parameterand isarealvector-valuedfunctioncalledthemagneticpotential, Letusrecallthatsuperconductivityintwoorthreedimensionalspacecanbe isacomplex-valuedfunction. and aredefinedby = , is =[0 ) .Here isarealvector-valuedfunction(calledmagneticpotential),

Tomus 46 (2010), 185–201

ESTIMATE OF THE HAUSDORFF MEASURE OF THE SINGULAR SET OF A SOLUTION

FOR A SEMI-LINEAR ELLIPTIC EQUATION ASSOCIATED WITH SUPERCONDUCTIVITY

Junichi Aramaki

Abstract. We study the boundedness of the Hausdorff measure of the singular set of any solution for a semi-linear elliptic equation in general dimensional Euclidean spaceRⁿ. In our previous paper, we have clarified the structures of the nodal set and singular set of a solution for the semi-linear elliptic equation.

In particular, we showed that the singular set is (n−2)-rectifiable. In this paper, we shall show that under some additive smoothness assumptions, the (n−2)-dimensional Hausdorff measure of singular set of any solution is locally finite.

1. Introduction We consider a semi-linear elliptic equation

(1.1) − ∇²_Aψ=f(|ψ|²)ψ in Ω

where Ω is a bounded domain inRⁿ and f is a real-valued, bounded function on R⁺= [0,∞). HereAis a real vector-valued function (called magnetic potential), ψ is a complex-valued function.∇A and∇²_A are defined by∇A=∇ −iA,∇ is the gradient operator and

∇²_Aψ= ∆ψ−i[2A· ∇ψ+ (divA)ψ]− |A|²ψ .

This type of operator is considered in Aramaki [1, 2, 3, 5] and Pan and Kwek [24].

Associated with the magnetic potentialA= (A1, A2, . . . , An), define an anti-sym- metricn×nmatrixB= (B_ij) called the magnetic vector field by

B_ij =∂_xiA_j−∂_xjA_i for i, j= 1,2, . . . , n .

Let us recall that superconductivity in two or three dimensional space can be described by a pair (ψ,A), whereψis a complex-valued function called the order parameter and Ais a real vector-valued function called the magnetic potential,

2000Mathematics Subject Classification: primary 82D55; secondary 47F05, 35J60.

Key words and phrases: singular set, semi-linear elliptic equation, Ginzburg-Landau system.

Received June 12, 2009, revised April 2010. Editor M. Feistauer.

which is a minimizer of the so-called Ginzburg-Landau functional. The Euler equation becomes

(1.2)

(−∇²_Aψ=κ²(1− |ψ|²)ψ in Ω, ν· ∇Aψ= 0 on ∂Ω

where Ω⊂Rⁿ withn= 2 or n= 3 is a bounded domain andν is the outer unit normal vector at∂Ω. It is well known that any solution of (1.2) satisfies|ψ| ≤1 in Ω. If we choose a bounded functionf on [0,∞) so thatf(t) =κ²(1−t) for|t| ≤1, the first equation of (1.2) is of the form (1.1).

In the superconductivity theory or the Landau-de Gennes model of liquid cristal, it is important to know the third critical fieldHc₃ orQc₃. It is associated with the lowest eigenvalue of the magnetic Schrödinger operator of type−∇²_qA, i.e., (1.3)

(−∇²_qAψ=µ(qA)ψ in Ω, ν· ∇qAψ= 0 on ∂Ω.

If we putf(t) =µ(qA) which is a constant, the first equation of (1.3) is also of form (1.1). For the superconductivity theory, see Lu and Pan [17], [18] and Pan [22]. For the theory of liquid cristal with A=nwhich is a unit vector field, see Pan [21]. Helffer and Mohamed [15] and Helffer and Morame [16] have extensively considered the eigenvalue problem for the magnetic Schrödinger operator of type

−∇²_A forn≥2.

In the equation (1.2), the nodal set{x∈Ω ;ψ(x) = 0}means the normal state there. Pan [23] has studied the structure of the nodal set and the singular set {x∈Ω ;ψ(x) = 0, ∇ψ(x) = 0} of any non-trivial solution of (1.1) in the three dimensional domain.

In the previous paper Aramaki [4], we showed that the nodal set and the singular set of any non-trivial solution of (1.1) in the general ndimensional domain are (n−1) and (n−2)-rectifiable, respectively.

For the second order linear elliptic equations with the real coefficients, there are many articles on the nodal set or the singular set. For example, see Garofalo and Lin [9], Han [11], [12] and Han et al. [13]. In particular, Hardt et al. [14] proved that for any non-trivial solution of a linear elliptic equation with real smooth coefficients, the (n−2)-dimensional Hausdorff measure of the singular set is locally finite.

However it seems that there are not many articles on the structure of the singular set of complex-valued solutions of equations of type (1.1) (cf. Elliot et al. [7]).

In this paper, we shall estimate the (n−2)-dimensional Hausdorff measure of singular set of any non-trivial complex-valued solutionψof (1.1).

We assume that

(H) A∈L^∞(Ω;Rⁿ),divA∈L^q_loc(Ω) withq > n/2 ifn≥4 andq≥2 ifn= 3, andB ∈L^∞(Ω;Rⁿ

2).

Our main result on the singular set is the following.

Theorem 1.1. Let Ω ⊂ Rⁿ (n ≥ 3) be a bounded domain, assume that the hypothesis (H) holds, and let ψ∈ W_loc^1,2(Ω;C) be any non-trivial complex-valued weak solution of (1.1)with∇Aψ∈W_loc^1,2(Ω;Cⁿ)andf_∞:=kf(|ψ|²)kL^∞(Ω)<∞.

Then there exists an integer M >0 depending onψ,f_∞ andkBkL^∞(Ω) such that if, in addition, A ∈C^M(Ω;Rⁿ), divA∈C^M(Ω) and f ∈ C^M([0,∞)), then for any Ω⁰ bΩ, there exists a constantC >0depending onM, ψ,f_∞,kAk_L^∞_(Ω),Ω⁰, C^M(Ω⁰⁰)norms ofA,divA,f(|ψ|²)for someΩ⁰bΩ⁰⁰bΩsuch that

(1.4) Hⁿ⁻² {x∈Ω⁰;ψ(x) = 0,∇ψ(x) = 0}

≤C whereHⁿ⁻² is the(n−2)-dimensional Hausdorff measure.

2. Preliminaries

In this section, we shall list up some propositions which are needed later and held under the hypothesis (H). All the propositions and theorem are found in [4]

(c.f. [23]).

At first, we have the regularity of the solution.

Proposition 2.1. Assume that the hypothesis(H)holds and letψ∈W_loc^1,2(Ω;C)be any weak solution of (1.1). Thenψ∈W_loc^2,q(Ω;C)∩C_loc^α (Ω;C)for some α∈(0,1), and for any B2R(x0)bΩand1< p≤q, there exists a constantC >0 depending on p, qandkAkL^∞(Ω) such that

R²kD²ψk_Lp(BR(x₀))+Rk∇ψk_Lp(BR(x₀))

≤C

kψk_Lp(B_R(x₀))+R²kf(|ψ|²)ψ−i(divA)ψk_Lp(B_R(x₀)) . Next, we state the doubling property of solutions. Let ψ 6≡ 0 be any weak solution of (1.1). For anyBr(x0)bΩ, we define some quantities.

I(x0, r) = Z

B_r(x₀)

{|∇Aψ|²−f(|ψ|²)|ψ|²}dx , (2.1)

H(x₀, r) = Z

∂Br(x0)

|ψ|²dSr, D(x₀, r) = Z

Br(x0)

|∇ψ|²dx , M(x0, r) =rI(x0, r)

H(x0, r), N(x0, r) = rD(x0, r)

H(x0, r) if H(x0, r)6= 0 wheredSr denotes the surface area of∂Br(x0). Then we have

Proposition 2.2. Assume that the conditions of Theorem 1.1 hold for any non-trivial weak solution ψ ∈ W^1,2(Ω;C). Then there exist r₀, c₀, N >0 where r₀ depends only on f_∞, and c0 and N depend only onΩ, ψ, f_∞ and kBk_L∞(Ω) such that for any 0< r≤r₀/2 with B2r(x₀)bΩ, we have the following.

(i) M(x0, r)≤c0, (ii)

Z

B_r(x₀)

|ψ|²dx≤r Z

∂B_r(x₀)

|ψ|²dS , and the doubling property:

(iii)

Z

B2r(x0)

|ψ|²dx≤4^N Z

Br(x0)

|ψ|²dx .

If ψ6≡0 is a weak solution of (1.1) and satisfies that R

∂Br(x0)|ψ|²dS_r= 0 for some 0< r≤r0/2 withB2r(x0)bΩ, it follow from (ii) that R

B_r(x₀)|ψ|²dx= 0.

Therefore, from the unique continuation theorem (cf. Aronszajn [6]) or the doubling property (iii) (cf. [9]),ψ≡0 in Ω. Thus we have

H(x0, r) = Z

∂Br(x0)

|ψ|²dSr6= 0

for any 0< r≤r0/2 withB2r(x0)bΩ, and so we see that M(x0, r) andN(x0, r) are well defined. From Proposition 2.2 (i) we see that M(x0, r) ≤ c0 for any 0< r≤r0/2 withB2r(x0)bΩ.

We get an important fact.

Proposition 2.3 ([4] or [23]). Assume that the conditions of Theorem 1.1 for any non-trivial weak solution ψ∈W_loc^1,2(Ω;C)of (1.1). Then we have

(2.2) lim

r→0M(x0, r) = lim

r→0N(x0, r) for any x0∈Ω and the limit is a non-negative integer.

From this proposition, we can define the vanishing order ofψatx₀∈Ω by Oψ(x₀) = lim

r→0M(x₀, r) = lim

r→0N(x₀, r). Of course, ifψis smooth enough, we see that

(D^αψ(x₀) = 0 for anyα with |α|<Oψ(x₀), D^βψ(x0)6= 0 for some β with|β|=Oψ(x0) whereD^α=∂^α=∂^|α|/∂x^α.

We note that the vanishing order of ψis uniformly bounded in Ω, i.e.,

(2.3) Oψ(x)≤c0 for x∈Ω

wherec0 is the constant as in Proposition 2.2 (i) and depends only onψ, Ω,f_∞ andkBk_L^∞_(Ω).

Next, we state the decomposition of the solution of (1.1).

Proposition 2.4 (cf. [4], [23] and [11]). Assume that the conditions of Theorem 1.1 hold for any non-trivial weak solutionψ of (1.1). Then for any0< R≤r0/2 with B2R(x0)bΩ, there exists an integerm≥0 such that we can write

(2.4) ψ(x+x0) =Pm(x) +φ(x), x∈BR(0)

where Pm is a non-zero, complex-valued homogeneous, harmonic polynomial of degree m, andφsatisfies

(2.5) |φ(x)| ≤C|x|^m+α in B_R(0)

for some α∈(0,1), and a constant C >0which depends only on m,kAkL^∞(Ω), kdivAkL²(B_2R(x₀)) andf_∞.

We call P_m = ψ_x0 the leading polynomial of ψ at x₀. We see that m is the vanishing order ofψatx₀, som=Oψ(x₀)≤c₀. Now we define the singular set of ψ by

S(ψ) ={x∈Ω;O_ψ(x)≥2}. In the previous paper [4], we showed

Theorem 2.5. Assume the the conditions of Theorem 1.1 hold for any non-trivial weak solution ψ ∈ W^1,2(Ω;C). Then for any Ω⁰ b Ω, S(ψ)∩Ω⁰ is countably (n−2)-rectifiable, more precisely, if we define

S_∗(ψ) ={x∈ S(ψ); the leading polynomial ofψatxis a polynomial of two variables after some rotation of coordinates}, then S(ψ)∩Ω⁰\ S_∗(ψ)is countably (n−3)-rectifiable. Thus

(2.6) Hⁿ⁻²(S(ψ)∩Ω⁰\ S_∗(ψ)) = 0. 3. Estimate of the singular set

In this section, we shall estimate the Hausdorff measure of the singular set of any non-trivial weak solution of (1.1). In addition to the hypothesis (H), we assume that for an integer M ≥1,

(K)_M A∈C^M(Ω;Rⁿ),divA∈C^M(Ω) andf ∈C^M([0,∞)).

In the following, for any given Ω⁰bΩ, we always choose

Ω⁰⁰={x∈Ω; dist (x, ∂Ω)>min(r0,dist (Ω⁰, ∂Ω))/3}, wherer0is as in Proposition 2.2, and define

(3.1) Λ(Ω⁰⁰) =|A|_CM(Ω⁰⁰)+|A|²_CM(Ω⁰⁰)+|divA|_CM(Ω⁰⁰)+|f(|ψ|²)|_CM(Ω⁰⁰), ifψ∈C_loc^M(Ω;C). We also use the notations

|D^jψ(x)|= X

|β|=j

|D^β_xψ(x)|,

|D^jψ(x)−D^jψ(y)|=|D^j(ψ(x)−ψ(y))|.

At first, we obtain the regularity of any solution of (1.1) under the hypotheses (H) and (K)_M.

Proposition 3.1 (Regularity). Addition to the hypothesis(H), assume that(K)_M holds for some integer M ≥1. Letψ∈W_loc^1,2(Ω;C) be any weak solution of (1.1).

Then ψ∈C_loc^M+1,α(Ω;C)for someα∈(0,1). Moreover, we have the Schauder type estimate: for anyΩ⁰ bΩ, there exists R0>0depending onn, M, α,ΩandΛ(Ω⁰⁰) such that for all0< R≤R0 andx0∈Ω⁰,

M+1

X

j=1

R^j sup

x∈BR(x₀)

|D^jψ(x)|

+R^M+1+α sup

x,y∈BR(x₀) x6=y

|D^M+1ψ(x)−D^M+1ψ(y)|

|x−y|^α ≤C sup

x∈B2R(x₀)

|ψ(x)|

where the constant C depends onn,M,α,Ω,Ω⁰ andΛ(Ω⁰⁰).

Proof. By Proposition 2.1, we see that ψ ∈ C_loc^α (Ω;C)∩W_loc^2,q(Ω;C) for some α ∈(0,1) andψ satisfies the equation (1.1). We note that A∈ C^M(Ω;Rⁿ),→ C_loc^M−1,α(Ω;Rⁿ). Similarly, divA,|A|²belong toC_loc^M−1,α(Ω). Sincef ∈C^M(R+),→ C_loc^M−1,1(R+), we havef(|ψ|²)ψ ∈C_loc^α (Ω;C). Therefore, it follows from Gilbarg and Trudinger [10, Theorem 9.19] thatψ∈C_loc^2,α(Ω;C). By the boot-strap method, we see thatψ∈C_loc^M+1,α(Ω;C).

Next, we shall get the estimate. In order to do so, we write (1.1) into the form:

(3.2) −∆ψ+ 2iA· ∇ψ+ i(divA) +|A|²

ψ=f(|ψ|²)ψ in Ω. We simply write 3R1= min r0,dist (Ω⁰, ∂Ω)

and we choose Ω⁰⁰ as above. Then for any 0 < R < R1 andx0 ∈Ω⁰, B2R(x0)⊂Ω⁰⁰. We shall apply the Schauder estimate [10, p.142] in B2R(x0). We use the following notations as in [10]. For g∈C^k,α Br(x0)

,

|g|^(σ)_k,α,B

r(x₀)=|g|^(σ)_k,B

r(x₀)+ [g]^(σ)_k,α,B

r(x₀), where

|g|^(σ)_k,B

r(x₀)=

k

X

j=0

sup

x∈Br(x₀)

d^j+σ_x |D^jg(x)|,

[g]^(σ)_k,α,B

r(x0)= sup

x,y∈B_r(x₀) x6=y

d^k+α+σ_x,y |D^kg(x)−D^kg(y)|

|x−y|^α ,

|g|^∗_k,α,Br(x0)=|g|⁽⁰⁾_k,α,B

r(x0), dx= dist x, ∂Br(x0)

, dx,y= min(dx, dy). It follows from the hypothesis (K)M that

|A|⁽¹⁾_M−1,α,B

2R(x₀)+|divA|⁽²⁾_M−1,α,B

2R(x₀)+||A|²|⁽²⁾_M−1,α,B

2R(x₀)

≤CΛ(Ω⁰⁰)<∞

for some constant Cdepending only on Ω. Therefore we can apply the result of [10] to (3.2) to get

(3.3) |ψ|^∗_M+1,α,B

2R(x₀)≤C |ψ|_0,B2R(x0)+|f(|ψ|²)ψ|⁽²⁾_M−1,α,B

2R(x₀)

whereC depends onn, M, α,Ω and Λ(Ω⁰⁰).

We estimate the last term in the right hand side of (3.3). Since ψ∈C^M+1,α(B2R(x0))⊂C^M+1,α(Ω⁰⁰) (M ≥1), we have

M−1

X

j=0

sup

x∈B2R(x₀)

d^j+2_x |D^j[f(|ψ(x)|²)ψ(x)]| ≤C

M−1

X

j=0

sup

x∈B2R(x₀)

d^j+2_x |D^jψ(x)|

≤C₁R²

M−1

X

j=0

sup

x∈B2R(x0)

d^j_x|D^jψ(x)|

where the constantC1depends on Λ(Ω⁰⁰). Similarly we can estimate sup

x,y∈B2R(x0) x6=y

d^M+1+α_x,y |D^M−1[f(|ψ(x)|²)ψ(x)]−D^M−1[f(|ψ(y)|²)ψ(y)]|

|x−y|^α

= sup

x,y∈B2R(x0) x6=y

d^M_x,y^+1+α|D^M−1[f(|ψ(x)|²)ψ(x)]−D^M−1[f(|ψ(y)|²)ψ(y)]|

|x−y|

× |x−y|^1−α

≤C2 M

X

j=0

sup

x,y∈B2R(x₀)

sup

z∈B2R(x₀)

d^M+1+α_x,y |D^jψ(z)||x−y|^1−α

≤C₃R²

M

X

j=0

sup

z∈B2R(x0)

d^j_z|D^jψ(z)|

where the constantC3also depends on Λ(Ω⁰⁰). Thus we see that

|f(|ψ|²)ψ|⁽²⁾_M−1,α,B

2R(x₀)≤C4R²|ψ|^∗_M+1,α,B

2R(x₀)

whereC4 depends on Λ(Ω⁰⁰). If we chooseR0>0 so thatCC4R²₀<1/2 whereC is as in (3.3) andR0< R1, then it follows from (3.3) that for all 0< R≤R0

|ψ|^∗_M+1,α,B

2R(x₀)≤C|ψ|_0,B2R(x0) where the constantC depends onn,M, α, Ω and Λ(Ω⁰⁰). Since

dx= dist (x, ∂B2R(x0))≥Rforx∈BR(x0), we obtain the conclusion.

We choose an integer M ≥1 in Theorem 1.1 so that

(3.4) M ≥2c²₀

wherec0 is the constant as in Proposition 2.2 (i). We note that it follows from (2.3) that the vanishing order ofψis uniformly bounded in Ω :Oψ(x)≤c0for allx∈Ω.

Let ψ be any non-trivial weak solution ψ of (1.1) and Ω⁰ bΩ. Then for all x0∈Ω⁰ and 0< R < R0 whereR0is as in Proposition 3.1,ψ has a decomposition in B2R(x0)bΩ⁰⁰:

(3.5) ψ(x+x0) =Pm(x) +φ(x), x∈BR(0)

where P_m is a non-zero complex-valued homogeneous, harmonic polynomial of degreemandφsatisfies (2.5).

We estimate the remainder termφ.

Lemma 3.2. Assume that the hypotheses(H) and(K)M hold. Then in the decom- position of ψ in (3.5),φ satisfies

|D^jφ(x)| ≤

(CR^m−j+α, j= 0,1, . . . , m , C , j=m+ 1, . . . , M+ 1 in BR(0)where the constant C depends onn, M, ψ,ΩandΛ(Ω⁰⁰).

Proof. SincePmis harmonic inRⁿ, we have

∆φ(x) =−∆ψ(x+x0) in BR(0).

Therefore we can apply the Schauder estimate as in the proof of Proposition 3.1, so we can get

M+1

X

j=0

R^j sup

x∈BR(0)

|D^jφ(x)|

≤Cn sup

x∈B_R(0)

|φ(x)|+

M−1

X

j=0

R^j+2 sup

x∈B_R(x₀)

|D^j∆ψ(x)|

+R^M+1+α sup

x,y∈BR(x0) x6=y

|D^j∆ψ(x)−D^j∆ψ(y)|

|x−y|^α

o (3.6)

whereC depends onn,M and Ω. We write the equation (1.1) into the form

∆ψ= 2iA· ∇ψ+i(divA)ψ+|A|²ψ−f(|ψ|²)ψ .

Then applying Proposition 3.1, we shall estimate the last two terms in (3.6). In the following we denote constants depending only on Ω and Λ(Ω⁰⁰) byC which may vary from line to line. For 0≤j≤M−1, we have

R^j+2 sup

x∈BR(x₀)

|D^j∆ψ(x)|

=R^j+2 sup

x∈BR(x0)

D^j[2iA· ∇ψ+i(divA)ψ+|A|²ψ−f(|ψ|²)ψ]

≤CR^j+2

j+1

X

k=0

sup

x∈BR(x0)

|D^kψ(x)|

≤CR

j+1

X

k=0

R^k sup

x∈BR(x₀)

|D^kψ(x)|

≤CR sup

x∈B2R(x₀)

|ψ(x)| ≤CR^m+1.

We can similarly estimate the last term in (3.6). Thus we get

M+1

X

j=0

R^j sup

x∈BR(0)

|D^jφ(x)| ≤C{ sup

x∈BR(0)

|φ(x)|+R^m+1} ≤CR^m+α.

Therefore, forj= 0,1, . . . , m, we have

|D^jφ(x)| ≤CR^m−j+α in B_R(0).

For j=m+ 1, . . . , M+ 1, sinceD^jP_m≡0, we see thatD^jφ(x) =D^jψ(x+x₀) in B_R(0). Thus we have

|D^jφ(x)| ≤ sup

x∈B_R(x₀)

|D^jψ(x)| ≤ sup

x∈Ω⁰⁰

|D^jψ(x)| ≤Cj in BR(0).

This completes the proof.

Now we show a property of a complex-valued harmonic polynomial.

Lemma 3.3(cf. [13]). LetP be a complex-valued non-zero homogeneous, harmonic polynomial of degreem≥2, and of two variables inRⁿ. Then there existδ_∗, r_∗>0 depending on P such that ifϕ∈C^2m2(B1(0);C)satisfies|ϕ−P|_C2m2(B1(0);C)< δ_∗, then

Hⁿ⁻²(|∇ϕ|⁻¹{0} ∩Br(0))≤c(n)(m−1)²rⁿ⁻² for all 0< r≤r∗.

Proof.

Step 1. It suffices to prove the case whereP andϕare real-valued.

In fact, assume that Lemma 3.3 holds for the case wherePandϕare real-valued.

LetP andϕbe complex-valued functions satisfying the hypotheses in the lemma.

Since either of<Por=Pis non-zero, let<P 6≡0. We chooseδ_∗andr_∗corresponding to <P. If |ϕ−P|_C2m2(B1(0);C) < δ_∗, then |<ϕ− <P|_C2m2(B1(0);C) < δ_∗. Since

|∇ϕ|⁻¹{0} ⊂ |∇<ϕ|⁻¹{0}, we get the conclusion.

Step 2. We shall show the lemma for the real case. Though the proof is identical as [13, Lemma 3.2], we introduce an outline of the proof. We choose a coordinates xe= (xe1,xe2, . . . ,exn)∈Rⁿ, and the polar coordinates ex1 =rcosθ,ex2 =rsinθ in R². By the hypothesis onP, we may assume that

P(x) =e r^mcosmθ . Then we have

D e^x1

P(ex) =mr^m−1cos(m−1)θ , D

e^x2

P(ex) =mr^m−1sin(m−1)θ . By the formulae:

cos(m−1)θ= 2^m−2

m−1

Y

r=1

sin

θ+(2r−1)π 2(m−1)

,

sin(m−1)θ= 2^m−2

m−1

Y

r=1

sin

θ+(r−1)π m−1

,

there exists 2(m−1) non-zero vectorsν_kⁱ ∈R² (i= 1,2, k= 1,2, . . . , m−1) such that

D e^xi

P(x) =e

m−1

Y

k=1

ν_kⁱ ·(ex₁,ex₂)

for i= 1,2. We note that

(3.7) det (ν_kⁱ, ν_l^j)6= 0 for (i, k)6= (j, l) andD

ex_iP(x) = 0 fore i= 3,4, . . . , n. Here forν_kⁱ = (νⁱ_k,1, ν_k,2ⁱ )∈R², det(ν_kⁱ, ν_l^j) = det ν_k,1ⁱ ν_l,1^j

ν_k,2ⁱ ν_l,2^j

!

. We take a change of coordinatesxe = Ox with an orthogonal matrixO= (oij) to be chosen. Letηi = (o1i, o2i)∈R²,i= 1,2, . . . , n. Then we get

D_xiP(x) =o_1i

m−1

Y

k=1

(η₁·ν_k¹)x₁+· · ·+ (η_n·ν_k¹)x_n

+o2i m−1

Y

k=1

(η1·ν_k²)x1+· · ·+ (ηn·ν_k²)xn

(3.8)

fori= 1,2, . . . , n. We note that ifDx_iP(x) does not vanish, it is a homogeneous polynomial of degreem−1, and that (3.8) contains only first two rows ofO. For any 1≤i < j≤nandp∈Rⁿ, we define two dimensional planes

Pij(p) =

(p₁, . . . , p_i−1, x_i, p_i+1, . . . , p_j−1, x_j, p_j+1, . . . , p_n) andPij =Pij(0). Then it follows from (3.8) that

(3.9) Dx_iP|_Pij D_xjP|_Pij

!

= o1i o2i

o_1j o_2j

! Qm−1

k=1 (ηi·ν_k¹)xi+ (ηj·ν_k¹)xj

Qm−1

k=1 (ηi·ν_k²)xi+ (ηj·ν_k²)xj

! . If we require that

(3.10) det(ηi, ηj) = det

o_1i o_1j o_2i o_2j

6= 0, then that DxiP =DxjP = 0 onPij is equivalent to

m−1

Y

k=1

(ηi·ν_k¹)xi+ (ηj·ν_k¹)xj

=

m−1

Y

k=1

(ηi·ν_k²)xi+ (ηj·ν_k²)xj

= 0. By (3.7) and (3.10), for any 1≤k, l≤m−1,

det

ηi·ν¹_k ηj·ν_k¹ ηi·ν²_l ηj·ν_l²

= det(ηi, ηj) det(ν_k¹, ν_l²)6= 0. Thus if we require that in the orthogonal matrixO,

det

o1i o2i

o1j o2j

6= 0 for all 1≤i < j≤n ,

we obtain that for fixed 1≤i < j≤n,fij = (DxiP, DxjP)|_Pij:R²→R²has only one zero at x_i =x_j = 0. If we replace (x_i, x_j) in (3.9) with (z_i, z_j)∈C², we see

thatf_ij:C²→C² has also only zero atz_i=z_j= 0. We apply [13, Theorem 4.1]

withn= 2 due to Hilbert’s Nullstellensatz. For fixed 1≤i < j ≤n, there exist δ_ij >0 andr_ij>0 depending onf_ij such that for any v∈C^M(B_1/2² (0);R²) with

|v−f_ij|_CM(B_1/2(0);R²)< δ_ij, we have (3.11) card v⁻¹{0} ∩B_r²

ij(0)

≤(m−1)²

where B_r²(0) denotes the ball centered at the origin with radius r in R² and M = 2(m−1)² which is independent of i, j. Put δ_∗ = ¹₂min_1≤i<j≤nδij and r_∗ = min_1≤i<j≤nr_ij. Moreover, we assume that ϕ ∈ C^2m2(B₁(0);R) satisfies

|ϕ−P|_C2m2(B₁(0);R)< δ_∗. If we taker_∗smaller if necessary, for any 0< r < r_∗and for anyp∈B_r(0),v_ij,p:= (D_xiϕ, D_xjϕ)|_Pij(p)satisfies|v_ij,p−f_ij|_CM(B²_1/2(0);R)<

2δ≤δ_ij. Hence from (3.11)

card (v⁻¹_ij,p{0} ∩B_r²(0))≤(m−1)². Since|∇ϕ|⁻¹{0} ∩Pij(p)⊂v⁻¹_ij,p{0}, if we set the projection

π_ij(x₁, . . . , x_n) = (x₁, . . . , x_i−1, x_i+1, . . . , x_j−1, x_j+1, . . . , x_n), then for anyq∈B_rⁿ⁻²(0)⊂Rⁿ⁻² and any 1≤i < j ≤n,

card |∇ϕ|⁻¹{0} ∩π_ij⁻¹(q)∩Br(0)

≤(m−1)².

By the general area-coarea formula (cf. Federer [8, 3.3.22] or Morgan [20, 3.13]), we have

Hⁿ⁻² |∇ϕ|⁻¹{0} ∩Br(0)

≤ X

1≤i<j≤n

Z

B_rⁿ⁻²(0)

card |∇ϕ|⁻¹{0} ∩π_ij⁻¹(q)∩Br(0)

dHⁿ⁻²q

≤c(n)(m−1)²rⁿ⁻². (3.12)

SinceM + 1 = 2(m−1)²+ 1≤2m², this completes the proof.

Now we can get the following

Proposition 3.4. Assume that the conditions of Theorem 1.1 hold. Then for any Ω⁰ bΩ and anyε >0, there existC(ε) =Cεⁿ⁻² and γ(ε) =γεⁿ⁻² whereC and γ depend on ψ,Ω⁰ andΛ(Ω⁰⁰), and a collection of finitely many balls{Br_i(xi)}i

with ri≤ε,xi∈ S(ψ)∩Ω⁰ such that

(3.13)

Hⁿ⁻²

S(ψ)∩Ω⁰\[

i

Br_i(xi)

≤C(ε),

X

i

r_iⁿ⁻²≤γ(ε). Proof. By (2.6), we haveHⁿ⁻² S(ψ)∩Ω⁰\ S∗(ψ)

= 0. Therefore (n−2)-spherical measure of S(ψ)∩Ω⁰\ S∗(ψ) is equal to zero (cf. Mattila [19, p. 75]). Thus for

any ε >0, there exist at most countably many balls {Bri(x_i)} withr_i ≤εand x_i∈ S(ψ)∩Ω⁰\ S∗(ψ) such that

X

i

rⁿ⁻²_i ≤γ(ε, ψ,Ω⁰) whereγ(ε, ψ,Ω⁰)→0 asε→0.

Then we shall show the following claim.

Claim: For any y∈S_∗(ψ)∩Ω⁰, there exist R=R(y, ψ,Ω⁰) andc=c(y, ψ,Ω⁰) withR≤R0whereR0is as in Proposition 3.1 such that for any 0< r < R,

Hⁿ⁻²(S(ψ)∩Ω⁰∩B_r(y))≤crⁿ⁻². Here Randc depend only ony,ψ, Ω⁰ but also on Λ(Ω⁰⁰).

We prove the claim. Lety∈ S_∗(ψ)∩Ω⁰. By the construction of Ω⁰⁰,B2R₀(y)bΩ⁰⁰. IfR < R0, it follows from Proposition 2.4 that we can write

ψ(x+y) =Pm(x) +φ(x) in BR(0)

wherePmis a non-zero homogeneous, harmonic polynomial of degreem≥2 of two variables after some rotation of coordinates, and from Lemma 3.2, we have

|D^jφ(x)| ≤

(CR^m−j+α j= 0,1, . . . , m , C j=m+ 1, . . . M+ 1

in B_R(0). If we chooseR=R(y, ψ,Ω⁰) small enough withR≤R₀, we have (3.14)

1 R^mφ

∗

C^M+1(B_R(0));C

< δ_∗ whereδ_∗is as in Lemma 3.3. In fact,

1 R^mφ

∗

C^M+1(BR(0);C)

=

M+1

X

j=0

R^j R^m sup

x∈BR(0)

|D^jφ(x)|

=

m

X

j=0

+

M+1

X

j=m+1

≤ChX^m

j=0

R^j

R^mR^m−j+α+

M+1

X

j=m+1

R^j R^m

i

≤CR^α

whereC depends ony,ψ, Ω and Λ(Ω⁰⁰). Thus if we chooseR >0 small enough, we get (3.14). That is to say, we have

1

R^m(ψ(·+y)−P_m)

∗

C^M+1(B_R(0);C)

< δ_∗. By scaling:x7→Rxand using the homogeneity ofPm, we have

1

R^mψ(y+Rx)−P_m(x) _CM+1(B

1(0);C)

< δ_∗.

Since from (3.4) and (2.3), M ≥2c²₀≥2m², we can apply Lemma 3.3 to _R¹mψ(y+ Rx) and get

c(n)(m−1)²rⁿ⁻²≥ Hⁿ⁻² {x;∇xψ(y+Rx) = 0} ∩B_r(0)

=Hⁿ⁻²nz−y

R ;∇_zψ(z) = 0o

∩nz−y R ;|z−y

R |< ro

= 1

Rⁿ⁻²Hⁿ⁻² |∇ψ|⁻¹{0} ∩Br(y)

for all 0< r < r_∗. Here we may assume thatR <1. Therefore, we get (3.15) Hⁿ⁻² |∇ψ|⁻¹{0} ∩B_r(y)

≤c(n)(m−1)²rⁿ⁻²

for all 0< r < Rr_∗. Since we can replaceRwith a smaller one, we can takeRr_∗≤ε.

Thus the claim holds.

Therefore, since

S∗(ψ)⊂ [

y∈S∗(ψ)

Br(y)(y), we have

S(ψ)∩Ω⁰= (S(ψ)∩Ω⁰\ S∗(ψ))∪ S∗(ψ)

⊂[

i

B_ri(x_i)∪ [

y∈S∗(ψ)

B_r(y)(y).

SinceS(ψ)∩Ω⁰is relatively compact, there exist finitely manyx_i∈ S(ψ)∩Ω⁰\S∗(ψ) (i= 1,2, . . . , k=k(ε, ψ)) andy_j∈ S∗(ψ) (j= 1,2, . . . , l=l(ε, ψ)) such that

S(ψ)∩Ω⁰⊂

k

[

i=1

B_ri(x_i)∪

l

[

j=1

B_sj(y_j) and

k

X

i=1

r_iⁿ⁻²≤

k

X

i=1

εⁿ⁻²:=γ(ε, ψ) =kεⁿ⁻². Thus it follows from the claim that

Hⁿ⁻²

S(ψ)∩Ω⁰\

k

[

i=1

Br_i(xi)

≤

l

X

j=1

Hⁿ⁻² S(ψ)∩Ω⁰∩Bs_j(yj)

≤C

l

X

j=1

sⁿ⁻²_j ≤C

l

X

j=1

εⁿ⁻²=Clεⁿ⁻².

This completes the proof.

Finally, we have

Theorem 3.5. Assume that the conditions in Theorem 1.1 hold. For anyΩ⁰bΩ, there exists a constant C >0 depending onψ,Ω⁰ andΛ(Ω⁰⁰)such that

Hⁿ⁻²(S(ψ)∩Ω⁰)≤C .

Proof. Let 0< R < R₀whereR₀is as in Proposition 3.1. Since{BR(x)}_x∈Ω⁰ is an open covering of a compact set Ω⁰, there exists finitely many pointsx1, . . . , xk₀ ∈Ω⁰ such that Ω⁰ ⊂Sk₀

i=1BR(xi). We put a collection of the balls φ0={BR(xi)}^k_i=1⁰ . Fix anyε >0. Then we have the following

Claim: There exist collections of ballsφ1, φ2, . . .such that for anyl≥1, (i) rad (B)≤(2ε)^lR₀for allB∈φ_lwhere rad (B) denotes the radius of the ball B.

(ii) The center ofB is contained in Ω⁰ for allB∈φ_l. (iii)P

B∈φl(rad (B))ⁿ⁻²≤γ(ε)^l. (iv)Hⁿ⁻² S(ψ)∩Ω⁰∩ S

B∈φl−1B∼S

B∈φlB

≤C(ε)γ(ε)^l−1whereγ(ε) and C(ε) are as in Proposition 3.4.

First, we show that the claim implies Theorem 3.5. In order to do so, we show that

S(ψ)∩Ω⁰⊂

∞

[

l=1

S(ψ)∩Ω⁰∩ [

B∈φl−1

B ∼ [

B∈φ_l

B

∪

∞

\

l=0

S(ψ)∩Ω⁰∩[^∞

j=l

[

B∈φj

B

. (3.16)

In fact, letp∈ S(ψ)∩Ω⁰ and assume that

(3.17) p /∈ [

B∈φl−1

B ∼ [

B∈φl

B

for alll≥1. Sincep∈S

B∈φ0B, clearlyp∈ S(ψ)∩Ω⁰∩ S∞ j=0

S

B∈φjB

. It suffices to show that for anyk≥0,

(3.18) p∈

k

\

l=0

S(ψ)∩Ω⁰∩[^∞

j=l

[

B∈φj

B

.

We show (3.18)_k by induction on k. When k = 1, by (3.17), p /∈ S

B∈φ0B

\ S

B∈φ1B

andp /∈ S

B∈φ1B

\ S

B∈φ0B

. Since p∈S

B∈φ0B, we see that p∈ S(ψ)∩Ω⁰∩ [

B∈φ1

B

⊂ S(ψ)∩Ω⁰∩[^∞

j=1

[

B∈φj

B .

Thus we have

p∈

1

\

l=0

S(ψ)∩Ω⁰∩[^∞

j=l

[

B∈φj

B

.

Therefore (3.18)1 holds. Assume that (3.18)j holds for j ≤ k (k ≥ 1). Then p∈ S(ψ)∩Ω⁰ andp∈S∞

j=k

S

B∈φjB. That is to say, for somej≥k,p∈S

B∈φjB.