1+ε0P(|∇u−∇v|)dx≤1. | (3.7) |
By (3.7) and (2.8), one finds that the norm k≤1C1C3, that is, 1C1C3≤1, which is contradictory to the condition 1C1C3>1. Therefore, the inequality (3.6) holds.
Step 2. Scaling procedures.
We assume that |μ|(B(x0,2R))=1, and we shall reduce to the case B(x0,2R)≡B(0,1) by a standard scaling argument. By letting
˜u(y):=u(x0+2Ry)2R,˜v(y):=v(x0+2Ry)2R,˜A(y,ξ):=(2R)n−θ0−1A(x0+2Ry,ξ),˜μ(y):=(2R)n−θ0μ(x0+2Ry) |
for y∈B(0,1), one has the following equations:
−div˜A(y,∇˜u)=˜μ on B(0,1),div˜A(y,∇˜v)=0 on B(0,1). |
With the definition of Radon measure (1.2), we find the following relation between |˜μ|(B(0,1)) and |μ|(B(x0,2R)), namely,
|˜μ|(B(0,1))=∫B(0,1)˜μ(y)dy = ∫B(0,1)(2R)n−θ0μ(x0+2Ry)dy=1(2R)n∫B(x0,2R)(2R)n−θ0μ(x)dx = |μ|(B(x0,2R))(2R)θ0. |
Next, we shall reduce the general case to the special case |μ|(B(0,1))=1. We define
M=[|˜μ|(B(0,1))]11+ε0=[|μ|(B(x0,2R))(2R)θ0]11+ε0. | (3.8) |
By (1.3), one has M≤1. Hence the new solution, coefficient, and datum become
ˉu:=˜uM,ˉv:=˜vM,ˉA(x,ξ):=˜A(x,Mξ)|˜μ|(B(0,1)),ˉμ:=˜μ|˜μ|(B(0,1)). |
Then we find that
−divˉA(x,∇ˉu)=ˉμ on B(0,1),divˉA(x,∇ˉv)=0 on B(0,1) |
hold in the weak sense and |ˉμ|(B(0,1))=1. Then by applying the result (3.6) in Step 1, one has
∫B(0,1)P(|∇ˉu(x0+2Ry)−∇ˉv(x0+2Ry)|)dy=∫B(0,1)P(|∇˜u(x0+2Ry)−∇˜v(x0+2Ry)|M)dy ≤ C. | (3.9) |
Considering (3.9) on B(x0,2R), then
∫B(0,1)P(|∇ˉu(x0+2Ry)−∇ˉv(x0+2Ry)|)dy=1(2R)n∫B(x0,2R)P(|∇u(x)−∇v(x)|M)dx, |
we apply Lemma 2.5 with λ replaced by M to obtain
−∫B(x0,2R)P(|∇u(x)−∇v(x)|)dx≤C1M1+ε0−∫B(x0,2R)P(|∇u(x)−∇v(x)|M)dx ≤ C|μ|(B(x0,2R))(2R)θ0, | (3.10) |
where C depends on n,ν, and C1. From (3.10), we complete the proof of Proposition 3.2.
Via a classical approach, we have the following estimate inspired by [10].
Lemma 3.3. Let v∈W1,P(Ω) be a weak solution to the Dirichlet problem (3.1) under the assumptions (A1), let P be an N-function satisfying Δ2(P,˜P)<∞, Assumption 2.4 and (2.12). There exist constants α∈(0,1] and C4≡C4(n,ν,L)≥1 such that the estimate
−∫B(x0,ρ)P(|∇v|)dx≤C4(ρR)−1+α−∫B(x0,R)P(|∇v|)dx | (3.11) |
holds whenever B(x0,ρ)⊂B(x0,R)⊂Ω.
We note that the estimate (3.11) is tenable by imitating the proof for the p-Laplacian equations.
Lemma 3.4. Let u∈W1,P(Ω) be a weak solution to (1.1) under the assumptions (A1)2. Let P be an N-function satisfying Δ2(P,˜P)<∞, Assumption 2.4, and (2.12). Then there exist constants ℓ≡ℓ(n,ν,L,CΔ2)≥1 and C≡C(n,ν,L,CΔ2,C1)≥1 such that
−∫B(x0,ρ)P(|∇u|)dx≤ℓ(ρR)−1+α−∫B(x0,R)P(|∇u|)dx+C(Rρ)n|μ|(B(x0,2R))(2R)θ0 |
holds whenever B(x0,ρ)⊂B(x0,R)⊂Ω.
Proof. From the triangle inequality and (2.2), we see that
−∫B(x0,ρ)P(|∇u|)dx≤CΔ2(−∫B(x0,ρ)P(|∇u−∇v|)dx+−∫B(x0,ρ)P(|∇v|)dx)≤CΔ2(Rρ)n−∫B(x0,R)P(|∇u−∇v|)dx+CΔ2−∫B(x0,ρ)P(|∇v|)dx. |
By applying (3.11), one carries out
−∫B(x0,ρ)P(|∇u|)dx≤CΔ2(Rρ)n−∫B(x0,R)P(|∇u−∇v|)dx+ CΔ2C4(ρR)−1+α−∫B(x0,R)P(|∇v|)dx. | (3.12) |
Using the triangle inequality |∇v|≤|∇u−∇v|+|∇u| again, the inequality (3.12) leads to
−∫B(x0,ρ)P(|∇u|)dx≤CΔ2[(Rρ)n+CΔ2C4(ρR)−1+α]−∫B(x0,R)P(|∇u−∇v|)dx+ ℓ(ρR)−1+α−∫B(x0,R)P(|∇u|)dx |
with ℓ=C2Δ2C4. It follows from (3.3) that
−∫B(x0,ρ)P(|∇u|)dx≤C[(Rρ)n+CΔ2C4(ρR)−1+α]|μ|(B(x0,2R))(2R)θ0+ ℓ(ρR)−1+α−∫B(x0,R)P(|∇u|)dx. |
Notice that ρ≤R with
CΔ2C4(ρR)−1+α<C(Rρ)n, |
we complete the proof of Lemma 3.4.
We also define w∈v+W1,P0(B(x0,R)) as the unique solution to the homogeneous Dirichlet problem with frozen coefficients
{divA(x0,∇w)=0 in B(x0,R),w=v on ∂B(x0,R). | (3.13) |
We have the following decay estimate.
Lemma 3.5. Let w∈W1,P(Ω) be a weak solution to (3.13) under the assumption (A1). Then there exist constants ˜α∈(0,1] and C≥1, both depending on n,ν,L, such that
−∫B(x0,ρ)P(|∇w−(∇w)B(x0,ρ)|)dx≤C(ρR)˜α−∫B(x0,R)P(|∇w−(∇w)B(x0,R)|)dx | (3.14) |
holds whenever B(x0,ρ)⊂B(x0,R)⊂Ω.
Notice that the conclusion (3.14) is inspired by [32].
Lemma 3.6. Under the assumptions (A1) and (A2) of Theorem 1.1, with v as in (3.1) and w as in (3.13), there exists a constant C≡C(n,ν,L) such that
−∫B(x0,R)P(|∇v−∇w|)dx≤CKω(R)−∫B(x0,R)P(|∇v|+s)dx | (3.15) |
for B(x0,R)⊂Ω, where K and ω(R) are given in the assumption (A1).
Proof. We test Eq (3.13)1 with v−w. Since both v and w are weak solutions, then the assumption (A1)2 gives us that
Cν−1∫B(x0,R)P(|∇v−∇w|)dx≤∫B(x0,R)⟨A(x0,∇v)−A(x0,∇w),∇v−∇w⟩dx=∫B(x0,R)⟨A(x0,∇v)−A(x,∇v),∇v−∇w⟩dx. |
By using (A1)3 with |x−x0|≤R and Young's inequality (2.9), we derive that
Cν−1∫B(x0,R)P(|∇v−∇w|)dx≤Kω(R)∫B(x0,R)P((|∇v|2+s2)12)(|∇v|2+s2)12|∇v−∇w|dx≤εKω(R)∫B(x0,R)P(|∇v−∇w|)dx+ CεKω(R)∫B(x0,R)˜P[P((|∇v|2+s2)12)(|∇v|2+s2)12]dx. | (3.16) |
Finally, Lemma 3.6 is proved by using the assumptions (A2), (3.16), and (2.3).
Lemma 3.7. Assume that u∈W1,P(Ω) is a weak solution to (1.1) satisfying (A1), (A2), and P is an N-function with Δ2(P,˜P)<∞, and satisfies Assumption 2.4, (2.12). Let w be defined in (3.13), and μ be a Radon measure that satisfies (M). There exists a constant C≡C(n,ν,L,s,CΔ2,C1) such that
−∫B(x0,R)P(|∇u−∇w|)dx≤C|μ|(B(x0,2R))(2R)θ0+CKω(R)−∫B(x0,2R)P(|∇u|+s)dx. | (3.17) |
The key to the proof of Lemma 3.7 is the triangle inequality as follows:
P(|∇u−∇w|)≤CΔ2[P(|∇u−∇v|)+P(|∇v−∇w|)] |
with (3.3), (3.15), and (3.2).
Corollary 3.8. Let u∈W1,P(Ω) be a weak solution to (1.1) under the assumptions (A1), (A2). Let P be an N-function satisfying Δ2(P,˜P)<∞, Assumption 2.4, and (2.12). Then there exists a constant C≡C(n,ν,L,CΔ2,C1)≥1 such that
−∫B(x0,ρ)P(|∇u|)dx≤C−∫B(x0,R)P(|∇u|)dx+C(Rρ)n[|μ|(B(x0,2R))(2R)θ0+Kω(R)−∫B(x0,2R)P(|∇u|)dx] |
holds whenever B(x0,ρ)⊂B(x0,R)⊂B(x0,2R)⊂Ω.
Our goal is to derive an oscillation decay estimate of ∇u. Based on Lemmas 3.5 and 3.7, we first involve the corresponding oscillation decay estimate (3.14) of ∇w, and then compare ∇u and ∇w by (3.17). We note that Lemma 2.10 and the triangle inequality play an essential role in the following lemma.
Lemma 3.9. Let u be a weak solution to (1.1) under the assumptions (A1), (A2), (M), Δ2(P,˜P)<∞, Assumption 2.4, and (2.12). Then there exists C≡C(n,ν,L,s,CΔ2,C1)>0 such that
−∫B(x0,ρ)P(|∇u−(∇u)B(x0,ρ)|)dx≤C(ρR)˜α−∫B(x0,2R)P(|∇u−(∇u)B(x0,2R)|)dx+ CK(Rρ)nω(R)−∫B(x0,2R)P(|∇u|+s)dx+ C(Rρ)n|μ|(B(x0,2R))(2R)θ0 |
for B(x0,ρ)⊂B(x0,2R)⊂Ω. Here the constant ˜α is introduced in Lemma 3.5.
In this section, we use the estimate established in Proposition 3.2 to derive the Caccioppoli-type inequality and the maximal estimate. First, the following Caccioppoli-type inequality gives a connection between ∇u and u.
Proposition 4.1. (Caccioppoli-type inequality) Let u∈W1,P(Ω) be a weak solution to (1.1) with measurable coefficients and satisfy (A1)1, (A1)2. Let μ be a Radon measure with (M). Suppose P is an N-function satisfying Assumption 2.4. Then there exists a constant C≡C(n,ν,L,C1,CΔ2) such that
−∫B(x0,R)P(|∇u|)dx≤C−∫B(x0,2R)P(|u−(u)B(x0,2R)|R)dx+C|μ|(B(x0,2R))(2R)θ0, | (4.1) |
where B(x0,2R)⊂Ω.
Proof. We may assume that (u)B(x0,2R)=0 as if u solves (1.1) also u−(u)B(x0,2R) dose. Let η∈C∞0(B(x0,2R)) such that 0≤η≤1, and
{η=1,inB(x0,R),|∇η|≤1R,inB(x0,2R)∖B(x0,R),η=0,otherwise. | (4.2) |
Let v∈u+W1,P0(B(x0,2R)) be the weak solution to (3.1). We choose a test function φ:=vη to (3.1), and obtain
∫B(x0,2R)⟨A(x,∇v),∇φ⟩dx=0. |
It is clear that
∫B(x0,2R)⟨A(x,∇v),η∇v⟩dx=−∫B(x0,2R)⟨A(x,∇v),v∇η⟩dx. | (4.3) |
By Lemma 2.6 and (4.2), we deduce that
∫B(x0,R)P(|∇v|)dx≤C∫B(x0,R)⟨A(x,∇v),∇v⟩dx≤C∫B(x0,2R)⟨A(x,∇v),η∇v⟩dx. | (4.4) |
By (4.3), (4.4), Cauchy-Schwartz inequality, and Lemma 2.6, one has
∫B(x0,R)P(|∇v|)dx≤∫B(x0,2R)P′(|∇v|)⋅|v||∇η|dx. |
According to Young's inequality (2.10), we derive that for ε>0, there exists ˜Cε such that
∫B(x0,R)P(|∇v|)dx≤ε∫B(x0,2R)˜P(P′(|∇v|))dx+˜Cε∫B(x0,2R)P(|v||∇η|)dx. |
By (2.3) and Lemma 2.13, we deduce that
∫B(x0,R)P(|∇v|)dx≤˜Cε∫B(x0,2R)∖B(x0,R)P(|v|R)dx. |
Dividing by |B(x0,R)|, one gives
−∫B(x0,R)P(|∇v|)dx≤2n˜Cε−∫B(x0,2R)P(|v|R)dx. | (4.5) |
Applying the triangle inequality with P(|∇u|)≤CΔ2(P(|∇u−∇v|)+P(|∇v|)) and (4.5), one has
−∫B(x0,R)P(|∇u|)dx≤2nCΔ2−∫B(x0,2R)P(|∇u−∇v|)dx+2nCΔ2˜Cε−∫B(x0,2R)P(|v|R)dx. | (4.6) |
In order to estimate the last term of (4.6), we use the triangle inequality again to obtain
−∫B(x0,2R)P(|v|R)dx≤CΔ2−∫B(x0,2R)P(|u−v|R)dx+CΔ2−∫B(x0,2R)P(|u|R)dx. | (4.7) |
Lemma 2.11 and the classical Hölder's inequality give us that
−∫B(x0,2R)P(|u−v|R)dx≤C(−∫B(x0,2R)Pθ1(|∇u−∇v|)dx)1θ1≤C−∫B(x0,2R)P(|∇u−∇v|)dx. | (4.8) |
Thus we combine (3.3) and (4.6)–(4.8), and conclude that
−∫B(x0,R)P(|∇u|)dx≤C|μ|(B(x0,2R))(2R)θ0+C−∫B(x0,2R)P(|u|R)dx. |
This establishes the Caccioppoli-type inequality (4.1).
Based on the definition of maximal functions in Section 2.4 and the control estimate in Lemma 3.4, we present the following pointwise estimate involving the maximal functions.
Proposition 4.2. (Maximal estimate) Let u∈W1,P(Ω) be a weak solution to (1.1) under (A1), (A2). Let P be an N-function with Δ2(P,˜P)<∞, Assumption 2.4, and (2.12). Let Radon measure μ satisfy (M). Then there exists a constant C≡C(n,ν,L,θ0,ω(⋅),CΔ2,C1) such that
˜M#,P1+ε0−α,R(u)(x0)+[MPθ1(1+ε0−α)θ1,R(∇u)(x0)]1θ1≤CR1+ε0Iμn−θ0−α(x0,2R)+C,R1+ε0−α−∫B(x0,R)P(|∇u|+s)dx. | (4.9) |
Here Iμn−θ0−α(x0,2R) is a Riesz potential that is introduced in (1.4). In (4.9), the constants ε0, α∈(0,1], θ1 and θ0 are given in Lemmas 2.5, 3.3, 2.11, and (1.3), respectively.
Proof. The key of the proof is to consider the radii R satisfying that R≤R0, where the quantity R0>0 is in dependence of the data n,ν,L,α, and ω(⋅). More precisely, by (A2), we shall choose R0 so that
ω(R0)≤δ, |
where δ will be a small quantity that will be reduced at several stages, as a decreasing function of the quantities n,ν,L, and also α. The proof of Proposition 4.2 is accomplished through two steps, to which the following content is devoted.
By (2.15) and (2.13) with θ=1+ε0−α and τ=(1+ε0−α)θ1, there holds
˜M#,P1+ε0−α,R(u)(x)≤C[MPθ1(1+ε0−α)θ1,R(∇u)(x)]1θ1. | (4.10) |
By using Hölder's inequality, we obtain
[MPθ1(1+ε0−α)θ1,R(∇u)(x)]1θ1=sup0<r≤R(r(1+ε0−α)θ1−∫B(x0,r)Pθ1(|∇u|)dx)1θ1≤sup0<r≤R(r1+ε0−α−∫B(x0,r)P(|∇u|)dx)=MP1+ε0−α,R(∇u)(x0). | (4.11) |
Then the inequality (4.9) will follow if we are able to show that
MP1+ε0−α,R(∇u)(x0)≤CR1+ε0Iμn−θ0−α(x0,2R)+CR1+ε0−α−∫B(x0,R)P(|∇u|+s)dx. | (4.12) |
Step 1. The case for small radii R≤R0.
We take 0<ρ≤r/2≤r≤R, and adopt the estimate in Lemma 3.4 with two radii ρ and r/2. There exists a constant C5≡C5(n,ν,L,CΔ2,C1) such that
−∫B(x0,ρ)P(|∇u|)dx≤C5(ρr)−1+α−∫B(x0,r)P(|∇u|)dx+C(rρ)n|μ|(B(x0,r))rθ0. | (4.13) |
Multiplying both sides of (4.13) by ρ1+ε0−α, and taking S=r/ρ, it follows that
ρ1+ε0−α−∫B(x0,ρ)P(|∇u|)dx≤C5S−ε0r1+ε0−α−∫B(x0,r)P(|∇u|)dx+ CSn+α−ε0−1r1+ε0|μ|(B(x0,r))rθ0+α |
for ρ≤r/2≤R/2. We choose the constant S≥2 large enough, which satisfies that
C5Sε0≤12, |
and take the supremum with 0<r≤R such that the following estimate holds
sup0<r≤R(ρ1+ε0−α−∫B(x0,ρ)P(|∇u|)dx)≤12sup0<r≤R(r1+ε0−α−∫B(x0,r)P(|∇u|)dx)+ CR1+ε0sup0<r≤R|μ|(B(x0,r))rθ0+α, | (4.14) |
where 0<r≤R is equivalent to 0<ρ≤R/S. By (2.13) and (4.14), we obtain
supρ≤R/S(ρ1+ε0−α−∫B(x0,ρ)P(|∇u|)dx)≤12MP1+ε0−α,R(∇u)(x0)+CR1+ε0sup0<r≤R|μ|(B(x0,r))rθ0+α | (4.15) |
with a constant C depending on n,ν,L,CΔ2,C1,S, and α.
On the other hand, we notice that
supR/S≤ρ≤R(ρ1+ε0−α−∫B(x0,ρ)P(|∇u|)dx)≤CSnR1+ε0−α−∫B(x0,R)P(|∇u|+s)dx. | (4.16) |
Recalling the constant S, and putting (4.15) and (4.16) together, we obtain the following:
MP1+ε0−α,R(∇u)(x0)≤12MP1+ε0−α,R(∇u)(x0)+CR1+ε0sup0<r≤R|μ|(B(x0,r))rθ0+α+ CR1+ε0−α−∫B(x0,R)P(|∇u|+s)dx. | (4.17) |
The definition of the supremum shows that for any ε>0, there is r∈(0,R] such that
sup0<r≤R|μ|(B(x0,r))rθ0+α≤|μ|(B(x0,r))rθ0+α+ε. | (4.18) |
This leads to
|μ|(B(x0,r))rθ0+α=|μ|(B(x0,r))rθ0+α1ln2∫2rrdρρ≤2θ0+αln2∫2rr|μ|(B(x,ρ))ρθ0+αdρρ. | (4.19) |
Since ε is arbitrary, and 0<r<2r≤2R, the preceding estimates (4.18) and (4.19) show that there exists a constant C≡C(n,θ0) such that
sup0<r≤R|μ|(B(x0,r))rθ0+α≤C∫2R0|μ|(B(x0,ρ))ρθ0+αdρρ=CIμn−θ0−α(x0,2R). | (4.20) |
Combining (4.17) and (4.20), we deduce the desired estimate (4.12) with R≤R0.
Step 2. Removing the condition R≤R0.
Our goal is to prove (4.12) without the restriction R≤R0. Taking R>R0 and recalling Definition (2.13), it is clear that
MP1+ε0−α,R(∇u)(x0)=sup0<r≤R(r1+ε0−α−∫B(x0,r)P(|∇u|)dx)≤sup0<r≤R0(r1+ε0−α−∫B(x0,r)P(|∇u|)dx)+supR0<r≤R(r1+ε0−α−∫B(x0,r)P(|∇u|)dx)≤MP1+ε0−α,R0(∇u)(x0)+(RR0)nR1+ε0−α−∫B(x0,R)P(|∇u|+s)dx. | (4.21) |
We apply (4.12) with radius R0, i.e.,
MP1+ε0−α,R0(∇u)(x0)≤CR1+ε00Iμn−θ0−α(x0,2R0)+ CR1+ε0−α0−∫B(x0,R0)P(|∇u|+s)dx. | (4.22) |
By the definition of Riesz potential (1.4), one has
R1+ε00Iμn−θ0−α(x0,2R0) = R1+ε00∫2R00|μ|(B(x0,ρ))ρθ0+αdρρ≤R1+ε0∫2R0|μ|(B(x0,ρ))ρθ0+αdρρ = R1+ε0Iμn−θ0−α(x0,2R). | (4.23) |
It is apparent to enlarge the integral by
R1+ε0−α0−∫B(x0,R0)P(|∇u|+s)dx≤(RR0)nR1+ε0−α−∫B(x0,R)P(|∇u|+s)dx. | (4.24) |
By using (4.21)–(4.24), we derive
MP1+ε0−α,R(∇u)(x0)≤R1+ε0Iμn−θ0−α(x0,2R)+(RR0)nR1+ε0−α−∫B(x0,R)P(|∇u|+s)dx. |
Since Ω is bounded, then (4.12) holds.
Combining (4.10)–(4.12), we obtain (4.9), which means the proof of Proposition 4.2 is completed.
By establishing the preceding technical tools and lemmas, we are in a position to present the proofs of the main theorems. We first have the following proof.
Proof of Theorem 1.1. We set a sequence of balls {Bi}∞i=0 by
Bi:=B(x0,Ri)=B(x0,R(2Λ)i), | (5.1) |
where 2Λ>1 will be chosen later. It is clear that Bi+1⊂Bi for every i≥0. We set two sequences {Ki}∞i=0 and {ki}∞i=0 by
Ki:=−∫BiP(|∇u−(∇u)Bi|)dx, ki:=P(|(∇u)Bi|)+P(s). | (5.2) |
We also introduce ˜k0 by
˜k0:=−∫B(x0,R)[P(|∇u|)+P(s)]dx. | (5.3) |
By (2.10), it is obvious that
k0=P(|(∇u)B0|)+P(s)=P(|−∫B(x0,R)∇udx|)+P(s)≤˜k0, | (5.4) |
as well as
K0=−∫B0P(|∇u−(∇u)B0|)dx≤C−∫B(x0,R)P(|∇u|)dx≤˜k0. | (5.5) |
Step 1. An estimate of the summation of Ki.
An application of Lemma 3.9 with B(x0,ρ)≡B(x0,R2Λ)⊂B(x0,R) shows that
−∫B(x0,R2Λ)P(|∇u−(∇u)B(x0,R2Λ)|)dx≤C(12Λ)˜α−∫B(x0,R)P(|∇u−(∇u)B(x0,R)|)dx+ C(2Λ)n|μ|(B(x0,2R))(2R)θ0+C(2Λ)nKω(R)−∫B(x0,R)P(|∇u|+s)dx, |
where the constants C depend on n,ν,L,s,CΔ2, and C1. Using (5.1), we choose Λ≡Λ(n,ν,L,s,CΔ2,C1)>1 large enough such that
C(12Λ)˜α≤14, |
where ˜α∈(0,1] is given in Lemma 3.5. By (2.2), it is clear that
−∫B(x0,R)P(|∇u|+s)dx≤C[−∫B(x0,R)P(|∇u−(∇u)B(x0,R)|)dx+P(|(∇u)B(x0,R)|)+P(s)]. |
Hence there exists a constant C6>0 depending on n,ν,L,K,s,CΔ2, and C1 such that
−∫B(x0,R2Λ)P(|∇u−(∇u)B(x0,R2Λ)|)dx≤(14+C6ω(R))−∫B(x0,R)P(|∇u−(∇u)B(x0,R)|)dx+ C|μ|(B(x0,2R))(2R)θ0+C6ω(R)[P(|(∇u)B(x0,R)|)+P(s)]. | (5.6) |
Using (A2), we take ˜R small enough to obtain
C6ω(˜R)≤14. |
It follows that if R≤˜R, then all Ri≤˜R. Applying the estimate (5.6) with R≡Ri−1, and noting that ω(⋅) is non-decreasing, it yields that
−∫BiP(|∇u−(∇u)Bi|)dx≤12−∫Bi−1P(|∇u−(∇u)Bi−1|)dx+C|μ|(2Bi−1)(2Ri−1)θ0+Cω(Ri−1)[P(|(∇u)Bi−1|)+P(s)], |
which can be simplified as
Ki≤12Ki−1+C|μ|(2Bi−1)(2Ri−1)θ0+Cω(Ri−1)ki−1. | (5.7) |
Via a summation, one deduces that
m∑i=1Ki≤12m−1∑i=0Ki+Cm−1∑i=0|μ|(2Bi)(2Ri)θ0+Cm−1∑i=0ω(Ri)ki |
for C≡C(n,ν,L,K,s,CΔ2,C1) and for every integer m. This implies that
m∑i=1Ki≤K0+2Cm−1∑i=0|μ|(2Bi)(2Ri)θ0+2Cm−1∑i=0ω(Ri)ki | (5.8) |
holds for every m∈N.
Step 2. An estimate of km+1.
Using (5.2), we have
km+1 := m∑i=0(ki+1−ki)+k0 ≤ m∑i=0|P(|(∇u)Bi+1|)−P(|(∇u)Bi|)|+k0. | (5.9) |
By using Lemma 2.9 with x=|(∇u)Bi+1|, y=|(∇u)Bi|, we estimate the difference as
|P(|(∇u)Bi+1|)−P(|(∇u)Bi|)|≤εCΔ2P(|(∇u)Bi|)+CP(||(∇u)Bi+1|−|(∇u)Bi||). |
Considering the triangle inequality ||x|−|y||≤|x−y| and Lemma 2.10, we have
P(||(∇u)Bi+1|−|(∇u)Bi||)≤CP(−∫Bi+1|∇u−(∇u)Bi|dx)≤C(2Λ)n−∫BiP(|∇u−(∇u)Bi|)dx. |
Hence there exists a constant C≡C(n,ν,L,s,CΔ2,C1) such that
|P(|(∇u)Bi+1|)−P(|(∇u)Bi|)|≤εCΔ2P(|(∇u)Bi|)+C−∫BiP(|∇u−(∇u)Bi|)dx. | (5.10) |
For each i, we choose ε=ε(i,n,CΔ2,Λ) small enough such that
εCΔ2≤1(2Λ)i(n+1). |
Then we have
εCΔ2P(|(∇u)Bi|)≤εCΔ2ki ≤ εCΔ2−∫Bi[P(|(∇u)|)+P(s)]dx≤1(2Λ)i(n+1)[(2Λ)i]n−∫B(x0,R)[P(|(∇u)|)+P(s)]dx=1(2Λ)i˜k0. | (5.11) |
Here we use the fact that the sum of geometric series is finite, i.e.,
m∑i=0(12Λ)i≤∞∑i=0(12Λ)i=11−12Λ. | (5.12) |
By combining (5.9)–(5.12), and (5.4), there exists a constant C such that
km+1≤Cm∑i=0[Ki+1(2Λ)i˜k0]+k0≤Cm∑i=0Ki+C˜k0. | (5.13) |
Making use of (5.13), (5.8), and (5.4), one derives that for every integer m≥1, there holds
km+1≤C[K0+m−1∑i=0|μ|(2Bi)(2Ri)θ0+m−1∑i=0ω(Ri)ki+˜k0]≤C[˜k0+m−1∑i=0|μ|(2Bi)(2Ri)θ0+m−1∑i=0ω(Ri)ki]. | (5.14) |
For the second term on the right side of (5.14), one has
m−1∑i=0|μ|(2Bi)(2Ri)θ0≤∞∑i=0|μ|(2Bi)(2Ri)θ0≤|μ|(B(x0,2R))(2R)θ0+∞∑i=0|μ|(2Bi+1)(2Ri+1)θ0. |
Using Λ>1, the method adopted in (4.20), and (1.4), we obtain that
m−1∑i=0|μ|(2Bi)(2Ri)θ0≤2θ0ln2∫4R2R|μ|(B(x0,ρ))ρθ0dρρ+(2Λ)θ0ln(2Λ)∞∑i=0∫2Ri2Ri+1|μ|(B(x0,ρ))ρθ0dρρ≤CIμn−θ0(x0,4R) | (5.15) |
holds with a constant C depending on n and θ0. Inserting (5.15) in (5.14), we obtain the following inequality:
km+1≤C(˜k0+Iμn−θ0(x0,4R))+Cm−1∑i=0ω(Ri)ki. | (5.16) |
Step 3. An induction approach.
By setting
J:=˜k0+Iμn−θ0(x0,4R)=−∫B(x0,R)[P(|∇u|)+P(s)]dx+Iμn−θ0(x0,4R), |
we shall use the mathematical induction to prove
km+1≤CJ. | (5.17) |
Initial Step. If m=−1, then by (5.4), we see that (5.17) is trivial. For the case m=0, (5.17) holds by using (5.13).
Inductive Step. Assuming that (5.17) is valid for any ˜m<m, we shall prove it for m+1. By (5.16), we have
km+1 ≤ CJ+Cm−1∑i=0ω(Ri)ki ≤ CJ+CJm−1∑i=0ω(Ri). |
Due to the fact that ω(⋅) is non-decreasing, we estimate
m−1∑i=0ω(Ri)≤ω(R0)+∞∑i=0ω(Ri+1)≤1ln2∫2RRω(ρ)dρρ+∞∑i=0ω(Ri+1)≤1ln2∫2RRω(ρ)dρρ+1ln(2Λ)∞∑i=0∫RiRi+1ω(ρ)dρρ≤(1ln2+1ln(2Λ))∫2R0ω(ρ)dρρ. |
Considering the fact that Λ>1 and the definition of d(⋅) in (A3), we have
m−1∑i=0ω(Ri)≤2d(2R)ln2. | (5.18) |
By applying (5.18), we complete the proof of the inequality (5.17).
For every Lebesgue point x0 of P(|(∇u)|), we let m→∞, and show that
P(|∇u(x0)|)+P(s)=limm→∞km+1≤C−∫B(x0,R)[P(|∇u|)+P(s)]dx+CIμn−θ0(x0,4R), |
where C depends on n,ν,L,K,s,θ0,CΔ2 and C1. Therefore, (1.6) has been proved.
Via a similar approach to the previous proof, we are in a position to prove Theorem 1.2.
Proof of Theorem 1.2. We introduce a sequence of concentric balls {˜Bi}∞i=0 by
˜Bi:=B(x0,ri)=B(x0,r(2H)i), |
where H>1 is a constant determined later, and r≤R<1. Hence ˜Bi+1⊂B(x0,ri2)⊂˜Bi for every i≥0. We define
Ai:=r1+ε0i−∫˜BiP(|u−(u)˜Bi|ri)dx, and ai:=P(|(u)˜Bi|). |
We also introduce ˜a0 by
˜a0:=r1+ε0−∫B(x0,r)P(|u|r)dx. |
By (2.10) and (2.5), it is obvious that
a0=P(|−∫˜B0udx|)≤−∫B(x0,r)P(|u|)dx≤C1r1+ε0−∫B(x0,r)P(|u|r)dx=C˜a0, | (5.19) |
as well as
A0=r1+ε0−∫˜B0P(|u−(u)˜B0|r)dx≤CΔ2r1+ε0[−∫˜B0P(|u|r)dx+P(|(u)˜B0|r)]≤C˜a0. | (5.20) |
One applies Lemma 2.11 and Hölder's inequality to obtain
\begin{equation*} A_{i+1}\, \leq\, Cr_{i+1}^{1+\varepsilon_0}\left[\rlap{-} \displaystyle {\int}_{\widetilde{B}_{i+1}}P^{\theta_1}(|\nabla u|)\, \mathrm{d}x\right]^{\frac{1}{\theta_1}} \leq\, C\, r_{i+1}^{1+\varepsilon_0}\rlap{-} \displaystyle {\int}_{\widetilde{B}_{i+1}}P(|\nabla u|)\, \mathrm{d}x. \end{equation*} |
Applying Lemma 3.4 with \rho\equiv r_{i+1} , R\equiv \frac{r_i}{2} , one has
\begin{equation} A_{i+1}\leq C\left[r_i\, r_{i+1}^{\varepsilon_0}\left(\frac{2r_{i+1}}{r_i}\right)^{\alpha}\rlap{-} \displaystyle {\int}_{B\left(x_0, \frac{r_i}{2}\right)}P(|\nabla u|)\, \mathrm{d}x+r_{i+1}^{1+\varepsilon_0}\left(\frac{r_i}{2r_{i+1}}\right)^n\frac{|\mu|(\widetilde{B}_{i})}{r_i^{\theta_0}}\right], \end{equation} | (5.21) |
where the constant C depends on n, \, \nu, \, L, \, C_{\Delta_2} , and C_1 . By Caccioppoli-type inequality (4.1) and Definition 2.3, we obtain
\begin{equation} \rlap{-} \displaystyle {\int}_{B\left(x_0, \frac{r_i}{2}\right)}P(|\nabla u|)\, \mathrm{d}x\leq C\rlap{-} \displaystyle {\int}_{\widetilde{B}_i}P\left(\frac{\big|u-(u)_{\widetilde{B}_i}\big|}{r_i}\right)\, \mathrm{d}x+C\, \frac{|\mu|(\widetilde{B}_{i})}{r_i^{\theta_0}}. \end{equation} | (5.22) |
Combining (5.21) and (5.22) with r_{i+1}\leq r_{i} , it follows that
\begin{eqnarray*} A_{i+1}&\leq& C\left\{\left(\frac{1}{H}\right)^{\alpha}r_i^{1+\varepsilon_0}\rlap{-} \displaystyle {\int}_{\widetilde{B}_i}P\left(\frac{\big|u-(u)_{\widetilde{B}_i}\big|}{r_i}\right)\, \mathrm{d}x+\left[r_i^{1+\varepsilon_0}\left(\frac{1}{H}\right)^{\alpha}+r_{i+1}^{1+\varepsilon_0}H^n\right]\frac{|\mu|(\widetilde{B}_{i})}{r_i^{\theta_0}}\right\}\\ &\leq&C_{7}\left(\frac{1}{H}\right)^{\alpha}A_i+C\, \frac{|\mu|(\widetilde{B}_{i})}{r_i^{\theta_0}}, \end{eqnarray*} |
where C_{7} depends on n, \, \nu, \, L, \, C_{\Delta_2}, \, C_1 , and R . By choosing H\equiv H(n, \, \nu, \, L, \, C_{\Delta_2}, \, C_1, \, R) large enough, one has
\begin{equation*} \left(\frac{1}{H}\right)^{\alpha}\leq\frac{1}{2\, C_{7}}, \end{equation*} |
which implies immediately that
\begin{equation*} A_{i+1}\leq\frac{1}{2}\, A_i+C\, \frac{|\mu|(\widetilde{B}_{i})}{r_i^{\theta_0}}. \end{equation*} |
We consider a summation with respect to i from 0 to m-1 , and deduce that
\begin{equation} \sum\limits_{i = 1}^{m}A_i \leq A_0+2\, C\sum\limits_{i = 0}^{m-1}\frac{|\mu|(\widetilde{B}_{i})}{r_i^{\theta_0}}. \end{equation} | (5.23) |
For every non-negative integer m , one writes
\begin{equation*} a_{m+1}\ : = \ \sum\limits_{i = 0}^{m}(a_{i+1}-a_i)+a_0\ \leq\ \sum\limits_{i = 0}^{m}\left|\, P\left(\, \big|(u)_{\widetilde{B}_{i+1}}\big|\, \right)-P\left(\, \big|(u)_{\widetilde{B}_i}\big|\, \right)\, \right|+a_0. \end{equation*} |
Adopting a similar approach as in the proof of (5.10), we have
\begin{eqnarray*} && \left|\, P\left(\, \big|(u)_{\widetilde{B}_{i+1}}\big|\, \right)-P\left(\, \big|(u)_{\widetilde{B}_i}\big|\, \right)\, \right|\\ &\leq& \varepsilon\, C_{\Delta_2}P\left(\, \big|(u)_{\widetilde{B}_i}\big|\, \right)+C\rlap{-} \displaystyle {\int}_{{\widetilde{B}}_i}P\left(\big|u-(u)_{\widetilde{B}_i}\big|\right)\, \mathrm{d}x\\ &\leq& \varepsilon\, C_{\Delta_2}P\left(\, \big|(u)_{\widetilde{B}_i}\big|\, \right)+C\, r_{i}^{1+\varepsilon_{0}}\rlap{-} \displaystyle {\int}_{{\widetilde{B}}_i}P\left(\frac{\big|u-(u)_{\widetilde{B}_i}\big|}{r_{i}}\right)\, \mathrm{d}x\\ & = & \varepsilon\, C_{\Delta_2}\, a_{i}+C\, A_{i}, \end{eqnarray*} |
and choose \varepsilon = (2H)^{-i(n+1)} sufficiently small. With the help of (5.19), it follows that there exists a constant C\equiv C(n, \, \nu, \, L, \, C_{\Delta_2}, \, C_1, \, R) such that
\begin{equation} a_{m+1}\, \leq\, C\sum\limits_{i = 0}^{m}\left[\frac{1}{(2H)^i}\, \widetilde{a}_0+A_i\right]+a_0\, \leq\, C\sum\limits_{i = 0}^{m}A_i+C\, \widetilde{a}_0. \end{equation} | (5.24) |
Analogous to (5.15), we obtain
\begin{equation} \sum\limits_{i = 0}^{m-1}\frac{|\mu|(\widetilde{B}_{i})}{r_i^{\theta_0}}\leq r^{\alpha}\, \sum\limits_{i = 0}^{m-1}\frac{|\mu|(\widetilde{B}_{i})}{r_i^{\theta_0+\alpha}}\leq C\, r^{\alpha}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x_0, 2r). \end{equation} | (5.25) |
Applying (5.23)–(5.25) and (5.20), one gets that for every integer m\geq 1 , there holds
\begin{eqnarray*} a_{m+1}&\leq& C\left[A_{0}+\widetilde{a}_0+\sum\limits_{i = 0}^{m-1}\frac{|\mu|(\widetilde{B}_{i})}{r_i^{\theta_0}}\right]\\ &\leq& C\, r^{1+\varepsilon_0}\rlap{-} \displaystyle {\int}_{B(x_0, r)}P\left(\frac{|u|}{r}\right)\, \mathrm{d}x+C\, r^{\alpha}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x_0, 2r). \end{eqnarray*} |
By the dominated convergence theorem, for every Lebesgue point x_{0}\in P(|u|) , there holds
\begin{equation} P(|u(x_0)|) = \lim\limits_{m\rightarrow \infty}a_{m+1}\leq C\, r^{1+\varepsilon_0}\rlap{-} \displaystyle {\int}_{B(x_0, r)}P\left(\frac{|u|}{r}\right)\, \mathrm{d}x+C\, r^{\alpha}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x_0, 2r), \end{equation} | (5.26) |
where constant C depends on n, \, \nu, \, L, \, \theta_0, \, C_{\Delta_2}, \, C_1 , and \mathrm{diam}(\Omega) .
By establishing (5.26), we are ready to prove the result of Theorem 1.3.
Proof of Theorem 1.3. For any real number g , we observe that if u is a weak solution to (1.1), then u-g is still a solution to (1.1). Let B(x_{0}, 2R)\subset\Omega . We consider x, \, y\in B\left(x_{0}, \frac{R}{2}\right) satisfying that r: = |x-y| < \frac{R}{4} . By (5.26), it follows that
\begin{eqnarray*} P\left(|u(x)-g|\right)&\leq& C\, r^{1+\varepsilon_0}\rlap{-} \displaystyle {\int}_{B(x, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta+C\, r^{\alpha}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x, 2r), \\ P\left(|u(y)-g|\right)&\leq& C\, r^{1+\varepsilon_0}\rlap{-} \displaystyle {\int}_{B(y, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta+C\, r^{\alpha}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(y, 2r). \end{eqnarray*} |
By (1.4), one has the following monotone property:
\begin{equation*} \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x, 2r)\leq \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x, 2R), \ \mbox{and}\ \mathbf{I}_{n-\theta_0-\alpha} ^\mu(y, 2r)\leq \mathbf{I}_{n-\theta_0-\alpha} ^\mu(y, 2R). \end{equation*} |
Via (2.2) and a direct computation, one has
\begin{eqnarray} P\left(\left|u(x)-u(y)\right|\right) &\leq& C_{\Delta_2}\left[P\left(\left|u(x)-g\right|\right)+P\left(\left|u(y)-g\right|\right)\right]\\[0.2cm] &\leq&C\, r^{1+\varepsilon_0}\left[\rlap{-} \displaystyle {\int}_{B(x, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta+\rlap{-} \displaystyle {\int}_{B(y, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta\right]\\[0.1cm] &&+\ C\, r^{\alpha}\left[\mathbf{I}_{n-\theta_0-\alpha} ^\mu(x, 2R)+\mathbf{I}_{n-\theta_0-\alpha} ^\mu(y, 2R)\right]. \end{eqnarray} | (5.27) |
We take g: = (u)_{B(x, 2r)} , and observe that B(x, r)\cup B(y, r)\subset B(x, 2r)\subset B\left(x, \frac{R}{2}\right) . Using Definition 2.3, we deduce that
\begin{equation*} \rlap{-} \displaystyle {\int}_{B(x, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta+\rlap{-} \displaystyle {\int}_{B(y, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta\leq C\, C_{\Delta_2}\rlap{-} \displaystyle {\int}_{B(x, 2r)}P\left(\frac{|u-(u)_{B(x, 2r)}|}{2r}\right)\, \mathrm{d}\zeta. \end{equation*} |
Since 2r < \frac{R}{2} , then (2.14) gives us that
\begin{equation*} \rlap{-} \displaystyle {\int}_{B(x, 2r)}P\left(\frac{|u-(u)_{B(x, 2r)}|}{2r}\right)\, \mathrm{d}\zeta \leq (2r)^{-(1+\varepsilon_0-\alpha)}\cdot\widetilde{M}_{1+\varepsilon_0-\alpha, \, \frac{R}{2}}^{\#, \, P}(u)(x). \end{equation*} |
Then it follows from Proposition 4.2 that
\begin{eqnarray} &&\!\rlap{-} \displaystyle {\int}_{B(x, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta+\rlap{-} \displaystyle {\int}_{B(y, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta\\[0.1cm] &\leq&\! C r^{-(1+\varepsilon_0-\alpha)} R^{1+\varepsilon_0}\left\{\mathbf{I}_{n-\theta_0-\alpha} ^\mu(x, 2R)+R^{-\alpha}\!\left[\rlap{-} \displaystyle {\int}_{B\left(x, \frac{R}{2}\right)}P(|\nabla u|)\, \mathrm{d}\zeta+P(s)\right]\right\}. \end{eqnarray} | (5.28) |
To estimate the last integral, we use Caccioppoli-type inequality (4.1) and (4.19) to obtain that
\begin{eqnarray} \rlap{-} \displaystyle {\int}_{B\left(x, \frac{R}{2}\right)}P\left(|\nabla u|\right)\mathrm{d}\zeta&\leq& C\, C_{\Delta_2}\rlap{-} \displaystyle {\int}_{B(x, R)}P\left(\frac{| u|}{R}\right)\mathrm{d}\zeta+C\, R^{\alpha}\, \frac{|\mu|(B(x, R))}{R^{\theta_0+\alpha}}\\ &\leq&C\rlap{-} \displaystyle {\int}_{B(x, R)}P\left(\frac{| u|}{R}\right)\mathrm{d}\zeta+C\, R^{\alpha}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x, 2R). \end{eqnarray} | (5.29) |
Substituting (5.29) into (5.28) and considering 0\leq\alpha < 1 , it yields that
\begin{eqnarray} &&\!\rlap{-} \displaystyle {\int}_{B(x, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta+\rlap{-} \displaystyle {\int}_{B(y, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta\\ &\leq&\! C r^{-(1+\varepsilon_0-\alpha)} R^{1+\varepsilon_0}\left\{\mathbf{I}_{n-\theta_0-\alpha} ^\mu(x, 2R)+R^{-\alpha}\left[\rlap{-} \displaystyle {\int}_{B(x, R)}P\left(\frac{| u|}{R}\right)\, \mathrm{d}\zeta+P(s)\right]\right\}. \end{eqnarray} | (5.30) |
Combining (5.27)–(5.30) together with B(x, R)\subset B(x_0, 2R) , there is a constant C such that
\begin{eqnarray} && P\left(|u(x)-u(y)|\right) \\[0.1cm] &\leq&C\, r^{\alpha}\left[\mathbf{I}_{n-\theta_0-\alpha} ^\mu(x, 2R)+\mathbf{I}_{n-\theta_0-\alpha} ^\mu(y, 2R)\right]+C\, r^{\alpha}\, R^{1+\varepsilon_0}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x, 2R)\\ &&+\ C\, r^{\alpha}R^{1+\varepsilon_0-\alpha}\left[\rlap{-} \displaystyle {\int}_{B(x_{0}, 2R)}P\left(\frac{|u|}{R}\right)\, \mathrm{d}\zeta+P(s)\right], \end{eqnarray} | (5.31) |
where the constant C depends on n, \, \nu, \, L, \, \theta_0, \, \omega(\cdot), \, C_{\Delta_2}, \, C_1 , and \mathrm{diam}(\Omega) . Since R < 1 and \alpha\in[0, 1) , then the estimate (5.31) is the desired interior Hölder estimate of Theorem 1.3.
In this work, we establish pointwise potential estimates of weak solutions to a class of elliptic equations in divergence form with measure data. Our primary result is to employ the Riesz potential to prove the pointwise estimates of the solutions. The key innovation of this paper manifests in the proof of Proposition 3.2, which enables the relationship between measure data and the Riesz potential in the Sobolev-Orlicz spaces. Furthermore, we obtain Hölder continuity estimates for the solutions by establishing the Caccioppoli-type inequality and the maximal estimate. This systematic approach extends the potential estimates of regularity for nonlinear elliptic equations in the existing literature.
Zhaoyue Sui: Conceptualization, methodology, writing–original draft preparation; Feng Zhou: Supervision, funding acquisition, project administration, writing–review and editing. All authors have read and approved the final version of the manuscript for publication.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors would like to thank the anonymous referee for their careful reading and valuable comments. The authors are supported by the National Natural Science Foundation of China (NNSF Grant No. 12001333) and the Shandong Provincial Natural Science Foundation (Grant No. ZR2020QA005).
The authors declare no conflicts of interest regarding the publication of this article.
[1] |
T. Kilpeläinen, J. Malý, Degenerate elliptic equations with measure data and nonlinear potentials, Ann. Scuola Norm. Sci., 19 (1992), 591–613. https://doi.org/10.1016/0022-1236(89)90005-0 doi: 10.1016/0022-1236(89)90005-0
![]() |
[2] |
T. Kilpeläinen, J. Malý, The wiener test and potential estimates for quasilinear elliptic equations, Acta Math., 172 (1994), 137–161. https://doi.org/10.1007/BF02392793 doi: 10.1007/BF02392793
![]() |
[3] |
N. Trudinger, X. Wang, On the week continuity of elliptic operators and applications to potential theory, Am. J. Math., 124 (2002), 369–410. https://doi.org/10.1353/AJM.2002.0012 doi: 10.1353/AJM.2002.0012
![]() |
[4] |
F. Duzaar, G. Mingione, Gradient estimates in non-linear potential theory, Rend. Lincei. Mat. Appl., 20 (2009), 179–190. https://doi.org/10.4171/RLM/540 doi: 10.4171/RLM/540
![]() |
[5] |
L. Hedberg, T. Wolff, Thin sets in nonlinear potential theory, Ann. I. Fourier, 33 (1983), 161–187. https://doi.org/10.5802/aif.944 doi: 10.5802/aif.944
![]() |
[6] |
T. Kuusi, G. Mingione, Linear potentials in nonlinear potential theory, Arch. Ration. Mech. Anal., 207 (2013), 215–246. https://doi.org/10.1007/s00205-012-0562-z doi: 10.1007/s00205-012-0562-z
![]() |
[7] |
G. Mingione, Gradient potential estimates, J. Eur. Math. Soc., 13 (2011), 459–486. https://doi.org/10.4171/JEMS/258 doi: 10.4171/JEMS/258
![]() |
[8] |
P. Baroni, Riesz potential estimates for a general class of quasilinear equations, Calc. Var., 53 (2015), 803–846. https://doi.org/10.1007/s00526-014-0768-z doi: 10.1007/s00526-014-0768-z
![]() |
[9] |
T. Kuusi, G. Mingione, Riesz potentials and nonlinear parabolic equations, Arch. Ration. Mech. Anal., 212 (2014), 727–780. https://doi.org/10.1007/s00205-013-0695-8 doi: 10.1007/s00205-013-0695-8
![]() |
[10] |
T. Kuusi, G. Mingione, Universal potential estimates, J. Funct. Anal., 262 (2021), 4205–4269. https://doi.org/10.1016/j.jfa.2012.02.018 doi: 10.1016/j.jfa.2012.02.018
![]() |
[11] |
F. Yao, M. Zheng, Gradient estimates via the Wolff potentials for a class of quasilinear elliptic equations, J. Math. Anal. Appl., 452 (2017), 926–940. https://doi.org/10.1016/j.jmaa.2017.03.037 doi: 10.1016/j.jmaa.2017.03.037
![]() |
[12] |
A. Benyaiche, I. Khlifi, Wolff potential estimates for supersolutions of equations with generalized Orlicz growth, Potential Anal., 58 (2023), 761–783. https://doi.org/10.1007/s11118-021-09958-5 doi: 10.1007/s11118-021-09958-5
![]() |
[13] |
V. Bögelein, J. Habermann, Gradient estimates via non-standard potentials and continuity, Ann. Fenn. Mathematici, 35 (2010), 741–678. https://doi.org/10.5186/aasfm.2010.3541 doi: 10.5186/aasfm.2010.3541
![]() |
[14] |
I. Chlebicka, F. Giannetti, A. Zatorska-Goldstein, Wolff potentials and local behavior of solutions to elliptic problems with Orlicz growth and measure data, Adv. Calc. Var., 17 (2024), 1293–1321. https://doi.org/10.1515/acv-2023-0005 doi: 10.1515/acv-2023-0005
![]() |
[15] |
I. Chlebicka, Y. Youn, A. Zatorska-Goldstein, Wolff potentials and measure data vectorial problems with Orlicz growth, Calc. Var., 62 (2023), 64. https://doi.org/10.1007/s00526-022-02402-5 doi: 10.1007/s00526-022-02402-5
![]() |
[16] |
I. Chlebicka, Y. Youn, A. Zatorska-Goldstein, Measure data systems with Orlicz growth, Ann. Mat., 204 (2025), 407–426. https://doi.org/10.1007/s10231-024-01489-1 doi: 10.1007/s10231-024-01489-1
![]() |
[17] |
T. Gkikas, Quasilinear elliptic equations involving measure valued absorption terms and measure data, J. Anal. Math., 153 (2024), 555–594. https://doi.org/10.1007/s11854-023-0321-0 doi: 10.1007/s11854-023-0321-0
![]() |
[18] |
P. Tran, N. Nguyen, A global fractional Caccioppoli-Type estimate for solutions to nonlinear elliptic problems with measure data, Stud. Math., 263 (2022), 323–338. https://doi.org/10.4064/sm201121-12-3 doi: 10.4064/sm201121-12-3
![]() |
[19] |
H. Cheng, F. Zhou, Besov estimates for sub-elliptic equations in the Heisenberg group, Adv. Pure Math., 14 (2024), 744–758. https://doi.org/10.4236/apm.2024.149039 doi: 10.4236/apm.2024.149039
![]() |
[20] |
T. Gkikas, Nonlinear nonlocal equations involving subcritical or power nonlinearities and measure data, Math. Eng., 6 (2024), 45–80. https://doi.org/10.3934/mine.2024003 doi: 10.3934/mine.2024003
![]() |
[21] |
F. Duzaar, G. Mingione, Gradient continuity estimates, Calc. Var. Partial Differ. Equ., 39 (2010), 379–418. https://doi.org/10.1007/s00526-010-0314-6 doi: 10.1007/s00526-010-0314-6
![]() |
[22] | J. Musielak, Orlicz spaces and modular spaces, Berlin, Heidelberg: Springer, 1983. https://doi.org/10.1007/BFb0072210 |
[23] |
L. Diening, F. Ettwein, Fractional estimates for non-differentiable elliptic systems with general growth, Forum Math., 20 (2008), 523–556. https://doi.org/10.1515/FORUM.2008.027 doi: 10.1515/FORUM.2008.027
![]() |
[24] |
G. Barletta, Existence and regularity results for nonlinear elliptic equations in Orlicz spaces, Nonlinear Differ. Equ. Appl., 31 (2024), 29. https://doi.org/10.1007/s00030-024-00922-x doi: 10.1007/s00030-024-00922-x
![]() |
[25] |
A. Cianchi, A sharp embedding theorem for Orlicz-Sobolev spaces, Indiana Univ. Math. J., 45 (1996), 39–66. https://doi.org/10.1512/IUMJ.1996.45.1958 doi: 10.1512/IUMJ.1996.45.1958
![]() |
[26] |
L. Diening, P. Kaplicky, S. Schwarzacher, Campanato estimates for the generalized Stokes system, Ann. Mat. Pura Appl., 193 (2014), 1779–1794. https://doi.org/10.1007/s10231-013-0355-5 doi: 10.1007/s10231-013-0355-5
![]() |
[27] |
L. Diening, P. Kaplicky, S. Schwarzacher, BMO estimates for the p-Laplacian, Nonlinear Anal. Theor., 75 (2012), 637–650. https://doi.org/10.1016/j.na.2011.08.065 doi: 10.1016/j.na.2011.08.065
![]() |
[28] | R. A. Adams, J. J. F. Fournier, Sobolev spaces, 2 Eds., New York: Academic Press, 2003. |
[29] | E. Giusti, Direct methods in the calculus of variations, World Scientific, 2003. https://doi.org/10.1142/5002 |
[30] | B. Simon, Harmonic analysis: A comprehensive course in analysis, Part 3, American Mathematical Society, 2015. |
[31] | J. Kinnunen, Sobolev spaces, Aalto University, 2025. |
[32] |
F. Duzaar, G. Mingione, Gradient estimates via non-linear potentials, Am. J. Math., 133 (2009), 1093–1149. https://doi.org/10.1353/AJM.2011.0023 doi: 10.1353/AJM.2011.0023
![]() |