Carlson's type theorem on removable sets for α-Holder continuous solutions is investigated for the quasilinear elliptic equations divA(x,u,∇u)=0, having degeneration ω in the Muckenhoupt class. In partial, when α is sufficiently small and the operator is weighted p-Laplacian, we show that the compact set E is removable if and only if the Hausdorff measure Λω−p+(p−1)α(E)=0.

1. Introduction

In this paper, we will consider questions of a removable singularity for the class of quasilinear elliptic equations of the form
div(A(x,u,∇u))=0,
where A(x,ξ,ζ)={A1(x,ξ,ζ),A2(x,ξ,ζ),…,An(x,ξ,ζ)}:D×ℝ×ℝn→ℝn are continuous with respect to ξ, continuously differentiable with respect to ζ functions. For ξ∈ℝ1, ζ∈ℝn, it is assumed that Aj(x,ξ,ζ);j=1,2,…n are measurable functions with respect to a variable x in the open domain D⊂ℝn. Let the following growth conditions be satisfied:
∑i,k=1n∂Ai∂ζk(x,ξ,ζ)ηiηk≥λω(x)|ζ|p-2|η|2,|∂Ai∂ζk(x,ξ,ζ)|≤λ-1ω(x)|ζ|p-2,k,i=1,2,…,n,
where 1<p<∞,λ∈(0,1). Throughout the paper, ω:ℝn→[0,∞] is a measurable function satisfying the doubling condition: for any ball B=B(x,r) with centre at a point x and of radius r>0, the inequality ∫2Bωdx≤C∫Bωdx
is satisfied, where the constant C is positive and does not depend on the ball B⊂ℝn. For the system of functions Aj(x,ξ,ζ);j=1,2,…n, we can write the following expression: Ai(x,u(x),∇u(x))=∫01∑k=1n∂Ai∂ζk(x,u(x),t∇u(x))uxkdt=∑k=1nbik(x)ω(x)|∇u(x)|p-2uxk,i=1,2,…,n,
where bik(x)=ω(x)-1|∇u(x)|2-p∫01∑k=1n(∂Ai∂ζk)(x,u(x),t∇u(x))dt;i,k=1,2,…,n. Therefore, (1.1) can be written in the form ∑i,k=1n∂∂xi(ω|∇u|p-2bik(x)∂u∂xk)=0.
By virtue of (1.2) and (1.4), the system of functions {bik(x)}i,k=1,2,…,n is bounded and measurable. Moreover, the condition of uniform ellipticity is satisfied: for a.e. x∈D,η∈ℝn there exist positive constants C1,C2 such that C1|η|2≤∑i,k=1nbik(x)ηiηk=∑i,k=1nω(x)-1|∇u(x)|2-p∫01∂Ai∂ζk(x,u(x),t∇u(x))ηiηkdt≤C2|η|2.

Denote by Cα(D),0<α≤1, the class of continuous in D functions f:D→ℝ satisfying the condition|f(x)-f(y)|≤K|x-y|α
with some K>0 not depending on the points x,y∈D. Denote by Wpω1(D) the space of measurable functions in D, which have the finite norm ‖u‖=‖u‖Lpω(D)+∑j=1n‖uxj‖Lpω(D),
where uxj are the derivatives of a function u∈Lpω(D) in the sense of the distribution theory, which belong to the space Lpω(D). The norm of the space Lpω(D) is given in the form ∥f∥Lpω(D)=(∫Dω|f|pdx)1/p for p≥1; ∥f∥Lpω(D)=esssupx∈D|f(x)| for p=∞. Denote by Ẇpω1(D) a subspace of the space Wpω1(D), where the class of functions C0∞(D) is an everywhere dense set. Denote by W̃pω1(D) the closure of the set of functions C∞(D) with respect to the norm Wpω1(D). The spaces Wpω1(D) and W̃pω1(D) coincide and are completely reflexive [1, 2] if the conditions 1<p<∞ and Ap-Muckenhoupt are fulfilled for ω:(∫Bω(x)dx)(∫Bω-1/(p-1)(x)dx)p-1≤Cp|B|p,
where |B| denotes the Lebesgue measure of an arbitrary ball B⊂ℝn. In the sequel, we will also use the A1-condition:∫B(x,ρ)ω(x)dx≤Cρninfx∈Bω.

Definition 1.1.

A function u∈Wpω1(D) is called a solution of (1.1) if the integral identity
∫DA(x,u,∇u)⋅∇φdx=0
is fulfilled for any test function φ∈Ẇpω1(D).

Definition 1.2.

Let E⊂⊂D be a compact subset of the bounded domain D⊂ℝn. One will say that the set E is removable for the class of Cα(D) of solutions of (1.1) if any solution of (1.1) in D∖E from the space Wpω,loc1(D∖E) belongs to the space Wpω1(D) throughout the domain D and is extendable inside the compactum E as solution.

Definition 1.3.

Let E⊂ℝn be a bounded closed subset, h:ℝ→(0,∞) a continuous function, and h(0)=0,μ some Radon measure. A finite system of balls {Bν=B(xν,rν)}ν=1,2,…,N, the radii of which do not exceed δ>0, covers the set E, that is, E⊂⋃νBν. Assume that Λμh,δ(E)=inf{∑νh(rν)μ(Bν)}, where the lower bound is taken with respect to all the mentioned balls.

Assume that
Λμh(E)=limδ→∞Λμh,δ(E).
In the case μ=dx,h(t)=t-p+(p-1)α, the number Λμh(E) is a Hausdorff measure of order n-p+(p-1)α of the set E. We will sometimes denote it by mesn-p+(p-1)α(E). Denote by Λω-p+(p-1)α(E) the term Λμh(r)(E) for h(t)=t-p+(p-1)α, dμ=ωdx.

By Carlson’s theorem [3], a necessary and sufficient condition for the compact set E to be removable in the class of harmonic outside E functions belonging to the class Cα(D) is expressed in terms of a Hausdorff measure of order n-2+α having the formmesn-2+αE=0,0<α<1.
(For the case α=1, the same result was proved in [4, 5].) In [6], the corresponding result was proved for a general linear elliptic equation with variable coefficients (see also [7, 8]). In [9], for the case p≥2, a sufficient condition was proved for a solution of the p-Laplace equation (A=|∇u|p-2∇u) to be removable in the class Cα(D) (0<α≤1) in the form mesn-p+(p-1)αE=0,0<α≤1.
Furthermore, a complete analogue of Carlson’s result was proved in [10], where the authors not only proved the necessity of condition (1.13), but also gave another proof of sufficiency that includes also range of exponent 1<p<2. Their approach was applied to the case of a metric measure space in [11].

It should be said that in [9] a somewhat general result was in fact obtained for the compact set E to be removable for the class Cα(D) of solutions of the equation ∑i=1n∂∂xi(ω|∇u|p-2∂u∂xi)=0
in the form of a sufficient condition Λω-p+(p-1)α(E)=0.
Note, in [9], the case p≥2 was considered and it was required that the function ω to satisfy the doubling condition.

The present paper continues the development of the approach of [9]. We show that condition (1.15) is also the necessary one for the compact set E to be removable. Moreover, imposing some restrictions on the degeneration function, we manage to make the proof embrace a range of the exponent 1<p<2.

We will use the following auxiliary statements.

Lemma 1.4 (see [<xref ref-type="bibr" rid="B2">12</xref>]).

Assume that a function u∈L1(D) satisfies the inequality
∫B(x,r)|u-(u)x,r|dx≤Mrn+α
for any ball B(x,r)⊂D, where α∈(0,1). Then, u∈Cα(D) and for any D′⊂⊂D the estimate
supD′|u|+supx,y∈D′|u(x)-u(y)||x-y|α≤C(M+‖u‖L1(D)),
where C=C(n,α,D′,D), is satisfied.

We also need the following analogue of the well-known Giaquinta’s lemma [13].

Lemma 1.5.

Let ϕ(t), ω(t) be nonnegative nondecreasing functions on [0,R]. Assume that s>0 is such that
ω(λr)ω(r)≥λs
for all r>0 and 0<λ<1. Suppose that
ϕ(ρ)≤A[ω(ρ)ω(r)(ρr)α+ε]ϕ(r)+Bω(r)rβ
for any 0<ρ<r<R, with A,B,α,β nonnegative constants and β<α. Then, for any γ∈(β,α), there exists a constant ε0=ε0(A,α,β,γ,s) such that if ε<ε0, then one has, far all 0<ρ<r<R,
ϕ(ρ)≤c[ω(ρ)ω(r)(ρr)γϕ(r)+Bω(ρ)ρβ],
where c=c(β,α,A,s,γ)>0.

Proof.

For τ∈(0,1) and r<R, we have
ϕ(τr)≤Aταω(τr)ω(r)[1+ετ-α-s]ϕ(r)+Bτ-sω(τr)rβ.
Choose τ<1 in such a way that 2Aτα=τγ and assume ε0τ-α-s≤1. Then, we get, for every r<R,
ϕ(τr)≤τγω(τr)ω(r)ϕ(r)+Bτ-sω(τr)rβ
and therefore, for all integers k>0,
ϕ(τk+1r)≤τγω(τk+1r)ω(τkr)ϕ(τkr)+Bτ-sω(τk+1r)τkβrβ≤τ(k+1)γω(τk+1r)ω(r)ϕ(r)+Bτ-sω(τk+1r)τkβrβ∑j=0kτj(γ-β)≤τ(k+1)γω(τk+1r)ω(r)ϕ(r)+Bτ-sτkβ1-τγ-βrβω(τk+1r).
Choosing k such that τk+2r<ρ≤τk+1r, the last inequality gives
ϕ(ρ)≤1τγ(ρr)γω(ρ)ω(r)ϕ(r)+Bτ-s(1-τγ-β)τs+2βρβω(ρ).
This proves Lemma 1.5.

We did not find the proof of the next inequality in the literature and therefore give here our proof.

Lemma 1.6.

Let 1≤p≤2. Let x,y∈ℝn be arbitrary points. Then, the estimate
||x|p-2x-|y|p-2y|≤22-p|x-y|p-1
is valid.

Proof.

Let us introduce the vector function
φ(θ)=|θx+(1-θ)y|p-2(θx+(1-θ)y),0≤θ≤1
acting from [0,1] into ℝn. Applying the methods of differential calculus for the vector function, we obtain
||x|p-2x-|y|p-2y|=|φ(1)-φ(0)|=|∫01dφdθdθ|=(p-1)|∫01|θx+(1-θ)y|p-2(x-y)dθ|≤(p-1)|x-y|∫01|θx+(1-θ)y|p-2dθ.

The set of points {l(θ)∈ℝn:l(θ)=θx+(1-θ)y;0≤θ≤1} in ℝn forms a segment of the straight line that connects the point x with the point y. We denote this segment by [x,y]. Let |dl| be a length element of this segment. It is obvious that |dl|=|x-y|dθ. Therefore, for the above integral expression, we have the estimate
≤(p-1)∫[x,y]|l(θ)|p-2|dl(θ)|=(p-1)∫0|x-y||dl(θ)|dist(l(θ),0)2-p.

To proceed with the estimation of this expression, we introduce into consideration the triangle, the base of which is the segment [x,y] and the vertex lies at the point 0. Now, the integration in the preceding estimate will be carried out with respect to the base of the triangle. It is not difficult to verify that the above integral expression takes a maximal value when the point 0 lies in the middle of the segment [x,y], which means that for it we have the estimate
≤(p-1)∫0|x-y|/2dss2-p=22-p|x-y|p-1.

To show that this is true, let us choose a new coordinate system, where the xn-axis is directed along the segment [x,y]. Let (u1,u2,…,un) be the coordinates of the the point 0 in the new coordinate system. Then, the preceding integral expression is equal to
(p-1)∫-|x-y|/2|x-y|/2(u12+u22+⋯+un-12+(un-xn)2)p-2dxn=(p-1)∫-|x-y|/2|x-y|/2|un-s|p-2ds≤2(p-1)∫0|x-y|/2sp-2ds=22-p|x-y|p-1.

The main result of this paper is contained in the following statements.

Theorem 1.7.

Let D⊂ℝn be a bounded domain, E⊂⊂D be a compact subset. Let 2<p<∞ and ω be a positive, locally integrable function satisfying condition (1.3) or 1<p<2 and let any of the following conditions be fulfilled for the function ω:

the function ω is integrable along any finite smooth n-1-dimensional surface and condition (1.9′) is fulfilled for it;

for any x∈D and sufficiently small ρ>0, the condition ∫B(x,ρ)ωdy≤Cρs is fulfilled for some s>n-p+1, where the constant C>0 does not depend on x.

Then, for a compact set E to be removable in the class Cα(D), 0<α≤1 of solutions of (1.1) in D∖E, u∈Wpω,loc1(D∖E), it is sufficient that condition (1.15) be fulfilled.

Here, we will use also the fact that a solution of generating equations of the form (1.1) is Hölderian. According to [14], when a weight ω belongs to the Muckenhoupt Ap-class, a solution of (1.1) belongs to the class Cκ(Dρ) in any subdomain Dρ={x∈D:dist(x,∂D)>ρ} of the domain D. For solutions, we have the estimate oscB(x,ρ)u≤C(ρr)κoscB(x,r)u,0<ρ<r,
where κ=κ(n,p,Cp,λ)∈(0,1] and C=C(n,p,Cp,λ). Let κ denote a maximal number κ=κ(n,p,Cp,λ), for which the estimate (1.31) holds for solutions of (1.1). The following statement is valid.

Theorem 1.8.

Let ω∈Ap, E⊂⊂D be a compact subset of the domain D. Let 0<α<κ be some number. In that case, if Λω-p+(p-1)α(E)>0, then the set E is not removable in the class of u∈Wpω,loc1(D∖E) solutions of (1.1) which belong to Cα(D).

The foregoing statements give rise to the following corollaries.

Corollary 1.9.

Let 0<α<κ, 2≤p<∞, ω∈Ap, or 1<p<2 and any of the following conditions be fulfilled:

the function ω satisfies condition (1.9′) and is integrable along any finite smooth n-1-dimensional surface;

for any x∈D and sufficiently small ρ>0, the condition ∫B(x,ρ)ωdy≤Cρs is fulfilled by some s>n-p+1, where the constant C>0 does not depend on x.

Then, for the compact set E to be removable for the class of Wpω1(D∖E) solutions of (1.1) belonging to Cα(D), it is necessary and sufficient that condition (1.15) be fulfilled.

Corollary 1.10.

Let 0<α≤1, 1<p<∞. Then, for the compact set E to be removable in the class Wp,loc1(D∖E) of solutions of the equation
∑i=1n∂∂xi(|∇u|p-2∂u∂xi)=0
belonging to the class Cα(D) throughout the domain D, it is necessary and sufficient that condition (1.13) be fulfilled.

2. Proof of the Main Results

In [9], the method of proving Theorem 1.7 was based on the application of an analogue of Landis-Gerver’s mean value theorem [15]. The restrictive condition p≥2 used in [9] was necessitated by the proof of Lemma 2.1 below (see also [15, Lemma 1]). Below, we prove a such type lemma for the case 1<p<2, ignoring some smoothness of the function f and making some additional assumptions for the function ω.

Lemma 2.1.

Let D be a bounded domain. Let 2≤p<∞ and the function ω:ℝn→[0,∞] satisfy condition (1.3) or 1<p<2, and let any of the following conditions be fulfilled for the function ω:

condition (1.9′) is fulfilled and ω is integrable along any finite smooth n-1-dimensional surface;

for any x∈D and sufficiently small ρ>0, the condition ∫B(x,ρ)ωdy≤Cρs, where the constant C>0 does not depend on x, is fulfilled for some s>n-p+1.

Assume that f:D→ℝ is a sufficiently smooth function (one can also assume the condition f(x)∈Cβ(D), where β≥min{p′,1}). Then, for any ε>0, there exist a finite number of balls {Bν}, ν=1,2,…,N, such that
∑ν=1N∫∂Bνω|∇f|p-1ds<ε.

Proof.

We will follow the same reasoning as that used in proving Lemma 1 in [9] (see also [3], Lemma 2.1). The set Of={x∈D:∇f(x)=0} is divided into two parts Of=Of′∪Of′′; here, Of′ is the set of points where ∇2f(x)≠0, and Of′′ is the set of points where ∇2f(x)=0. Let 1<p<2. Then, for the set Of′, our reasoning is as follows. By virtue of the implicit function theorem, the set Of′ lies on a countable quantity of smooth n-1-dimensional surfaces {Sj};j=1,2,…. Let x∈Sj be a fixed point on the j-th surface. For sufficiently small r>0, we have
∫S(x,r)ω|∇f|p-1ds≤2C1rp-1ω(S(x,r)),
where C1=supD|∇2f| and ω(S(x,r)) is integral omega over the n-1 dimensional surface S(x,r). By virtue of Fubini’s formula,
∫rx2rx(∫|y-x|=tωdsy)dt≤∫|y-x|<2rxω(y)dy.
Let E be the set of points t∈(rx,2rx), for which the following condition is fulfilled:
∫|y-x|=tωds>2rx(∫|y-x|<2rxω(x)dx).
Then
2rx(∫|y-x|<2rxω(y)dy)μ1(E)≤∫E(∫|y-x|=tωds)dt≤∫rx2rx(∫|y-x|=tωds)dt≤∫|y-x|<2rxω(y)dy,
whence for the one-dimensional Lebesgue measure of the set E we obtain the estimate μ1(E)≤rx/2. Hence, by virtue of the doubling condition, there exists a point tx∈(rx,2rx), for which
∫|y-x|=txωds<2rx(∫|y-x|<2rxω(y)dy)≤2Crx(∫|y-x|<rxω(y)dy)≤4Ctx(∫|y-x|<txω(x)dx).
Then, for sufficiently small tj>0, for any x∈Sj, it can be assumed that there exists a number ρx∈(tj,2tj), for which
ω(S(x,ρx))≤4Cρxω(B(x,ρx)).
Therefore,
∫S(x,ρx)ω|∇f|p-1ds≤4C1ρxp-1ω(S(x,ρx))≤16CC1ρxp-2ω(B(x,ρx)).
For the surface Sj, from the system of balls {B(x,ρx);x∈Sj}, we can extract, by virtue of Besicovtich theorem [16], a subcovering {B(xν,ρν);xν∈Sj,ν∈ℕ} with finite intersections:
∑νχB(xν,ρν)(x)≤Cn,⋃νB(xν,ρν)⊃Sj.
Therefore and by construction, for xν∈Sj,ν=1,2,…, we have ρν∈(tj,2tj). Thus,
∑ν∫S(xν,ρν)ω|∇f|p-1ds≤∑ν16CC1ρνp-2ω(B(xν,ρν))≤∑ν16CC1C2ρνn+p-2infx∈B(xν,ρν)ω≤∑ν32CC1C2ρνp-1ω(B(xν,ρν)∩Sj)≤32⋅2p-1Ctjp-1ω(Sj),j=1,2,….
Here, we have used the sufficient smallness of tj, condition (1.9′), and the inequality ρνn-1infx∈B(xν,ρν)ω≤2ω(B(xν,ρν)∩Sj). Choosing now tj, 32·2p-1CC1C2tjp-1ω(Sj)≤ε/2j, where ε>0 is an arbitrary number, we obtain ∑ν∫S(xν,ρν)ω|∇f|p-1ds<ε/2j, whence, after summing the inequalities over all surfaces Sj,j∈ℕ, we find
∑j∑xν∈Sj∫S(xν,ρν)ω|∇f|p-1ds<ε.

In the case of the second condition 1<p<2, using ∫B(x,ρ)ωdy≤Cρs, we immediately pass from the inequality (2.8) to (2.10) as
ω(S(x,ρx))≤4Cρxω(B(x,ρx))≤4CC4ρxn-p=8CC4ρx2-pmesn-1(B(x,ρx)∩Sj).
Due to the latter inequality, an estimate analogous to (2.10) will have the form
∑ν∫S(xν,ρν)ω|∇f|p-1ds≤∑ν2C1ρνp-2ω(B(xν,ρν))≤∑ν4C1ρνp-2+s,∑ν8C1ρνs-n+p-1mesn-1(B(xν,ρν)∩Sj)≤16⋅2p-1Ctjs-n+p-1ω(Sj),j=1,2,….
After choosing tj sufficiently small and taking the condition s>n-p+1 into account, we can make the right-hand part smaller than ε/2j.

In the case p≥2, the whole reasoning of [9] is applicable. Note that only instead of the inequality (2.10) we will have
∑ν∫S(xν,ρν)ω|∇f|p-1ds≤C∑νρνp-1ω(B(xν,ρν))≤C1tjp-2ω(Stjj),
where Stjj is the tj neighborhood of the surface Sj. After choosing a sufficiently small tj, we can make the right-hand part of this inequality smaller than ε/2j. This is possible because the n-dimensional Lebesgue measure of the surface Sj is equal to zero.

Now, it remains to obtain the covering for the set of points Of′′. Let 1<p<2. Let us decompose Of′′=Of′′′∪Of′′′′, where Of′′′ is the set of points Of′′, for which ∇3f≠0. Here, we repeat the reasoning for Of′. As above, the set O′′′′ is divided into two parts. In one part, we have ∇4f(x)≠0, to which we apply the same reasoning as for Of′. The second part of O′′′′, where ∇4f(x)=0, is again divided into two parts. At the k-th step, when k(p-1)≥1 and t>0 is sufficiently small, this process yields the estimate
∑ν∫S(xν,ρν)ω|∇f|p-1ds≤η∑νCρνk(p-1)ω(S(xν,ρν))≤η∑ν2Cρνk(p-1)-1ω(B(xν,ρν))≤2ηCC3ω(D),
where η>0 is arbitrary.

Note that, in the case p≥2, the foregoing estimate gives the desired results immediately at the first step (k=1).

Remark 2.2.

It is not difficult to verify that under the assumptions of Lemma 2.1, instead of the condition of sufficient smoothness it sufficed to assume that f(x)∈Cβ(D), where β≥min{p′,1}.

Applying approaches similar to those in [15, Theorem 2.2, page 128] and [9], we prove the following analogue of Landis-Gerver’s lemma.

Lemma 2.3.

Let 2≤p<∞ and the function ω:ℝn→[0,∞] satisfy condition (1.3) or 1<p<2 and let any of the following conditions be fulfilled for the function ω:

condition (1.9′) is fulfilled and ω is integrable along any finite smooth n-1-dimensional surface;

for any x∈D and sufficiently small ρ>0, the condition ∫B(x,ρ)ωdy≤Cρs is fulfilled by some s>n-p+1, where the constant C>0 does not depend on x.

Let D be some domain lying in the spherical layer B(x0,2r)∖B̅(x0,r) and having limit points on the surfaces of the spheres S(x0,2r) and S(x0,r). Let ∑i,k=1naik(x)ηiηk be the quadratic form, the coefficients of which are well defined and continuously differentiable in the domain D and for which the inequalities
λ|η|2≤∑i,k=1naik(x)ηiηk≤λ-1|η|2
are fulfilled for any x∈D,η∈ℝn for some λ∈(0,1). Assume that f:D→ℝ is a sufficiently smooth function.

Then, there exists a piecewise-smooth surface Σ, separating, in the domain D, the surfaces of the spheres S(x0,r) and S(x0,2r) and being such that
∫Σω|∇f|p-2|∂f∂ν|ds≤K(oscDf)p-1ω(D)rp,
where ∂f/∂ν=∑j=1naij(x)fxinj is conormal derivative on Σ, n=(n1,n2,…,nn) is unit orthogonal vector to the surface Σ, and the constant K depends on p,λ, and the dimension n.

Proof.

It suffices to consider the case r=1. Indeed, after the change of variables x=ry, the function f:D→R transforms to the function f̃:D̃→R, where f̃(y)=f(ry). Also, |∇yf̃|=|∇xf|r,∂f̃/∂vy=(∂f/∂vx)r,ω(D̃)=ω(D)r-n. D̃ lies in the spherical layer B(0,2)∖B(0,1). It suffices to show that ∂f̃/∂vy=(∂f/∂vx)r. Indeed, let a sufficiently small element of the surface Σ satisfy the equation φ(x)=0 in coordinates x. Then, after the change of variables, this equation takes the form φ̃(y)=0, where φ̃(y)=φ(ry). In other words, the normals of the surfaces Σ and Σ̃ are related by nx=∇xφ/|∇xφ|=∇yφ̃/|∇yφ̃|=ny. Therefore,
∂f̃∂vy=∑i,k=1nãik(y)∂f̃∂yiñk=r∑i,k=1naik(x)∂f∂xink=r∂f∂vx,ãik(y)=aik(ry),i,k=1,2,…,n.
Applying these equalities, from the estimate
∫Σ|∇yf̃|p-2|∂f̃∂vy|dsy≤K(oscD̃f̃)p-1ω(D̃),
we obtain (2.17).

Let us now prove (2.19). Following the notation and reasoning of [15] (see also [9]), we assume that ε=ω(D)(oscf)p-1. For this ε, we find the corresponding balls Q1,Q2,…,QN of Lemma 2.1 and remove them from the domain D. Assume that D*=D∖⋃m=1NQm and intersect D* with the closed layer (1+1/4)≤|x|≤(1+3/4). Denote this intersection by D′. On the closed set D′, we have ∇f≠0. Let us choose some δ-neighborhood Dδ′ with δ<1/4 so small that in Dδ′ we would have |∇f|>α>0. We consider on Dδ′ the system of equations
dxidt=∑k=1naik(x)∂f∂xk,i=1,2,…,n.
In Dδ′, there are no stationary points of the system (2.20), and at every point x∈Dδ′ the direction of the field forms with the direction of the gradient an angle different from the straight angle. Let l(x) be the vector of the field at the point x. Then, using cos(l(x),∇f)=(∑k=1naik(x)(∂f/∂xk),∇f)/|∑k=1naik(x)(∂f/∂xk)||∇f|>λ|∇f|2/λ-1|∇f|2=λ2, we obtain
|∂f∂l|>λ2|∇f|>γα>0,γ=λ2.
From this inequality, it follows that in Dδ′ there are no closed trajectories and all the trajectories have the uniformly bounded length.

Let some surface S be tangential, at each of its points, to the field direction. Then,
∫Sω|∇f|p-2|∂f∂v|ds=0,
since the integrand is identically zero. We will use this fact in constructing the needed surface Σ. The base of Σ consists of ruled surfaces, while the generatrices are the trajectories of the system (2.20). Note that they will add nothing to the integral in which we are interested. These surfaces will have the form of fine tubes which will cover the entire D′. Let us insert partitions into some of the tubes. The integral over these partitions will not any longer be equal to zero, but we can make it infinitesimal. The construction of tubes practically repeats that given in [15, pages 129–132].

Denote by E the intersection of D′ with the sphere S(1+3/4)0. Let N be the set of points x∈E, where the direction of the field of the system (2.20) is tangential to the sphere S(1+3/4)0. At the points x∈N, we have ∂f/∂v=0, where ∂/∂v is the derivative with respect to the conormal to the sphere S(1+3/4)0. Cover N by a set G, open on the sphere S(1+3/4)0 and being such that
∫Gω|∇f|p-2|∂f∂v|ds≤ω(D)(oscf)p-1.
Put E′=E∖G. At the points x∈E′, the direction of the field is transversal to the sphere. Cover E′ on the sphere S(1+3/4)0 by a finite number of uncovered domains with piecewise-smooth boundaries. We will call them cells. We will choose their diameters so small that at the points of the cells the field would be transversal to the sphere and the bundle of trajectories passing through each of the cells would diverge by δ/2n at most. The surface with trajectories lying inside the ball |x|<(1+3/4) and passing through the cell boundary will be called a tube. Thus, we obtain a finite number of tubes. We will call a tube a through tube if, without intersecting this tube, we can connect by a broken line a point of its corresponding cell with a point of the sphere S(1+1/4)-(δ/2)0 within the limits of the intersection of D′ with a spherical layer 1+1/4-δ<|x|<1+3/4. Such through tubes are denoted by T1,T2,…,Ts. If every through tube is partitioned, then the spheres S10 and S20 are separated in D by the set-theoretic sum of nonthrough tubes, partitions T1,T2,…,Ts, the spheres S1,S2,…,SN, and the set G on the sphere S(1+3/4)0.

Let us now take care to choose partitions in such a way that the integral ∫ω|∇f|p-2|∂f/∂v|ds over them would have the value which we need. Denote by Ui the domain bounded by Ti. Choose any trajectory on this tube. Denote it by Li. The length μ1Li of the curve Li satisfies the inequality μ1Li>1/2. Introduce, on Li, the parameter l which is the length of the arc counted from S(1+1/4)0. Denote by σi(l) the section Ui with a hypersurface which is orthogonal, at the point l, to the trajectory Li. Let the diameter at the beginning of the tube be so small that ∫Li(∫σi(l)ωds)dl≤2ω(Ui). Then, the set H of points l∈Li, where ∫σi(l)ωds>8ω(Ui), satisfies the inequality μ1Li<1/4. Thus, for E=Li∖H, the inequality μ1Li>1/4 is valid and
∫σi(l)ωds<8ω(Ui)forl∈E.
At the points of the curve Li, the derivative ∂f/∂l preserves the sign and therefore
∫E|∂f∂l|dl≤∫Li|∂f∂l|dl<oscDδ′f.
Hence, using μ1Li>1/4 and the mean value theorem, we see that there exists a point l0∈E such that |∂f/∂l|l=l0≤4oscf. On the other hand, since, by virtue of (2.21), |∂f/∂l|l=l0≥γ|∇f|l=l0, we have |∇f|p-1|l=l0≤(4oscf)p-1γ1-p. This together with (2.24) gives the estimate
(|∇f|p-1|l=l0)∫σi(l0)ωds≤C(p,γ)ω(Ui)(oscf)p-1.
Let us now choose a cell diameter so small that
∫σi(l0)|∇f|p-1ωds≤2C(p,γ)ω(Ui)(oscf)p-1.
This can be done since the derivatives ∂f/∂xk,k=1,2,…,n, are uniformly continuous. Therefore,
∑i=1s∫σi(l0)|∇f|p-1ωds≤4C(p,γ)ω(Ui)(oscf)p-1.
Denote by Σ the set-theoretic sum of all nonthrough tubes, all σi(l0), all spheres Si, and the set G on the sphere S(1+3/4)r0. Then, from Lemma 2.1 and (2.22)–(2.28), we obtain
∫Σ|∇f|p-2|∂f∂v|ωds≤C(p,n,γ)ω(Ui)(oscf)p-1.

Lemma 2.3 is proved.

In this paper, we give the complete proof of Theorem 1.7. Some part of the proof of sufficiency is in fact identical to the proof given in [9]. The method of proving Theorem 1.8 is analogous to the method [10], where the nonweight case was considered.

Proof of Theorem <xref ref-type="statement" rid="thm1.1">1.7</xref> (Approximation).

Let Λω-p+(p-1)α(E)=0 and u∈Cα(D); let u∈Wpω,loc1(D∖E) be a solution of (1.1). Denote by uj a mean value of the function u with smooth kernel ρ with finite support, u(j)=ρ1/j*u=jn∫ℝnρ((x-y)j)u(y)dy,∫ℝnρ(x)dx=1,j∈ℕ. Then, it is obvious that u(j)∈C∞(D), j=1,2,…. Moreover, uj→u uniformly in any subdomain G̅⊂D. Also, for any open set E′⊃E contained in G, u(j)→u in the norm of the space Wpω1(G∖E′) (see [2, 16]). Since, by condition (1.15), we have mesnE=0, it can be assumed that mesnE′<η, where η>0 is an arbitrary number.

Let ε>0 be an arbitrary number. Cover the set E by a finite system of balls {Bν}ν=1,2,…,N, ⋃ν=1NBν⊃E such that diamBν<δ,
∑ν=1Nrν-p+(p-1)αω(Bν)<ε.
Assume that the number δ=δ(ε,η) is so small that the set Γ′′=⋃ν=1N(4Bν) lies in E′.

For every ν, there exists, by virtue of Lemma 2.3 and inequality (1.31), a piecewise-smooth surface γν(j),ν=1,2,…,N,j=1,2,…, separating the surfaces of the spheres ∂(2Bν) and ∂(4Bν), such that
∫γνjω|∇u(j)|p-2|∂u(j)∂v|ds≤Krν-p(osc2Bνu(j))p-1ω(4Bν).
Denote by Γν(j) the interiority of the surface γν(j). Then, Γ(j)=⋃νΓν(j)⊃Γ′=⋃ν(2Bν). Assume that σν(j)=Γ(j)∩γν. Let σν(j)≠∅ for some ν. Then, for ν, the inequality (2.31) implies the estimate
∫σν(j)ω|∇u(j)|p-2|∂u(j)∂v|ds≤Krν-p(osc2Bνu(j))p-1ω(4Bν).

It is obvious that the set G∖Γ′̅ is a strictly interior subdomain of the domain D∖E. Thus, we have the identity
∑i,k=1n∫G∖Γ′ω|∇u|p-2bikuxkψxidx=0,
for any ψ∈C01(D∖Γ′). From this, by virtue of the convergence ∥u(j)-u∥Wpω1(G∖Γ′)→0 as j→∞ and the fact that the ∥bik(j)∥ satisfies (2.16) by some λ>0, we find
∑i,k=1n∫G∖Γ′ω|∇u(j)|p-2bik(j)uxk(j)ψxidx=δj.
This follows from the fact that the integrand is a system of equi-integrable functions: for any subset g⊂D∖Γ′, we have
∑i,k=1n∫gω|bik(j)|∇u(j)|p-2uxi(j)ψxk|dx≤C∫gω|∇u(j)|p-1|∇ψ|dx≤2C(∫gω|∇u|pdx)1/p′(∫gω|∇ψ|pdx)1/p⟶0
as mesng→0. Here and in the sequel, speaking in general, we denote by δj different sequences tending to zero as j→∞.

Green’s Formulae for Approximations

Let now φ∈C01(D) be an arbitrary function. Assume that ψ=φξ(d(x)/τ), where 0≤ξ(s)≤1 is an infinitely differentiable function equal to zero for s≤0 and to one for s≥1 and τ>0 is a parameter, forallφ∈C01(D), d(x)=dist(x,Γj). It is obvious that ψ∈C01(D∖Γj). Then, (2.34) implies, for j=1,2,…,
∑i,k=1n∫D∖Γ′ω|∇u(j)|p-2bik(j)uxk(j)φxiξdx+∑i,k=1n1τ∫D∖Γ′ω|∇u(j)|p-2bik(j)uxk(j)dxiξ′(d(x)τ)φdx=δj.
By virtue of the majorant Lebesgue theorem, for τ→0, the first summand in (2.36) tends to the limit ∑i,k=1n∫D∖Γ(j)ω|∇u(j)|p-2bik(j)uxk(j)φxidx. Let us now find the limit of the second summand. Applying the Federer formula, we have
∑i,k=1n1τ∫D∖Γ′ξ′(d(x)τ)ω|∇u(j)|p-2bik(j)uxk(j)φxidx=∑i,k=1n1τ∫0τ(∫{d(x)=t∩(D∖Γ′)}φω|∇u(j)|p-2bik(j)uxkjdxi|∇d|dst)ξ′(tτ)dt.
Applying the mean value theorem, for some t0∈(0,τ), we obtain
=∑i,k=1n(∫d(x)=t0φω|∇uj|p-2bik(j)uxkdxi|∇d|dst0)(1τ∫0τξ′(tτ)dt)⟶∑i,k=1n∫d(x)=t0φω|∇u(j)|p-2bik(j)uxk(j)dxi|∇d|dst0⟶∫∂Γ(j)φω|∇u(j)|p-2∂u(j)∂vdsasτ⟶0.
Taking this limit relation into account, from (2.36), we obtain, as τ→0, the following equality which is Green’ formula for approximation function uj:
∑i,k=1n∫D∖Γ(j)ω|∇u(j)|p-2bik(j)uxk(j)φxidx=∫∂Γ(j)φω|∇u(j)|p-2∂u(j)∂vds+δj.
Whence, in view of the inequality |φ(x)|≤∥φ∥C(D), x∈D, we have
|∑i,k=1n∫D∖Γ(j)ω|∇u(j)|p-2bik(j)uxk(j)φxidx|≤‖φ‖C(D)∫∂Γ(j)ω|∇u(j)|p-2|∂u(j)∂v|ds+δj≤‖φ‖C(D)∑ν∫γνω|∇u(j)|p-2|∂u(j)∂v|ds+δj.

Using the convergence ∥u(j)-u∥Wpω1(G∖Γ′)→0(j→∞), Lemma 1.6, conditions (1.2), and Hölder inequality, we have the estimate for 1<p<2:
|∑i,k=1n∫D∖Γ(j)ωbik(j)(|∇u(j)|p-2uxi(j)-|∇u|p-2uxi)φxkdx|≤C∫D∖Γ′ω|∇(u(j)-u)|p-1|∇φ|dx≤C(∫D∖Γ′ω|∇(u(j)-u)|pdx)1/p′(∫D∖Γ′ω|∇φ|pdx)1/p=δj⟶0asj⟶∞.
An analogous estimate for 2≤p<∞ has the form
|∑i,k=1n∫D∖Γ(j)ωbik(j)(|∇u(j)|p-2uxi(j)-|∇u|p-2uxi)φxkdx|≤C∫D∖Γ′ω|∇(u(j)-u)|(|∇u|p-2+|∇u(j)|p-2)|∇φ|dx≤C(∫D∖Γ′ω|∇(u(j)-u)|pdx)1/p(∫D∖Γ′ω(|∇u|p+|∇u(j)|p)dx)(p-2)/p×(∫Dω|∇φ|pdx)1/(p-2)≤2C‖∇φ‖Lpω(D)p′‖∇u‖Lpω(D∖Γ′)p-2‖∇(u(j)-u)‖Lpω(D∖Γ′)=δj⟶0asj⟶∞.

The Belongness<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M587"><mml:mi> </mml:mi><mml:mi>u</mml:mi><mml:mo>∈</mml:mo><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>D</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>

Taking into account (2.41) and (2.42), the estimate (2.32), and the uniform convergence u(j)→u in G, convergence a.e. bik(j)→bik, we find
|∑i,k=1n∫D∖Γ(j)ω|∇u|p-2bikuxkφxidx|≤C‖φ‖C(D)∑νrν-p(osc2Bνu(j))p-1ω(Bν)+δj≤C‖φ‖C(D)∑νrν-p+(p-1)αω(Bν)+δj.
Therefore,
∑i,k=1n∫D∖Γjω|∇u|p-2bikuxkφxidx=O(ε)+δj.
Taking into account the density of the class of functions C01(D) in the space Ẇpω1(D) and the fact that u∈Wpω,loc1(D∖E), we also come to the same equality (2.44) for any function φ∈Ẇpω1(D). Assuming now that, in (2.44) φ=uξp, where ξ∈C0∞(D) is a positive function equal to one in G, since Γj⊂Γ′′ and the integrand is positive, we obtain
λ∫G∖Γ′′ω|∇u|pdx≤∑i,k=1n∫D∖Γjωξ|∇u|p-2bikuxiuxkdx≤∫D∖Γjωξp-1|∇u|p-1|∇ξ||u|dx+O(ε)+βj.
Whence, by means of Young’s inequality, we derive
∫D∖Γ′′ω|∇u|pdx≤C∫Dω|∇ξ|p|u|pdx=O(1).
Then, by virtue of the arbitrariness of ε,η (mesnE=0), we obtain u∈Wpω1(D).

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Let us return to relation (2.44), from which, in view of u∈Wpω1(D) and Γ′′⊂E′, we have
∑i,k=1n∫D∖E′ω|∇u|p-2bikuxkφxidx=O(ε)
for any φ∈Ẇpω1(D). By virtue of the arbitrariness of ε,η, we find
∑i,k=1n∫Dω|∇u|p-2bikuxkφxidx=0,
that is, the function u∈Wpω1(D) is a solution of (1.5) throughout the domain D and thereby of (1.1), too.

Theorem 1.7 is proved.

Proof of Theorem <xref ref-type="statement" rid="thm1.2">1.8</xref>.

Let Λω-p+(p-1)α(E)>0 for some compact set E⊂D. Let us use the recent results for a Frosman type lemma with measure [11, 17] and follow the reasoning of the original paper [3]. We come to the following conclusion. There exists a Radon measure μ with a support on the set E, such that μ(E)>0 and for any ball B=B(x,r) we have
μ(B)≤Cr-p+(p-1)αω(B).

Let u∈Ẇpω1 be a solution of the equation
div(A(x,u,∇u))=μ
in the domain D=B(0,R), where B is a sufficiently large ball. The solution of (2.50) is understood in the sense as follows: the integral identity
∫DA(x,u,∇u)⋅∇φdx=∫Dφdμ
is fulfilled for any test function φ∈Ẇpω1(D). Such a solution exists by virtue of μ∈(Wpω1(D))*. Let us show the latter inclusion.

By virtue of ω∈Ap for q>p and the fact that q is sufficiently close to p, inequality (2.49) implies for 0<ρ<r:
r1-n(μ(B(x,ρ)))1/q(σ(B(x,ρ)))1/p′≤Cρα(p-1)/q-(n-1)(1-p/q)(σ(B(x,ρ)))(p-1)(1/p-1/q),
where σ(B(x,ρ))=∫B(x,ρ)ω-1/(p-1)(y)dy. This inequality defines the constant in the Adams inequality
(∫B(0,r)|u|qdμ)1/q≤Cp,q(r)(∫B(0,r)ω|∇u|pdx)1/p,
as
Cp,q(r)=supx∈B(0,r),0<ρ<4rρ1-n(μ(B(x,ρ)))1/q(σ(B(x,ρ)))1/p′.
Then, by virtue of (37′), we have
Cp,q(r)≤Crα(p-1)/q-(n-1)(1-p/q)(σ(B(0,r)))(p-1)(1/p-1/q).
Whence, by virtue of Hölder inequality, we obtain
∫B(0,r)|u|dμ≤Cp,q(r)(μ(B(0,r)))1/q′(∫B(0,r)ω|∇u|pdx)1/p.
Taking into account inequalities (2.49), (37′′) and the Ap-condition, for any function u∈C01(B(0,r)), we have
∫B(0,r)|u|dμ≤C(ω(B(0,r)))1/p′r1-p+(p-1)α(∫B(0,r)ω|∇u|pdx)1/p,
whence it follows that μ∈(Wpω1(D))*.

Let us, following ideas of [10], show that u(x)∈Cα(D). Let h∈Wpω1(B(x0,r)) be a solution of the equation
div(A(x,h,∇h))=0
with the condition h-u∈Ẇpω1(B(x0,r)). Then, for it, we have the integral identity
∫B(x0,r)(A(x,u,∇u)-A(x,h,∇h))⋅∇vdx=∫B(x0,r)vdμ,
where v=u-h. The integrand in the left-hand part of (2.56) is positive. Therefore, for ρ<r/2, we have
∫B(x0,ρ)(A(x,u,∇u)⋅∇u+A(x,h,∇h)⋅∇h)dx=∫B(x0,ρ)(A(x,h,∇h)∇u+A(x,u,∇u)∇h)dx+∫B(x0,r)vdμ.
By virtue of Young’s inequality and conditions (1.2), (1.3), from (2.57), we find
∫B(x0,ρ)ω|∇u|pdx≤C[∫B(x0,ρ)ω|∇h|pdx+∫B(x0,r)vdμ];C=C(n,p,λ).
From (2.55), we obtain
∫B(x0,r)A(x,h,∇h)⋅∇vdx=0,
whence by virtue of Hölder inequality, we find
∫B(x0,r)ω|∇h|pdx≤C∫B(x0,r)ω|∇u|pdx;C=C(n,p,λ).
For the first summand of (2.58), we have the following estimates. According to [14], there exists a positive number κ=κ(n,p,λ,Cp)∈(0,1) such that for the solution of (1.1) the inequality
oscB(x0,r1)h≤C(r1r2)κoscB(x0,r2)h,
where C=C(n,p,Cp,λ,κ), is fulfilled for any r1<r2. If we take into account the Caccioppoli type estimate (see [14])
∫B(x0,r1)ω|∇h|pdx≤C(r2-r1)p(oscB(x0,r2)h)pω(B(x0,r2)),
then, by virtue of Moser’s inequality, we obtain
(supB(x0,r2)h)p≤Cω(B(x0,2r2))∫B(x0,2r2)ω(h-hr2-)pdx.
From (2.60), we derive the estimate
∫B(x0,ρ)ω|∇h|pdx≤C(ρr)-p+pκω(B(x0,ρ))ω(B(x0,r))∫B(x0,r)ω|∇h|pdx.
Indeed,
∫B(x0,ρ)ω|∇h|pdx≤Cρ∫B(x0,ρ)ω|∇h|p-1|h-hρ-|pdx≤Cρp∫B(x0,ρ)ω|h-hρ-|pdx≤Cρp(oscB(x0,ρ)h)pω(B(x0,ρ))≤Cρp(ρr)κp(oscB(x0,r/2)h)pω(B(x0,ρ))≤Cρp(ρr)κpω(B(x0,ρ))(1ω(B(x0,r))∫B(x0,r)|h-hr-|pdx)≤Cρp(ρr)-p+pκω(B(x0,ρ))ω(B(x0,r))∫B(x0,r)ω|∇h|pdx,
where hr- is the lower bound of the function h in the ball B(x0,r). Inequality (2.63) is proved.

Using the estimate (2.63) in (2.58), by virtue of (41′), we have for 0<ρ<r/2∫B(x0,ρ)ω|∇u|pdx≤C(ρr)-p+pκω(B(x0,ρ))ω(B(x0,r))∫B(x0,r)ω|∇u|pdx+∫B(x0,r)vdμ.

Now, let us derive an estimate for the last summand in (2.65). To this end, we use inequality (38′) to obtain
(∫B(x0,r)vdμ)≤C(p,n,Cp)(∫B(x0,r)ω|∇v|pdx)1/p,
where for the constant we have the estimate
C(p,n,Cp)≤Cr1-p+(p-1)α(ω(B(x0,r)))1/p′
By virtue of (2.65) and (2.67), we find
∫B(x0,ρ)ω|∇u|pdx≤C(ρr)-p+pκω(B(x0,ρ))ω(B(x0,r))∫B(x0,r)ω|∇u|pdx+C[ω(B(x0,r))r(1-α)p]1/p′(∫B(x0,r)ω|∇u|pdx)1/p,
whence, by virtue of Young’s inequality, we obtain
∫B(x0,ρ)ω|∇u|pdx≤C[(ρr)-p+pκω(B(x0,ρ))ω(B(x0,r))+ε]∫B(x0,r)ω|∇u|pdx+Cεω(B(x0,r))r-p+pα.
Assuming that 0<α<κ, from (2.69) and Lemma 1.5, we obtain the estimate
∫B(x0,ρ)ω|∇u|pdx≤C(ρr)-p+pαω(B(x0,ρ))ω(B(x0,r))∫B(x0,r)ω|∇u|pdx+Cω(B(x0,ρ))ρ-p+pα.
This inequality implies
∫B(x0,ρ)|∇u|dx≤(∫B(x0,ρ)ω|∇u|pdx)1/p(∫B(x0,ρ)ω-(1/(p-1))dx)1/p′≤C(∫B(x0,ρ)ωdx)1/p(∫B(x0,ρ)ω-(1/(p-1))dx)1/p′ρ-1+α≤Cρn-1+α,
whence, by virtue of the Poincaré inequality, we obtain
∫B(x0,ρ)|u-(u)ρ|pdx≤Cρn+α,
where (u)ρ is average of the function u with respect to the ball B(x0,ρ).

By (2.72) and Campanato’s Lemma 1.4, we find u∈Cα.

Theorem 1.8 is proved.

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