Zeros for the Gradients of Weakly 𝐴-Harmonic Tensors

Abstract

The Caccioppoli inequality of weakly A-harmonic tensors has been proved, which can be used to consider the weak reverse Hölder inequality, regularity property, and zeros of weakly A-harmonic tensors.

1. Introduction

In this paper, we consider the 𝐴-harmonic equation for differential forms 𝑑^∗𝐴(𝑥,𝑑𝑢)=0, (1) where 𝐴∶Ω×∧^𝑙(𝐑^𝑛)→∧^𝑙+1(𝐑^𝑛) satisfies the conditions 𝐴(𝑥,𝜉)≤𝛽𝜉^𝑝−1,𝐴(𝑥,𝜉),𝜉≥𝛼𝜉^𝑝 (2) for almost every 𝑥∈Ω and all 𝜉∈∧^𝑙(𝐑^𝑛). Here, 𝛼,𝛽>0 are constants and 1<𝑝<∞ is a fixed exponent associated with (1). 𝑢∈𝑊^1,𝑝_loc(Ω,∧^𝑙−1) is an 𝐴-harmonic tensor in Ω if 𝑢 satisfies (1) in Ω.

There has been remarkable work [1–10] in the study of (1). When 𝑢 is a 0-form, that is, 𝑢 is a function, (1) is equivalent to div𝐴(𝑥,∇𝑢)=0. (3) Lots of results have been obtained in recent years about different versions of the 𝐴-harmonic equation; see [11–15].

In 1995, Stroffolini [16] first introduced weakly 𝐴-harmonic tensors and gave the higher integrability result of weakly 𝐴-harmonic tensors. The word weak means that the integrable exponent 𝑟 of 𝑢 is smaller than the natural exponent 𝑝. In 2010, Gao and Wang [17] gave an alternative proof of the higher integrability result of weakly 𝐴-harmonic tensors by introducing the definition of weak 𝑊𝑇₂-class of differential forms.

Definition 1 (see [16, 17]). A very weak solution to (1) (also called weakly 𝐴-harmonic tensor) is an element 𝑢 of the Sobolev space 𝑊^1,𝑟_loc(Ω,∧^𝑙−1) with max{1,𝑝−1}≤𝑟<𝑝 such that ⟨𝐴(_Ω𝑥,𝑑𝑢),𝑑𝜑⟩𝑑𝑥=0 (4) for all 𝜑∈𝑊^{1,(𝑟/(𝑟−𝑝+1))}(Ω,∧^𝑙−1) with compact support.
Under some conditions, the present paper proves that almost every zero for the gradients of weakly 𝐴-harmonic tensor 𝑢 has infinite order. To do this, we need to give the Caccioppoli inequality and the weak reverse Hölder inequality of weakly 𝐴-harmonic tensors.

We keep using the traditional notation.

Let Ω be a connected open subset of 𝐑^𝑛, let 𝑒₁,𝑒₂,…,𝑒_𝑛 be the standard unit basis of 𝐑^𝑛, and let ∧^𝑙=∧^𝑙(𝐑^𝑛) be the linear space of 𝑙-covectors, spanned by the exterior products 𝑒_𝐼=𝑒_𝑖1∧𝑒_𝑖2∧⋯∧𝑒_𝑖𝑙, corresponding to all ordered 𝑙-tuples 𝐼=(𝑖₁,𝑖₂,…,𝑖_𝑙), 1≤𝑖₁<𝑖₂<⋯<𝑖_𝑙≤𝑛, 𝑙=0,1,…,𝑛. Let 𝐑=𝐑¹. The Grassman algebra ∧=⊕∧^𝑙 is a graded algebra with respect to the exterior products. For 𝛼=∑𝛼^𝐼𝑒_𝐼∈∧ and 𝛽=∑𝛽^𝐼𝑒_𝐼∈∧, the inner product in ∧ is given by ⟨𝛼,𝛽⟩=∑𝛼^𝐼𝛽^𝐼 with summation over all 𝑙-tuples 𝐼=(𝑖₁,𝑖₂,…,𝑖_𝑙) and all integers 𝑙=0,1,…,𝑛. The Hodge star operator ⋆∶∧→∧ is denoted by the rules ⋆1=𝑒₁∧𝑒₂∧⋯∧𝑒_𝑛 and 𝛼∧⋆𝛽=𝛽∧⋆𝛼=⟨𝛼,𝛽⟩(⋆1) for all 𝛼,𝛽∈∧. The norm of 𝛼∈∧ is given by the formula |𝛼|²=⟨𝛼,𝛼⟩=⋆(𝛼∧⋆𝛼)∈∧⁰=𝐑. The Hodge star is an isometric isomorphism on ∧ with ⋆∶∧^𝑙→∧^𝑛−𝑙 and ⋆⋆(−1)^{𝑙(𝑛−𝑙)}∶∧^𝑙→∧^𝑙. Balls are denoted by 𝐵 and 𝜌𝐵 is the ball with the same center as 𝐵 and with diam(𝜌𝐵)=𝜌diam(𝐵). We do not distinguish balls from cubes throughout this paper. The 𝑛-dimensional Lebesgue measure of a set 𝐸⊆𝐑^𝑛 is denoted by |𝐸|.

Differential forms are important generalizations of real functions and distributions; note that a 0-form is the usual function in 𝐑^𝑛. A differential 𝑙-form 𝜔 on Ω is a Schwartz distribution on 𝜔 with values in ∧^𝑙(𝐑^𝑛). We use 𝐷^′(Ω,∧^𝑙) to denote the space of all differential 𝑙-forms 𝜔(𝑥)=∑_𝐼𝜔_𝐼(𝑥)𝑑𝑥_𝐼=∑𝜔_{𝑖1𝑖2,…,𝑖𝑙}(𝑥)𝑑𝑥_𝑖1∧𝑑𝑥_𝑖2∧⋯∧𝑑𝑥_𝑖𝑙. We write 𝐿^𝑝(Ω,∧^𝑙) for the 𝑙-forms with 𝜔_𝐼∈𝐿^𝑝(Ω,𝐑) for all ordered 𝑙-tuples 𝐼. Thus, 𝐿^𝑝(Ω,∧^𝑙) is a Banach space with norm ‖𝜔‖_𝑝,Ω=_Ω|𝜔(𝑥)|^𝑝𝑑𝑥^1/𝑝=_Ω𝜔_𝐼(𝑥)^2𝑝/2𝑑𝑥^1/𝑝. (5) For 𝜔∈𝐷^′(Ω,∧^𝑙), the vector-valued differential form ∇𝜔=(𝜕𝜔/𝜕𝑥₁,…,𝜕𝜔/𝜕𝑥_𝑛) consists of differential forms 𝜕𝜔/𝜕𝑥_𝑖∈𝐷^′(Ω,∧^𝑙) where the partial differentiations are applied to the coefficients of 𝜔. As usual, 𝑊^1,𝑝(Ω,∧^𝑙) is used to denote the Sobolev space of 𝑙-forms, which equals 𝐿^𝑝(Ω,∧^𝑙)⋂𝐿^𝑝₁(Ω,∧^𝑙) with norm ‖𝜔‖_𝑊^1,𝑝_(Ω,∧𝑙)=‖𝜔‖_𝑊^1,𝑝_(Ω,∧𝑙)=diam(Ω)⁻¹‖𝜔‖_𝑝,Ω+‖∇𝜔‖_𝑝,Ω. (6) The notations 𝑊^1,𝑝_loc(Ω,𝐑) and 𝑊^1,𝑝_loc(Ω,∧^𝑙) are self-explanatory. We denote the exterior derivative by 𝑑∶𝐷^′(Ω,∧^𝑙)→𝐷^′(Ω,∧^𝑙+1) for 𝑙=0,1,…,𝑛. Its formal adjoint operator 𝑑^⋆∶𝐷^′(Ω,∧^𝑙+1)→𝐷^′(Ω,∧^𝑙) is given by 𝑑^⋆=(−1)^𝑛𝑙+1⋆𝑑⋆ on 𝐷^′(Ω,∧^𝑙+1),  𝑙=0,1,…,𝑛. A differential 𝑙-form 𝑢∈𝐷^′(Ω,∧^𝑙) is called a closed form if 𝑑𝑢=0 in Ω. It is called exact if there exists a differential form 𝛼∈𝐷^′(Ω,∧^𝑙−1) such that 𝑢=𝑑𝛼. Poincaré Lemma implies that exact forms are closed.

From [1, 18], if 𝐷⊂𝐑^𝑛 is a bounded, convex domain, to each 𝑦∈𝐷 there corresponds a linear operator 𝐾_𝑦∶𝐶^∞(𝐷,∧^𝑙)→𝐶^∞(𝐷,∧^𝑙−1) defined by 𝐾_𝑦𝜔𝑥;𝜉₁,…,𝜉_𝑙−1=¹₀𝑡^𝑙−1𝜔𝑡𝑥+𝑦−𝑡𝑦;𝑥−𝑦,𝜉₁,…,𝜉_𝑙−1𝑑𝑡 (7) and a decomposition 𝜔=𝑑(𝐾_𝑦𝜔)+𝐾_𝑦(𝑑𝜔). A homotopy operator 𝑇∶𝐶^∞(𝐷,∧^𝑙)→𝐶^∞(𝐷,∧^𝑙−1) is defined by averaging 𝐾_𝑦 over all points 𝑦 in 𝐷; that is, 𝑇𝜔=_𝐷𝜑𝑦𝐾_𝑦𝜔𝑑𝑦, (8) where 𝜑∈𝐶^∞₀(𝐷) is normalized by ∫_𝐷𝜑(𝑦)𝑑𝑦=1. Then, there is also a decomposition 𝜔=𝑑(𝑇𝜔)+𝑇(𝑑𝜔). (9) The 𝑙-form 𝜔_𝐷∈𝐷^′(𝐷,∧^𝑙) is defined by 𝜔_𝐷=|𝐷|⁻¹_𝐷𝜔𝑦𝑑𝑦if𝑑()𝑙=0𝑇𝜔if𝑙=1,2,…,𝑛 (10) for all 𝜔∈𝐿^𝑝(𝐷,∧^𝑙). Clearly, 𝜔_𝐷 is a closed form and for 𝑙>0, 𝜔_𝐷 is an exact form.

2. The Caccioppoli Inequality of Weakly 𝐴-Harmonic Tensors

We need the following elementary inequality.

Lemma 2 (see [19]). Suppose 𝑋 and 𝑌 are vectors of an inner product space. Then |𝑋|^−𝜀𝑋−|𝑌|^−𝜀𝑌≤1+𝜀21−𝜀^𝜀|𝑋−𝑌|^1−𝜀 (11) for 0≤𝜀<1, and |𝑋|^𝜀𝑋−|𝑌|^𝜀𝑌≤(1+𝜀)(|𝑌|+|𝑋−𝑌|)^𝜀|𝑋−𝑌| (12) for 𝜀≥0.

Next is the caccioppoli inequality of weakly 𝐴-harmonic tensors.

Theorem 3. Let 𝑢∈𝐷^′(Ω,∧^𝑙−1) be a weakly 𝐴-harmonic tensor in a domain Ω∈𝐑^𝑛 and 𝑑𝑢∈𝐿^𝑟(𝐷,∧^𝑙), 𝑙=1,…,𝑛. Then, there exists a constant 𝐶, independent of 𝑢, such that _𝐵𝜌|𝑑𝑢|^𝑟𝛽𝑑𝑥≤𝐶𝑛,𝑝,𝛼_𝐵𝑅𝑢−𝑐𝑅−𝜌^𝑟𝑑𝑥 (13) for all balls 𝐵⊂Ω and all closed forms 𝑐, where 0<𝜌<𝑅.

Proof . Let 𝑢∈𝑊^1,𝑟_loc(Ω,∧^𝑙−1) be a very weak solution of (1). Fix 𝑅₀∶𝑅₀≤𝑑=dist(𝑥₀,𝜕Ω) for all 𝑥₀∈Ω. Let 𝐵_𝑅=𝐵_𝑅(𝑥₀)⊂⊂Ω and 0<𝑅/2≤𝜏<𝑡≤𝑅 be arbitrarily fixed cube. Fix a cutoff function 𝜂(𝑥)∈𝐶^∞₀(𝐵_𝑅) such that supp𝜂⊂𝐵_𝑡, 0≤𝜂≤1, |∇𝜂|≤𝐶(𝑛)/(𝑡−𝜏), and 𝜂≡1 on 𝐵_𝜏. Consider the exact form of 𝜂(𝑢−𝑐), where 𝑐∈𝐷^′(Ω,∧^𝑙−1) with 𝑑𝑐=0. With the aid of the Hodge decomposition [18], 𝑑𝜂(𝑢−𝑐)^𝑟−𝑝𝑑𝜂(𝑢−𝑐)=𝑑𝜑+ℎ, (14) where 𝑑𝜑, ℎ∈𝐿^{𝑟/(𝑟−𝑝+1)}(𝐵_𝑡,∧^𝑙), and ‖ℎ‖_{𝑟/(𝑟−𝑝+1)}≤𝐶(𝑛)𝑝−𝑟𝑑(𝜂(𝑢−𝑐))_𝑟^{𝑟−𝑝+1}. (15) Then we have 𝑑𝜑_{𝑟/(𝑟−𝑝+1)}≤𝑑𝜂(𝑢−𝑐)|^𝑟−𝑝𝑑𝜂(𝑢−𝑐)_{𝑟/(𝑟−𝑝+1)}+‖ℎ‖_{𝑟/(𝑟−𝑝+1)}≤𝑑𝜂(𝑢−𝑐)_𝑟^{𝑟−𝑝+1}+𝐶(𝑛)𝑝−𝑟𝑑𝜂(𝑢−𝑐)_𝑟^{𝑟−𝑝+1}≤𝐶(𝑛)𝑑𝜂(𝑢−𝑐)_𝑟^{𝑟−𝑝+1}. (16) We can use 𝜑∈𝑊^{1,𝑟/(𝑟−𝑝+1)}(Ω,∧^𝑙−1) as a test function for (4). Then, by Definition 1, _𝐵𝑡𝐴(𝑥,𝑑𝑢),𝑑𝜂(𝑢−𝑐)^𝑟−𝑝𝑑𝜂(𝑢−𝑐)−ℎ𝑑𝑥=0. (17) Let 𝐸=𝑑𝜂(𝑢−𝑐)^𝑟−𝑝𝑑𝜂()−(𝑢−𝑐𝜂𝑑𝑢−𝑐)^𝑟−𝑝𝜂𝑑(𝑢−𝑐); (18) using Lemma 2 yields |𝐸|≤2^𝑝−𝑟𝑝−𝑟+1)𝑟−𝑝+1(𝑢−𝑐𝑑𝜂^{𝑟−𝑝+1}. (19) Then (17) becomes _𝐵𝑡𝐴(𝑥,𝑑𝑢),𝜂𝑑(𝑢−𝑐)^𝑟−𝑝(≤𝜂𝑑𝑢−𝑐)𝑑𝑥_𝐵𝑡⟨𝐴(𝑥,𝑑𝑢),ℎ⟩𝑑𝑥−_𝐵𝑡⟨𝐴(𝑥,𝑑𝑢),𝐸⟩𝑑𝑥. (20) Noticing that 𝑐 satisfies 𝑑𝑐=0, then by the condition (2) we get _𝐵𝑡𝐴(𝑥,𝑑𝑢),𝜂𝑑(𝑢−𝑐)^𝑟−𝑝(=𝜂𝑑𝑢−𝑐)𝑑𝑥_𝐵𝑡𝐴(𝑥,𝑑𝑢),𝜂𝑑𝑢^𝑟−𝑝𝜂𝑑𝑢𝑑𝑥≥𝛼_𝐵𝜏|𝑑𝑢|^𝑟𝑑𝑥. (21) Combining the above inequality with (20), we get 𝛼_𝐵𝜏|𝑑𝑢|^𝑟≤𝑑𝑥_𝐵𝑡⟨𝐴(𝑥,𝑑𝑢),ℎ⟩𝑑𝑥−_𝐵𝑡⟨𝐴(𝑥,𝑑𝑢),𝐸⟩𝑑𝑥=𝐼₁+𝐼₂. (22) In the following we will estimate the right side of (22). By (2), the Hölder inequality, and (15), 𝐼₁≤_𝐵𝑡|𝐴(𝑥,𝑑𝑢)||ℎ|𝑑𝑥≤𝛽_𝐵𝑡|𝑑𝑢|^𝑝−1|ℎ|𝑑𝑥≤𝛽_𝐵𝑡|𝑑𝑢|^{𝑟(𝑝−1)/𝑟}_𝐵𝑡|ℎ|^{𝑟/(𝑟−𝑝+1)}𝑑𝑥^{(𝑟−𝑝+1)/𝑟}≤𝛽𝐶(𝑛)𝑝−𝑟_𝐵𝑡|𝑑𝑢|^{𝑟(𝑝−1)/𝑟}×_𝐵𝑡𝑑𝜂^𝑝(𝑢−𝑐)^𝑟𝑑𝑥^{(𝑟−𝑝+1)/𝑟}. (23) For _𝐵𝑡𝑑𝜂^𝑝(𝑢−𝑐)^𝑟𝑑𝑥^{(𝑟−𝑝+1)/𝑟}=_𝐵𝑡𝜂^𝑝𝑑𝑢+𝑝𝜂^𝑝−1(𝑢−𝑐)𝑑𝜂^𝑟𝑑𝑥^{(𝑟−𝑝+1)/𝑟}≤𝐶𝑝,𝑟_𝐵𝑡𝜂𝑑𝑢^𝑟𝑑𝑥^{(𝑟−𝑝+1)/𝑟}+𝐶𝑝,𝑟_𝐵𝑡(𝑢−𝑐)𝑑𝜂^𝑟𝑑𝑥^{(𝑟−𝑝+1)/𝑟}≤𝐶𝑝,𝑟_𝐵𝑡|𝑑𝑢|^𝑟𝑑𝑥^{(𝑟−𝑝+1)/𝑟}+𝐶𝑛,𝑝,𝑟_𝐵𝑡𝑢−𝑐𝑡−𝜏^𝑟𝑑𝑥^{(𝑟−𝑝+1)/𝑟}, (24) (23) and (24) with Young’s inequality yield 𝐼₁×≤𝛽𝐶𝑛,𝑝,𝑟𝑝−𝑟_𝐵𝑡|𝑑𝑢|^𝑟𝑑𝑥+𝛽𝐶𝑛,𝑝,𝑟𝑝−𝑟𝜀_𝐵𝑡|𝑑𝑢|^𝑟𝑑𝑥+𝛽𝐶𝜀,𝑛,𝑝,𝑟𝑝−𝑟_𝐵𝑡𝑢−𝑐𝑡−𝜏^𝑟𝑑𝑥. (25) Next we estimate 𝐼₂. By (2), the Hölder inequality, (19), and Young’s inequality, 𝐼₂≤_𝐵𝑡|𝐴(𝑥,𝑑𝑢)||𝐸|𝑑𝑥≤2^𝑝−𝑟𝑝−𝑟+1𝛽𝑟−𝑝+1_𝐵𝑡|𝑑𝑢|^𝑝−1(𝑢−𝑐)𝑑𝜂^{𝑟−𝑝+1}𝑑𝑥≤𝛽𝐶𝑛,𝑝,𝑟_𝐵𝑡|𝑑𝑢|^𝑟𝑑𝑥^{(𝑝−1)/𝑟}×_𝐵𝑡(𝑢−𝑐)𝑑𝜂^𝑟𝑑𝑥^{(𝑟−𝑝+1)/𝑟}≤𝜀𝛽𝐶𝑛,𝑝,𝑟_𝐵𝑡|𝑑𝑢|^𝑟×𝑑𝑥+𝛽𝐶𝑛,𝑝,𝑟,𝜀_𝐵𝑡𝑢−𝑐𝑡−𝜏^𝑟𝑑𝑥. (26) Combining (22), (25), and (26), we get 𝛼_𝐵𝜏|𝑑𝑢|^𝑟()𝑑𝑥≤𝛽𝐶𝑛,𝑝,𝑟𝑝−𝑟1+𝜀+𝜀_𝐵𝑡|𝑑𝑢|^𝑟𝑑𝑥+𝛽𝐶𝜀,𝑛,𝑝,𝑟𝑝−𝑟+1_𝐵𝑡𝑢−𝑐𝑡−𝜏^𝑟𝑑𝑥. (27) Let 𝑝−𝑟 and 𝜀 small enough to let 𝛽𝐶𝑛,𝑝,𝑟𝑝−𝑟(1+𝜀)+𝜀𝛼=𝜃<1; (28) then we have _𝐵𝜏|𝑑𝑢|^𝑟𝑑𝑥≤𝜃_𝐵𝑡|𝑑𝑢|^𝑟𝛽𝑑𝑥+𝐶𝑛,𝑝,𝑟,𝛼_𝐵𝑡𝑢−𝑐𝑡−𝜏^𝑟𝑑𝑥. (29) Next we will refine the inequality (29). Let 0<𝜌,𝑡<𝑅,𝑓(𝑡)=_𝐵𝑡|𝑑𝑢|^𝑟𝑑𝑥,𝐴=_𝐵𝑅𝑢−𝑐𝑅−𝜌^𝑟𝑑𝑥. (30) Choosing 𝜀∈(0,1) satisfied 𝜀^𝑟>𝜃. Let 𝑡₀=𝜌,𝑡_𝑖+1=𝑡_𝑖+(1−𝜀)𝜀^𝑖𝑅−𝜌,𝑖=0,1,2,… (31) then when 𝑘→∞, 𝑡_𝑘→𝑅. We deduce from (29) that 𝑓𝜌=𝑓𝑡₀≤𝜃𝑓𝑡₁+𝐶_𝐵𝑡1𝑢−𝑐(1−𝜀)𝜀⁰𝑅−𝜌^𝑟𝑡𝑑𝑥≤𝜃𝑓₁+𝐶𝐴(1−𝜀)^𝑟≤𝜃^𝑘𝑓𝑡_𝑘+𝐶𝐴(1−𝜀)^𝑟𝑘−1_𝑖=0𝜃𝜀^−𝑟𝑖. (32) Let 𝑘→∞ yield _𝐵𝜌|𝑑𝑢|^𝑟𝛽𝑑𝑥≤𝐶𝑛,𝑝,𝑟,𝛼_𝐵𝑅𝑢−𝑐𝑅−𝜌^𝑟𝑑𝑥. (33) Finally, in our case, 𝑟 is sufficiently close to 𝑝; we can estimate 𝐶(𝑛,𝑝,𝑟,𝛽/𝛼) independently of 𝑟.

Especially, let 𝜌=𝑅/2, 𝑐=𝑢_𝐵𝑅, and then (13) becomes _𝐵𝑅/2|𝑑𝑢|^𝑟𝛽𝑑𝑥≤𝐶𝑛,𝑝,𝛼_𝐵𝑅𝑢−𝑢_𝐵𝑅𝑅^𝑟𝑑𝑥, (34) or _𝐵𝑅/2|𝑑𝑢|^𝑟𝑑𝑥^1/𝑟≤𝐶𝑛,𝑝,𝛽/𝛼𝑅_𝐵𝑅𝑢−𝑢_𝐵𝑅^𝑟𝑑𝑥^1/𝑟. (35)

3. Zeros for the Gradients of Weakly 𝐴-Harmonic Tensors

We need the following Poincaré inequality.

Lemma 4 (see [16]). Let 𝐷 be a cube or a ball, and 𝜔∈𝐿^𝑠(𝐷,∧^𝑙) with 𝑑𝜔∈𝐿^𝑠(𝐷,∧^𝑙+1). Then 1diam𝐷_𝐷𝜔−𝜔_𝐷^𝑠1/𝑠≤𝐶(𝑛,𝑠)_𝐷|𝑑𝜔|^{𝑛𝑠/(𝑛+𝑠−1)(𝑛+𝑠−1)/𝑛𝑠}. (36) Here we denote by ⨍_𝐷 the integral mean over 𝐷.

Using the Caccioppoli inequality (13) and Lemma 4, we can get the weak-reverse Hölder inequality of weakly 𝐴-harmonic tensors.

Theorem 5. Let 𝑢∈𝐷^′(Ω,∧^𝑙−1) be a weakly 𝐴-harmonic tensor in a domain Ω∈𝐑^𝑛, and 𝑑𝑢∈𝐿^𝑟(𝐷,∧^𝑙), 𝑙=1,…,𝑛. Then there exists a constant 𝐶, independent of 𝑢 and 𝑅, such that _𝐵𝑅/2|𝑑𝑢|^𝑟𝑑𝑥^1/𝑟𝛽≤𝐶𝑛,𝑝,𝑟,𝛼_𝐵𝑅|𝑑𝑢|^{𝑛𝑟/(𝑛+𝑟−1)}𝑑𝑥^{(𝑛+𝑟−1)/𝑛𝑟} (37) for all balls 𝐵⊂Ω.

Proof. By Lemma 4, _𝐵𝑅𝑢−𝑢_𝑐^𝑟𝑑𝑥^1/𝑟≤𝐶(𝑛,𝑟)𝑅_𝐵𝑅|𝑑𝑢|^{𝑛𝑟/(𝑛+𝑟−1)}𝑑𝑥^{(𝑛+𝑟−1)/𝑛𝑟}. (38) Then, by (35), we get _𝐵𝑅/2|𝑑𝑢|^𝑟𝑑𝑥^1/𝑟𝛽≤𝐶𝑛,𝑝,𝑟,𝛼_𝐵𝑅|𝑑𝑢|^{𝑛𝑟/(𝑛+𝑟−1)}𝑑𝑥^{(𝑛+𝑟−1)/𝑛𝑟}. (39)

Next we consider the main results of this paper.

Definition 6 (see [20]). A point 𝑥₀∈Ω is said to be an essential zero of a function ℎ∈𝐿¹_loc(Ω) if lim_𝑅→01𝑅^𝑛_{𝑄(𝑥0,𝑅)}|ℎ(𝑥)|𝑑𝑥=0, (40) where 𝑄(𝑥₀,𝑅) denotes the cube centered at 𝑥₀ of side length 2𝑅. The order of the essential zero is defined to be 𝑁𝑥₀=sup𝛼∶lim_𝑅→01𝑅^𝑛+𝛼_{𝑄(𝑥0,𝑅)}|ℎ(𝑥)|𝑑𝑥=0. (41)

Lemma 7 (see [20]). Let ℎ∈𝐿¹_loc(Ω) satisfy the weak-reverse Hölder inequality 1|𝑄|_𝑄ℎ^𝑝𝑑𝑥≤𝐴_𝑝1|2𝑄|_2𝑄ℎ𝑑𝑥^𝑝 (42) for all cubes 𝑄⊂2𝑄⊂Ω and some 1<𝑝<∞, with a constant 𝐴_𝑃 independent of the cube. Then almost every zero of ℎ has infinite order.

Theorem 8. There exist exponents 𝑟₀=𝑟₀(𝑛,𝑝,𝛽/𝛼)∈(1,2); if 𝑟>𝑟₀, then for the weakly 𝐴-harmonic tensor 𝑢∈𝑊^1,𝑟_loc(Ω,∧^𝑙−1) almost every zero of 𝑑𝑢 has infinite order.

Proof. By the weak-reverse Hölder inequality (37) and Lemma 7, we get the desired result.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are supported by NSFC (11371050) and NSF of Hebei Province (A2013209278).