We consider a nondivergent elliptic equation of second order whose leading coefficients are from some weight space. The sufficient condition of removability of a compact with respect to this equation in the weight space of Hölder functions was found.

Let D be a bounded domain situated in n-dimensional Euclidean space En of the points x=(x1,…,xn),n≥3,andlet∂D be its boundary. Consider in D the following elliptic equation:Lu=∑i,j=1naij(x)uij+∑i=1nbi(x)ui+c(x)u=0,
in supposition that ∥aij(x)∥ is a real symmetric matrix, moreover ω(x) is a positive measurable function satisfying the doubling condition: for concentric balls BRx or R and 2R radius, there exists such a constant γω(BRx)≥γω(B2Rx),
where for the measurable sets E. ω(E) means ∫Eω(y)dyγ|ξ|2ω(x)≤∑i,j=1naij(x)ξiξj≤γ-1ω(x)|ξ|2;ξ∈En,x∈D,aij(x)∈Cω1(D¯);i,j,1,…,n,|bi(x)|≤b0;-b0≤c(x)≤0;i=1,…,n;x∈D.
Here ui=∂u/∂xi,uij=∂2u/∂xi∂xj;i,j=1,…,n;γ∈(0,1] and b0≥0 are constants. Besides we will suppose that the lower coefficients of the operator ℒ are measurable functions in D. Let λ∈(0,1) be a number. Denote by C0,λ(D) a Banach space of the functions u(x) defined in D with the finite norm;‖u‖Cωλ(D)=supx∈Dω(x)|u(x)|+supx,y∈Dx≠y|u(x)-u(y)|ω|x-y|λ.

The compact E⊂D¯ is called removable with respect to (1) in the space Cωλ(D) if fromLu=0,x∈D∖E,u∣∂D∖E=0,u(x)∈Cωλ(D),
it follows that u(x)≡0 in D.

The aim of the given paper is finding sufficient condition of removability of a compact with respect to (1) in the space Cωλ(D). This problem have been investigated by many researchers. For the Laplace equation the corresponding result was found by Carleson [1]. Concerning the second-order elliptic equations of divergent structure, we show in this direction the papers [2, 3]. For a class of nondivergent elliptic equations of the second order with discontinuous coefficients the removability condition for a compact in the space Cλ(D) was found in [4]. mention also papers [5–7] in which the conditions of removability for a compact in the space of continuous functions have been obtained. The removable sets of solutions of the second-order elliptic and parabolic equations in nondivergent form were considered in [8–10]. In [11], Kilpelӓinen and Zhong have studied the divergent quasilinear equation without minor members and proved the removability of a compact. Removable sets for pointwise solutions of elliptic partial differential equations were found by Diederich [12]. Removable singularities of solutions of linear partial differential equations were considered in [13]. Removable sets at the boundary for subharmonic functions have been investigated by Dahlberg [14]. Denote by BR(z) and SR(z) the ball {x:|x-z|<R} and the sphere

{x:|x-z|=R} of radius R with the center at the point z∈En respectively. We’ll need the following generalization of mean value theorem belonging to Gerver and Landis [15] in weight case.

Lemma 1.

Let the domain G be situated between the spheres SR(0)and S2R(0), moreover let the intersection ∂G∩{x:R<|x|<2R}be a smooth surface. Further, let in G¯ the uniformly positive definite matrix ∥aij(x)∥;i,j=1,…,n and the function u(x)∈C2(G)∩Cω1(G¯) be given. Then there exists the piecewise smooth surface Σ dividing in G the spheres SR(0) and S2R(0)such that ∫Σω|∂u∂ν|ds≤KoscuG⋅ω(G)R2.
Here K>0 is a constant depending only on the matrix ∥aij(x)∥and n, and ∂u/∂ν is a derivative by a conormal determined by the equality∂u(x)∂ν=∑i,j=1naij(x)∂u(x)∂xicos(n¯,xj)1/2,
where cos(n¯,xj);j=1,…,n are direction cosines of a unit external normal vector to Σ.

Proof.

Let G⊂Rn be a bounded domain f(x)∈C2(G). Then there exists a finite number of balls {Brνxν},ν=1,2,…,N which cover Qf and such that if we denote by Sν, the surface of νth ball, then
∑ν=1N∫Sνω(x)|∇f|dx<ε.

Decompose Of into two parts: Of=Of′∪Of′′, where Of′ is a set of points Of for which ∇2f≠0,Of′′ is a set of points for which ∇2f=0.

The set Of′ has n-dimensional Lebesgue measure equal zero, as on the known implicit function theorem, the Of′ lies on a denumerable number of surfaces of dimension n-1. If we use the absolute continuity of integral
ω(D)=∫Dω(x)dx
with respect to Lebesque measure D and the above said, we get that the set Of′ may be included into the set D for which ω(D)<η,η>0 will be chosen later. Let for each point x∈O′f, there exist such rx that Brxx and B6rxx are contained in D⊂G. Then
∫5rx6rxdr∫Srxω(σ)dσ≤ω(B6rxx),
therefore there exists such 5rx≤t≤6rx that
rx∫Srxω(σ)dσ≤ω(B6rxx).
Then
∫Srxω(σ)|∇f|dσ≤Ct∫Srxω(σ)dσ≤(6C)r(rx∫Srxω(σ)dσ)≤(6C)ω(B6rxx)≤(6C)γ-3ω(Brxx)≤c0ω(Bt/5x),
where C=supD|∇2f|,α=diamG,c0=(6C)γ-3.

Now by a Banach process [4, page 126] from the ball system {Bt/5x} we choose such a denumerable number of notintersecting balls {Btν/5xν},ν=1,2,…,N that the ball of five-times greater radius {Btνxν} cover the whole Of′ set. We again denote these balls by {Btν/5xν},ν=1,2,…,N and their surface by Sν′. Then by virtue of (5)
∑ν=1∞∫Sν′ω(σ)|∇f|dσ≤C0ω(G)<C0η.
Now let x∈Of′′. Then
∫5rx6rxdr∫Srxω(σ)dσ≤ω(B6rxx).
Therefore there exists such 5rx≤t≤6rx that
rx∫Srxω(σ)dσ≤ω(B6rxx).

Assign arbitrary η>0. By virtue of that |∇f|Stx≤η·t, for sufficiently small t we have
∫Stxω(σ)|∇f|dσ≤ηt∫Stxω(σ)dσ≤(2η)(rx∫Stxω(σ)dσ)≤(2η)ω(B2rxx)≤(6C)≤(2η)γ-1ω(Bt/5x)≤ηC1ω(Bt/5x).
Again by means of Banach process and by virtue of (43) we get
∑ν=1N∫Sνnω(σ)|∇f|dσ≤η⋅C1ω(D),
where Sνn is the surface of balls in the second covering.

Combining the spherical surfaces Sν′ and Sν′′ we get that the open balls system covers the closed set Of. Then a finite subcovering may be choosing from it. Let them be the balls B1,B2,…,Bx and their surfaces are S1,S2,…,SN. We get from inequalities (4) and (7)
∑ν=1N∫Sν|∇f|ω(σ)dσ≤[C1ω(D¯)+C0]η.

Put now ε=[C1ω(D¯)+C0]η.

Following [2], assume
ε=ω(D)(oscuG)R2,
and according to Lemma 1 for a given ε we will find the balls B1,B2,…,Bx and exclude them from the domain G. Put D*=D∖⋃ν=1NBν intersect with G* a closed spherical layer
R(1+14)≤|x|≤R(1+14).
We denote the intersection by G′. We can assume that the function u(x) is defined in some δ vicinity Gδ′ of set G′. Take δ<R/4 so that
oscuGδ′≤2oscGu.

On a closed set G′ we have ∇f≠0. Consider on Gδ′ the equation system
dxdt=ux.

Let some surface S touches the direction of the field at each its point, then
∫S|∂u∂n|dσ=0,
since ∂u/∂n is identically equal to zero at S.

We will use it in constructing the needed surface of Σ. Tubular surfaces whose generators will be the trajectories of the system (50) constitute the basis of Σ.

They will add nothing to the integral we are interested in. These surfaces will have the form of thin tubes that cover G′. Then we shall put partitions to some of these tubes. Lets construct tubes. Denote by E the intersection of G′ with sphere |x|=R(1+3/4).

Let N be a set of points E. Where field direction of system (50) touches the sphere |x|=R(1+3/4). Cover N with such an open on the sphere |x|=R(1+3/4) set F that
∫Fω(x)|∂u∂n|dσ≤ω(G)(oscuD)R2.
It will be possible if on N(∂u/∂R)≡0.

Put E′=E∖F. Cover E′ on the sphere by a finite number of open domains with piece-wise smooth boundaries. We shall call them cells. We shall control their diameters in estimation of integrals that we need. The surface remarked by the trajectories lying in the ball |x|≤(7/4)R and passing through the bounds of cells we shall call tube.

So, we obtained a finite number of tubes. The tube is called open if not interesting, this tube one can join by a broken line the point of its corresponding cell with a spherical layer (5/4)R-δ<|x|<(7/4)R. Choose the diameters of cells so small that the trajectory beams passing through each cell could differ no more than δ/2n.

By choose of cells diameters the tubes will be contained in
54R-δ<|x|<54R.

Let also the cell diameter be chosen so small that the surface that is orthogonal to one trajectory of the tube intersects the other trajectories of the tube at an angle more than π/4.

Cut off the open tube by the hypersurface in the place where it has been imbedded into the layer
54R-δ2<|x|<54R
at first so that the edges of this tube be embedded into this layer.

Denote these cutoff tubes by T1,T2,…,TS. If each open tube is divided with a partition, then a set-theoretical sum of closed tubes, tubes T1,T2,…,TS their partitions spheres S1,S2,…,SN, and the set F on the sphere |x|=(7/4)R divides the spheres |x|=R and |x|=2R. Note that ∫Sω|∂u/∂n|dσ along the surface of each tube equals to zero, since ∂u/∂n identically equals to zero.

Now we have to choose partitions so that the integral ∫Sω|∂u/∂n|dσ was of the desired value. Denote by Ui the domain bounded by Ti with corresponding cell and hypersurface cutting off this tube. We have Ui∩Uj=∅ and therefore
∑i=1mω(Ui)<2ω(D).

Consider a tube Ti and corresponding domain Ui. Choose any trajectory on this tube. Denote it by Li. The length μiLi of the curve Li satisfies the inequality
μiLi≥R2.

Let introduce on Li a parameter l (length of the arc), counted from the cell. By σi(l) denote the cross-section by Ui hypersurface passing thought the point, corresponding to l and orthogonal to the trajectory Li at this point. Let the diameter of cells be so small
∫Lidl∫σi(l)ω(x)dσ<2ω(Ui).

Then by Chebyshev inequality a set H points l∈Li where
∫σi(l)ω(x)dσ>8Rω(Ui)
satisfies the inequality μiH<R/4 and hence by virtue of (55) for E=Li∖H it is valid and
μ1E>R/4.

At the points of the curve Li the derivative ∂u/∂l preserves its sign, and therefore
∫E|∂u∂l|dl≤∫Li|∂u∂l|dl≤oscuDδ′.

Hence, by using (65) and a mean value theorem for one variable function we find that there exists l0∈E‖∂u∂l‖l=l0≤4RoscuDδ′.

But on the other hand
‖∂u∂l‖l=l0=|∇u|l=l0.

Together with (67) it gives
|∇u|l=l0∫σi(l0)ω(x)dσ≤8R4Rω(Ui)(oscuD).

Now, let the diameter of cells be still so small that
∫σi(l0)ω(x)|∇u|dσ≤16⋅4Rω(Ui)(oscuD),
(we can do it, since the derivatives ∂u/∂xiare uniformly continuous). Therefore according to (53)
∑i=1S∫σi(l0)ω(x)|∇u|dσ≤16⋅4Rω(Ui)(oscuD).

Define by Σ a set-theoretical sum of all closed tubes, all open tubes Ti, all σi(l0), all spheres Si and sets F on the sphere |x|=(7/4)R. Then, we get by (4), (49), (51), and (73)
∫Σω(x)|∂u∂n|dσ≤Kω(D)(oscuD)Rp.

Then, we get by (4), (49), (51), (73)
∫Σω(x)|∂u∂n|dσ≤Kω(D)(oscuD)Rp.

The lemma is proved.

Denote by W2,ω1(D) the Banach space of the functions u(x) defined in D with the finite norm‖u‖W2,ω1(D)=(∫Dω(u2+∑i=1nui2)dx)1/2,
and let Wo2,ω1(D) be a completion of C0∞(D) by the norm of the space W2,ω1(D).

By mHs(A) we will denote the Hausdorff measure of the set A of order s>0. Further, everywhere the notation C(⋯) means that the positive constant C depends only on the content of brackets.

Theorem 2.

Let D be a bounded domain in En and let E⊂D¯ be a compact. If with respect to the coefficients of the operator ℒ the conditions (3)–(5) are fulfilled, then for removability of the compact E with respect to the (1) in the space Cωλ(D) it sufficies that
mHn-2+λ(E)=0.

Proof.

At first we show that without loss of generality we can suppose the condition ∂D∈C1 is fulfilled. Suppose that the condition (43) provides the removability of the compact E for the domains, whose boundary is the surface of the class C1, but ∂D∈C1, and by fulfilling (43) the compact E is not removable. Then the problem (7) has a nontrivial solution u(x), moreover u|E=f(x) and f(x)≠0. We always can suppose the lowest coefficients of the operator ℒ is infinitely differentiable in D. Moreover, without loss of generality, we'll suppose that the coefficients of the operator ℒ are extended to a ball B⊃D¯ with saving the conditions (3)–(5). Let f+(x)=max{f(x),0},f-(x)=min{f(x),0}, and u±(x) be generalized by Wiener (see [15]) solutions of the boundary value problems
Lu±=0,x∈D∖E,u±∣∂D∖E=0,u±∣E=f±.

Evidently, u(x)=u+(x)+u-(x). Further, let D′ be such a domain that ∂D′∈C1,D¯⊂D′,D′¯⊂B,and ϑ±(x) be solutions of the problems
Lϑ±=0,x∈D′∖E,ϑ±∣∂D′=0,ϑ±∣E=f±,ϑ±(x)∈Cωλ(D′).
By the maximum principle for x∈D,
0≤u+(x)≤ϑ+(x),ϑ-(x)≤u-(x)≤0.
But according to our supposition, ϑ+(x)≡ϑ-(x)≡0. Hence, it follows that u(x)≡0. So, we'll suppose that ∂D∈C1. Now, let u(x) be a solution of the problem (7), and the condition (43) be fulfilled. Give an arbitrary ɛ>0. Then there exists a sufficiently small positive number δ and a system of the balls {Brk(xk)},k=1,2,…, such that rk<δ,E⊂⋃k=1∞Brk(xk) and
∑k=1∞rkn-2+λ<ε.

Consider a system of the spheres {B2rk(xk)}, and let Dk=D∩B2rk(xk),k=1,2,…. Without loss of generality we can suppose that the cover {B2rk(xk)} has a finite multiplicity a0(n). By the Landis-Gerver theorem, for every k, there exists a piece-wise smooth surface Σk dividing in Dk the spheres Srk(xk) and S2rk(xk), such that
∫Σkω|∂u∂ν|ds≤KoscuDkω(Dk)rk2.
Since u(x)∈Cωλ(D), there exists a constant H1>0 depending only on the function u(x) such that
oscuDkω≤H1(2rk)λ.
Besides,
ω(Dk)≤mesnB2rk(xk)=Ωn2nrkn;k=1,2,…,
where Ωn=mesnB1(0). Using (49) and (50) in (48), we get
∫Σkω|∂u∂ν|ds≤C1rkn-2+λ;k=1,2,…,
where C1=KH12n+λ.

Let DΣ be an open set situated in D∖E whose boundary consists of unification of Σ and Γ, where Σ=⋃k=1∞Σk,Γ=∂D∖⋃k=1∞Dk+,Dk+ is a part of Dk remaining after the removing of points situated between Σ and S2rk(xk);k=1,2,…. Denote by DΣ′ the arbitrary connected component DΣ, and by ℳ we denote the elliptic operator of divergent structure
M=∑i,j=1n∂∂xi(aij(x)∂∂xj).

According to Green formula for any functions z(x) and W(x) belonging to the intersection C2(DΣ′)∩C1(D¯Σ′), we have
∫DΣ′(zMβ-βMz)dx=∫∂DΣ′(z∂β∂ν-β∂z∂ν)ds.

Since ∂D∈C1, then u(x)∈C1(DΣ′)∩C1(DΣ′¯)(x)∈C1(D¯Σ′) (see [16]). From (53) choosing the functions z=1,β=ωu2, we have
∫DΣ′M(ωu2)dx=2∫∂DΣ′ωu∂u∂νds+∫∂DΣωxiu2ds.

But |u(x)|≤M<∞ for x∈D¯. Let us put the condition
ωxi<cω.

By virtue of condition (52) and ∫∂DΣωu2ds<C3Mε,subject to (51) and (47), we conclude
∫DΣ′M(ωu2)dx≤2Ma0∑k=1∞∫Σkω|∂u∂ν|ds+∫DΣ′ωu2dx≤2Ma0C1∑k=1∞rkn-2+α+εMc2<C3ε,
where C3=2Ma0C1.

On the other hand
M(ωu2)=6uωM(u)+2∑i,j=1nωaijuiuj+(2u+1)∑i,j=1naijuxjωxi+∑i,j=1n∂aij∂xiuωxj+∑i,j=1naijuωxixj
and besides,
Mu=Lu+∑i=1ndi(x)ui-c(x)u,
where
di(x)=∑j=1n∂aij(x)∂xj-bi(x),i=1,…,n.
It is evident that by virtue of conditions (4) and (5) |di(x)|≤d0<∞;i=1,…,n. Thus, from (55) we obtain
6∫DΣ′uω∑i=1ndi(x)uidx-6∫DΣ′u2c(x)dx+2∫DΣ′∑i,j=1nω(x)aijuiujdx+(2u+1)∫DΣ′∑i,j=1naijujωxidx+∫DΣ′∑i,j=1n∂aij∂xjuωxidx+|∇u|2dx+∫DΣ′∑i,j=1naijuωxixjdx<C3ε.

Hence, for any α>0 it follows that
2γ∫DΣ′ω|∇u|2dx<6d0∫DΣ′ω|u||ui|dx+6∫DΣ′u2ω(x)+(2u+1)∫DΣ′aijujωxidx+d0∫DΣ′uωxi2dx+∫DΣ′aijuωxixj+C3ε≤6d0ε∫DΣ′|u|2dx+6d0ε2∫DΣ′ω2|∇u|2dx+(2n+1)∫DΣ′ujωdx+d0∫DΣ′uωdx+γC4ε≤6d0εMmesnD+(2M+1)γεmesnD+d0Mω(D)+γC4Mω(D)+C3ε.

If we take into account that
|ωxixj|<C4ω(x),
then from here we have that
∫DΣ′ω2|∇u|2dx≤C5,
where C5=(6d0+(2M+1))MmesnD+(d0M+γC4M)ω(D)+C3/γ. Without loss of generality we assume that ε≤1. Hence we have∫Dω2|∇u|2dx≤C6.

Thus u(x)∈W2,ω1(D). From the boundary condition and mesn-1(∂D∩E)=0 we get u(x)∈W2,ω1(D). Now, let σ≥2 be a number which will be chosen later, DΣ+={x:x∈DΣ′,u(x)>0}. Without loss of generality, we suppose that the set DΣ+ is not empty. Supposing in (53) z=1,β=ωuσ, we get
∫DΣ+M(ωuσ)dx=σ∫∂DΣ+(ωνuσ+σuσ-1∂u∂ν)ds≤Mσ∫∂DΣ+ωds+σMσ-1∫∂DΣ+|∂u∂ν|ds≤C5(a0,M,σ,C1)ε.

But, on the other hand,
M(uσ)=∑i,j=1n∂∂xi(aij∂ωuσ∂xj)=∑i,j=1n∂∂xi(aijω(σuσ-1∂u∂xj)+∑i,j=1n∂∂xi(aijωxi∂uσ∂xj))=∑i,j=1n∂∂xi(aijωσuσ-1∂u∂xj)+∑i,j=1n∂∂xi(aijσuσ-1ωx∂u∂xj)=σωuσ-1M(u)+σω∂∂xi(aijuσ-1∂u∂xj)+σuσ-1∂∂xi(aijω∂u∂xj)+β=σωuσ-1M(u)+σωuσ-1∂∂xi(aij∂u∂xj)+σωaijuxj(σ-1)uσ-2uxi+σuσ-1ωxi(aij∂u∂xj)+σuσ-1ω∂∂xi(aij∂u∂xj)+β=3σωuσ-1M(u)+σ(σ-1)aijuxiuxjuσ-2ω+σuσ-1ωxiaijuxj+β=σ∫DΣ+di(x)uxiuωdx-σ(σ-1)∫DΣ+uσω(x)c(x)dx+σ(σ-1)∫DΣ+∑i,j=1nuσ-2ω(x)aijuiujdx+(2u+1)∫DΣ+∑i,j=1naijujωxjuσ-1.

Hence, we conclude
σ(σ-1)∫DΣ+ω2uσ-2|∇u|2dx≤d0∫DΣ+uσ-1ωuidx≤d0∫DΣ+uσ-1ωuidx≤d0ε2∫DΣ+uσdx.

Let D+={x:x∈D,u(x)>0},D1+ an arbitrary connected component of D+. Subject to the arbitrariness of ɛ from (65) we get
(σ-1)γ∫D1+ωuσ-2|∇u|2dx≤d0∫D1+ωuσ-1∑i=1n|ui|dx.

Thus, for anyμ>0(σ-1)γ∫D1+ωuσ-2|∇u|2dx≤d0μ2∫D1+ωuσ-2(∑i=1n|ui|)2dx+d02μ∫D1+ωuσdx≤d0μn2∫D1+ωuσ-2|∇u|2dx+d02μ∫D1+ωuσdx..

But, on the other hand,
I=-σ∑i=1n∫D1+xiωuσ-1uidx=-∑i=1n∫D1+xiω(uσ)idx=n∫D1+ωuσdx.
and besides, for anyβ>0I=σβ2∫D1+r2ωuσdx+σ2β∫D1+uσ-2ω2|∇u|2dx.

Then
I≤σβ2∫D1+r2ωuσdx+σ2β∫D1+ω2|∇u|2uσ-2dx,
where r=|x|. Denote by k(D) the quantity supx∈D|x|. Without loss of generality we’ll suppose that k(D)=1. Then
I≤σ2β∫D1+ωuσdx+σ2β∫D1+ω2uσ-2|∇u|2dx.

Thus,
(n-σβ2)∫D1+ωuσdx+σ2β∫D1+ω2uσ-2|∇u|2dx.

Now, choosing β=n/σ, we finally obtain
∫D1+ωuσdx≤σ2n2∫D1+ω2uσ-2|∇u|2dx.

Subject to (73) in (67), we conclude
(σ-1)γ∫D1+ω2uσ-2|∇u|2dx≤(d0εn2+d0σ22εn2)∫D1+ω2uσ-2|∇u|2dx.

Now choose μ such that
(σ-1)γ>d0μn2+d0σ22μn2.

Then from (73)–(75) it will follow that u(x)≡0 in D1+, and thus u(x)≡0 in D. Suppose that μ=(σ-1)γ/d0n. Then (75) is equivalent to the condition
n>(σσ-1)2(d0γ)2.

At first, suppose that
n>(d0γ)2.

Let’s choose and fix such a big σ≥2 that by fulfilling (77) the inequality (76) is true. Thus, the theorem is proved, if with respect to n the condition (77) is fulfilled. Show that it is true for any n≥3. For that, at first, note that if k(D)≠1, then condition (77) will take the form
n>(d0k(D)γ)2.

Now, let the condition (77) be not fulfilled. Denote by k the least natural number for which
n+k>(d0γ)2.

Consider (n+k)-dimensional semicylinder D′=D×(-δ0,δ0)×⋯×(-δ0,δ0), where the number δ0>0 will be chosen later. Since ω(D)=1, then ω(D′)≤1+δ0k. Let’s choose and fix δ0 so small that along with the condition (79), the condition
n+k>(d0ω(D′)γ)2
was fulfilled too.

Let
y=(x1,…,xn,xn+1,…,xn+k),E′=E×[-δ0,δ0]×⋯×[-δ0,δ0]︸ktimes.

Consider on the domain D′ the equation
Lϑ′=∑i,j=1naij(x)ϑij+∑i=1k∂2ϑ∂xn+i2+∑i=1nbi(x)ϑi+c(x)ϑ=0.

It is easy to see that the function ϑ(y)=u(x) is a solution of (82) in D′∖E′. Besides, mHn+k-2+λ(E′)=(2δ0)kmHn-2+λ(E)=0, the function ϑ(y) vanishes on (∂D×[-δ0,δ0]×··×[-δ0,δ0]︸ktimes)∖E′ and ∂ϑ/∂ν′=0 at xn+i=±δ0,i=1,…,k, where ∂/∂ν′ is a derivative by the conormal generated by the operator ℒ′. Noting that γ(ℒ′)=γ(ℒ),d0(ℒ′)=d0(ℒ) and subject to the condition (80), from the proved above we conclude that ϑ(y)≡0, that is, D′. The theorem is proved.

Remark 3.

As is seen from the proof, the assertion of the theorem remains valid if instead of the condition (4) it is required that the coefficients aij(x)(i,j=1,…,n) have to satisfy in domain D the uniform Lipschitz condition with weight.

CarlesonL.MoiseevE. I.On Neumann problem in piecewise smooth domainsMoiseevE. I.On existence and non-existence boundary sets for the Neumann problemLandisE. M.To question on uniqueness of solution of the first boundary value problem for elliptic and parabolic equations of the second orderKondratyevV. A.LandisE. M.Qualitative theory of linear partial differential equations of second orderMamedovI. T.On exceptional sets of solutions of Dirichlet problem for elliptic equations of second order with discontinuous coefficientsGerverM. L.LandisE. M.One generalization of a theorem on wean value for multivariable functionsKayzerV.MullerB.Removable sets for heat conductionMamedovaV. A.On removable sets of solutions of boundary value problems for elliptic equations of the second orderGadjievT. S.MamedovaV. A.On removable sets of solutions of second order elliptic and parabolic equations in nondivergent formKilpeläinenT.ZhongX.Removable sets for continuous solutions of quasilinear elliptic equationsDiederichJ.Removable sets for pointwise solutions of elliptic partial differential equationsHarveyR.PolkingJ.Removable singularities of solutions of linear partial differential equationsDahlbergB. E. J.On exceptional sets at the boundary for subharmonic functionsLandisE. M.GilbargD.TrudingerN. S.