CONTINUATION AT THE BOUNDARY Luis Escauriaza · 2008. 9. 28. · Luis Escauriaza Universidad del...

C1,α DOMAINS AND UNIQUE

CONTINUATION AT THE BOUNDARY

Luis Escauriaza

Universidad del Paıs Vasco

Abstract.

It is shown that the square of a nonconstant harmonic function u which eithervanishes continuously on an open subset V contained in the boundary of a Dini

domain or whose normal derivative vanishes on an open subset V in the boundary of

a C1,1 domain in Rd, satisfies the doubling property with respect to balls centeredat points Q ∈ V . Under any of the above conditions, the module of the gradient

of u is a B2(dσ)-weight when restricted to V , and the Hausdorff dimension of the

set of points Q ∈ V : ∇u(Q) = 0 is less or equal than d − 2. These results aregeneralized to solutions to elliptic operators with Lipschitz second order coefficients

and bounded coefficients in the lower order terms.

Introduction

Bourgain and Wolff [4] have constructed a counterexample of a C1-harmonicfunction u in Rd+ = X ∈ Rd : Xd ≥ 0 , d ≥ 3, such that u and its gradient vanishsimultaneously on a set of positive measure on ∂Rd+. This counterexample has beengeneralized to any C1,α domain D in Rd, for 0 < α < 1 and d ≥ 3 [13]. On theother hand, the following conjecture still remains an open question:

If u is a harmonic function on a connected Lipschitz domain D in Rd, d ≥ 3,vanishing continuously on an open set V contained in the boundary of D, ∂D, andwhose normal derivative vanishes on a subset of V with positive surface measure,then u is identically zero on D.

When u is non-negative, the comparison principle for nonnegative harmonicfunctions vanishing continuously on an open subset of ∂D, [5], implies that thenormal derivative of u is pointwise comparable almost everywhere to the densityof harmonic measure with respect to surface measure dσ on any compact subset Kcontained in V , and it is well known that harmonic measure is mutually absolutelycontinuous with respect to dσ. Therefore, the answer to the conjecture in this caseis positive.

The conjecture is well known to hold when the dimension of the ambient spaceis 2, for in this case the function f(z) = ux − iuy is analytic in D and from thesubharmonicity of log |f | on D

log |f(X)| ≤∫∂D

log |f(Q)| dωX(Q) for all X in D,

The first author is partially supported by a grant from the Royal Swedish Academy of Sciencesand the second author is supported by Universidad del Paıs Vasco grant UPV 127.310 EA-099/94

Typeset by AMS-TEX

1

2 VILHELM ADOLFSSON

where dωX denotes the harmonic measure for D at X. But if the gradient of uis zero on a set of positive surface measure contained in ∂D, the mutual absolutecontinuity of harmonic measure and surface measure implies that f is identicallyzero on a subset of ∂D with positive harmonic measure. Thus, the above integralis identically equal to −∞, and f is identically zero in D.

We recall that a non-negative function w defined on ∂D is a B2(dσ)-weightprovided that there is a constant C such that for all Q ∈ ∂D and r > 0 thefollowing holds(

1σ(∆r(Q))

∫∆r(Q)

w2 dσ

) 12

≤ C1

σ(∆r(Q))

∫∆r(Q)

w dσ,

where ∆r(Q) = ∂D ∩Br(Q).In [2], the following reduction of the conjectured was obtained.

Theorem 1. Let D be a Lipschitz domain in Rd, d ≥ 2, Q0 ∈ ∂D, and u bea harmonic function in D vanishing continuously on ∆6(Q0). Assume that thereexists a constant M , possibly depending on u, such that for all Q ∈ ∆3(Q0) and0 < r < 1 we have

(0.1)∫

Γ2r(Q)

u2 dX ≤M

∫Γr(Q)

u2 dX

Then, there exists a constant C depending on M , the Lipschitz character of D andd, such that for all Q ∈ ∆2(Q0) and 0 < r < 1

(0.2)

(1

σ(∆r(Q))

∫∆r(Q)

∣∣∣∣ ∂u∂N∣∣∣∣2 dσ

) 12

≤ C1

σ(∆r(Q))

∫∆r(Q)

∣∣∣∣ ∂u∂N∣∣∣∣ dσ.

In particular, the absolute value of the normal derivative of u is a B2(dσ)-weightwhen restricted to ∆2(Q0).

Also, from the arguments used in [2] to prove the above theorem and standardboundary estimates for harmonic functions, one can obtain the following.

Lemma 1. Let D be a Lipschitz domain in Rd, d ≥ 2, and u be a harmonicfunction in D vanishing continuously on an open set V contained in ∂D and whosenormal derivative vanishes on a subset E of V with positive surface measure. Then,if Q ∈ V is a density point of E, u vanishes to infinite order at Q; i.e., for all n > 0there exits Cn > 0 such that∫

Γr(Q)

|u| dX ≤ Cnrn for all 0 < r < 1,

where Γr(Q) = D ∩Br(Q).

It is well known that the doubling condition (0.1) implies that a non zeroharmonic function u as in theorem 1 cannot vanish to infinity order at pointsQ ∈ ∆2(Q0), and that the B2(dσ) condition (0.2) on the normal derivative of uover ∆2(Q0) implies that the set Q ∈ ∆2(Q0) : ∇u(Q) = 0 has zero surface mea-sure. Therefore, as it was already pointed out in [2], to obtain a positive answer

C1,α DOMAINS AND UNIQUE CONTINUATION AT THE BOUNDARY 3

to the conjecture suffices to show that the doubling property (0.1) holds, which isalready a classical method in problems related to unique continuation [6], [10].

In [2], [8] it was shown that the conjecture holds when the domain D is convex,showing that (0.1) holds with a constant M depending on d and u. Also, F. H. Lin[11] has proved that (0.1) holds when the domain D is C1,1 and with a constantM depending on u, d, and the C1,1 character of D. In fact in the last workthe author shows using the doubling property and Federer’s dimension reductionargument [12], that when D is a C1,1 domain the Hausdorff dimension of the setQ ∈ ∆2(Q0) : ∇u(Q) = 0 is less or equal than d− 2.

In the first version of this paper the results in theorems 2 and 3 did not includethe full generality of all the Dini domains because of a minor technicality. After thispaper was sent for publication, I. Kukavica [9] extended theorem 2 in the originalversion to all Dini domains. Since the changes needed in the first version to makethis extension were minimal, we have included them in this new version.

In [8] the reader will find some applications to control theory of a positive answerto this conjecture.

Definition. Let θ be a nonnegative and nondecreasing function defined on [0,+∞)verifying ∫ 4

0

θ(r)r

dr <∞.

A connected domainD in Rd is a Dini domain if for each pointQ on the boundaryof D there is a coordinate system (x, xd), x ∈ Rd−1, xd ∈ R, so that with respectto this coordinate system Q = (0, 0), and a ball B centered at Q and a Lipschitzfunction ϕ : Rd−1 −→ R verifying

(1) ‖∇ϕ‖L∞(Rd−1) ≤ m for some m > 0.(2) |∇ϕ(x)−∇ϕ(y) | ≤ θ(|x− y|) for all x, y ∈ Rd−1.(3) B ∩D = B ∩ (x, xd) : xd > ϕ(x) ,B ∩ ∂D = B ∩ (x, xd) : xd = ϕ(x) .

In this work, the following results are proved.

Theorem 2. Let D be a Dini domain in Rd. Then, if Q0 ∈ ∂D and u is anonconstant harmonic function in D vanishing continuously on ∆6(Q0), there existr0 > 0 depending on d and the module of continuity of θ at r = 0, and a constantM depending on d, the value of the above integral, and u such that the doublingproperty (0.1) holds for all Q ∈ ∆3(Q0) and 0 < r < r0.

Theorem 3. Let D be a Dini domain in Rd, Q0 ∈ ∂D, and u be a nonconstantharmonic function in D vanishing continuously on ∆6(Q0). Then, the absolutevalue of the normal derivative of u is a B2(dσ)-weight when restricted to ∆2(Q0),and the Hausdorff dimension of the set Q ∈ ∆2(Q0) : ∇u(Q) = 0 is less or equalthan d− 2.

The proof of the doubling property follows from a suitable modification of thearguments used in [6] to prove that a solution to a divergence form elliptic operatorwith Lipschitz coefficients in the unit ball of Rd satisfies the doubling property inthe interior of the ball. The B2(dσ) condition follows from theorem 1 and the laststatement from the doubling property and the methods developed in [11]. In thispaper and for completeness we have included the proof given in [11] of this laststatement.

4 VILHELM ADOLFSSON

In the case that the domain D is of class C1 the methods we use yield thefollowing partial answer to the conjecture.

Theorem 4. Let D be a connected C1 domain in Rd and u be a harmonic functionin D vanishing continuously on an open subset V of ∂D. Assume that for somepoint Q in V , u satisfies

(0.3)∫

Γr(Q)

u2 dX ≤ C exp(− 1rβ

)for all 0 < r < 1

and for some absolute constants β and C > 0, then u must be identically zero onD.

A natural question arising from the above conjecture is the following:If u is a harmonic function on a connected Lipschitz domain D in Rd whose

normal derivative vanishes almost everywhere on an open subset V of ∂D, andwhose gradient vanishes on a subset of V with positive surface measure, then umust be identically constant on D.

In this case we have the following analog of theorem 1.

Theorem 5. Let D be a Lipschitz domain in Rd, d ≥ 2, Q0 ∈ ∂D, and u be aharmonic function in D whose normal derivative vanishes almost everywhere on∆6(Q0). Assume that there exists a constant M , possibly depending on u, such thatfor all Q ∈ ∆3(Q0), l ∈ R and 0 < r < 1 we have

(0.4)∫

Γ2r(Q)

(u− l)2 dX ≤M

∫Γr(Q)

(u− l)2 dX

Then, there exists a constant C depending on M , the Lipschitz character of D andd, such that for all Q ∈ ∆2(Q0) and 0 < r < 1

(0.5)

(1

σ(∆r(Q))

∫∆r(Q)

|∇u|2 dσ

) 12

≤ C1

σ(∆r(Q))

∫∆r(Q)

|∇u| dσ.

In particular, |∇u| is a B2(dσ)-weight when restricted to ∆2(Q0).

Hence, as in the previous case it suffices to establish the doubling property atthe boundary to derive that a harmonic function u as above must be constant onD.

In this case, we are only able to show that the above conjecture holds in the caseof C1,1 domains, obtaining the following analog of theorems 2 and 3.

Theorem 6. Let D be a C1,1 domain in Rd, Q0 ∈ ∂D and u be a nonconstantharmonic function in D whose normal derivative vanishes continuously on ∆6(Q0).Then, there exist r0 depending on the C1,1 character of D and d, and a constantM depending on d, the C1,1 character of D and u, such that the doubling property(0.4) holds for all Q ∈ ∆2(Q0), l ∈ R and 0 < r < r0.


Theorem 7. Let D, Q0 ∈ ∂D and u be as in the previous theorem. Then, thelength of the gradient of u is a B2(dσ)-weight when restricted to ∆2(Q0) and theHausdorff dimension of the set Q ∈ ∆2(Q0) : ∇u(Q) = 0 is less or equal thand− 2.

In fact, in the remarks after the proofs of the theorems we will show that theorems1, 2, 3 and 4 hold when we replace the Laplace operator for an elliptic operatorin divergence form with Lipschitz coefficients and bounded coefficients in the lowerorder terms. We will also show that theorems 5, 6 and 7 hold when we replacethe Laplace operator for an elliptic operator in divergence form with Lipschitzcoefficients, bounded coefficients in the first order term and with a zero coefficientin the term of zero order.

In section 1 we show how the methods developed in [2] yield lemma 1 and provetwo auxiliary theorems that we need to prove theorems 2 and 6. In section 2 weprove theorems 2, 3 and 4, in section 3 we prove theorems 5, 6 and 7 and in section4 some of these results are extended to parabolic operators.

Acknowledgments:. The authors wish to thank Carlos E. Kenig for all the conversa-tions we had related to this work, and for having introduced us to these problems.

1. Proof of the auxiliary theorems

Proof of Lemma 1. Without loss of generality we may assume that V = ∆6(Q0).We let u be a harmonic function as in theorem 1 and Q ∈ ∆2(Q0) denote a densitypoint of the set E = Q ∈ ∆2(Q0) : ∇u(Q) = 0 ; i.e.,

(1.1) limr→0

σ(∆r(Q) ∩ E)σ(∆r(Q))

= 1.

In [2] (See also lemma 3 in this paper) it was shown that for all n > 0 there existsCn depending on the Lipschitz character of D, d and n such that for 0 < r < 1 thefollowing holds

(1.2)∫

Γr(Q)

|u| dX ≤ Cnr2

∫∆2r(Q)

| ∂u∂N

(Q)| dσ + 2−n∫

Γ2r(Q)

|u| dX.

On the other hand, the first integral on the right hand side of the last inequalitycan be bounded by

σ(∆2r(Q) \ E)1/2(∫

∆2r(Q)

∣∣∣∣ ∂u∂N∣∣∣∣2 dσ

)1/2

.

Let β denote a vector field supported in Γ4r(Q) with |∇β| ≤ r−1, β ·N ≥ C−1

on ∆2r(Q) for some constant C depending on the Lipschitz character of D andβ ·N ≥ 0 on ∆4r(Q), where N denotes the exterior unit normal to D at points of∂D.

Integrating the Rellich-Necas identity

div(β|∇u|2) = 2div((β.∇u)∇u) +O(|∇β||∇u|2)

6 VILHELM ADOLFSSON

over Γ2r(Q) and since ∇u = ∂u∂N N almost everywhere in ∆5(Q0), we obtain

∫∆2r(Q)

∣∣∣∣ ∂u∂N∣∣∣∣2 dσ ≤ Cr−1

∫Γ4r(Q)

|∇u|2 dX,

and from Cacciopoli’s inequality and standard estimates for harmonic functionsvanishing on the boundary [7] we have for some constant C depending on theLipschitz character of D

(∫∆2r(Q)

∣∣∣∣ ∂u∂N∣∣∣∣2 dσ

)1/2

≤ Cr−d+32

∫Γ6r(Q)

|u| dX.

From the above chain of inequalities

∫Γr(Q)

|u| dX ≤

[Cn

(σ(∆2r(Q) \ E)σ(∆2r(Q)

)1/2

+ 2−n]∫

Γ6r(Q)

|u| dX.

Then, from (1.1) and choosing n large enough we find that for all ε > 0 there existsr(ε) > 0 such that∫

Γr(Q)

|u| dX ≤ ε

∫Γ6r(Q)

|u| dX for 0 < r < r(ε),

and this is well known to imply that∫Γr(Q)

|u| dX = O(rn) for all n > 0,

which proves the lemma.

In the next theorems ∆r and Γr will denote respectively ∆r(0) and Γr(0), andwe will consider an operator L in divergence form

(1.3) Lu =d∑

i,j=1

Di(aij(X)Dju) +d∑i=1

bi(X)Diu+ c(X)u,

where the coefficient matrix A(X) = (aij(X)) is symmetric and the coefficientsverify the following properties:

λ|ξ|2 ≤d∑

i,j=1

aij(X)ξiξj ≤ λ−1|ξ|2 for all ξ and X in Rd.(i)

d∑i=1

|bi(X)|+ |c(X)| ≤ K for all X in Rd.(ii)


Theorem 8. Let Ω be a Lipschitz domain in Rd with 0 ∈ ∂Ω and u be a non-constant solution to Lu = 0 in Γ1 vanishing continuously on ∆1, and where theoperator L is as in (1.3) and satisfies the following assumptions:

(i) A(0) = I and A(Q)Q.N(Q) ≥ 0 for almost every Q ∈ ∆1, where N(Q)denotes the exterior unit normal vector to ∂Ω.

(ii) There is a nondecreasing function f : (0,+∞) −→ (0,+∞) such that forall X in Rd

|∇A(X)| ≤ f(|X|)/|X| and |A(X)−A(0)| ≤ f(|X|),

and ∫ 2

0

f(r)r

dr <∞

Then, there exist r0 depending on λ, d, K and the Lipschitz character of Ω, anda constant C depending on λ, d, K, the Lipschitz character of Ω , the value of theabove integral and u, such that the following holds∫

Γ2r

u2 dX ≤ C

∫Γr

u2 dX for all 0 < r < r0.

We will need the following well known inequalities.

Lemma 2. Let Ω be a Lipschitz domain with 0 ∈ ∂Ω. Then, there exists a con-stant C depending on d and the Lipschitz character of Ω such that the followinginequalities hold for all functions w ∈W 1,2(Γr)∫

Γr

w2 dX ≤ C

[r2∫

Γr

|∇w|2 dX + r

∫∂Br∩Ω

w2 dσ

]∫

Γr

|w||∇w| dX ≤ C

[r

∫Γr

|∇w|2 dX +∫∂Br∩Ω

w2 dσ

]

Proof. We may assume that Ω = X ∈ Rd : xd > ϕ(x) where ϕ is a Lipschitzfunction on Rd−1 with ϕ(0) = 0 and consider the vector field

β(X) = (x1, . . . , xd−1, xd − ϕ(x) + x∇ϕ(x)).

Then, div(β) = d and β.N(Q) = 0 on ∂Ω and the inequalities follow from Holder’sinequality and the identity∫

Γr

div(β)w2 dX =1r

∫∂Br∩Ω

β(Q).Qw2 dσ − 2∫

Γr

wβ.∇w dX.

Proof of Theorem 8. In this proof we will consider the function µ and vector fieldβ defined as

µ(X) = A(X)X.X/|X|2 and β(X) = A(X)X/µ(X)

8 VILHELM ADOLFSSON

and from the hypothesis in theorem 8 we have with |X| = r

λ ≤ µ(X) ≤ λ−1 |∇µ(X)| = O(f(r)/r)(1.4)

|β(X)| = O(r) Diβj = δij +O(f(r)),(1.5)

and where the constants depend on λ and d. For u as in theorem 8 and 0 < r < 1we define the following functions

(1.6)H(r) =

1rd−1

∫∂Br∩Ω

µu2 dσ, A(r) =1

rd−2

∫Γr

A∇u.∇u dX,

D(r) =1

rd−2

∫Γr

A∇u.∇u+ udiv(A∇u) dX, and N(r) =D(r)H(r)

.

Differentiating H(r) we have from (1.4)

H ′(r) =2

rd−1

∫∂Br∩Ω

µu∂u

∂rdσ +O(f(r)/r)H(r).

We recall that the conormal derivative associated to the operator L on ∂Br isgiven by ∂u

∂ν = A∇u. Q|Q| . Now, since α = A(X)X/|X| − µ(X)X/|X| is a tangentialvector field on ∂Br with |∇α(X)| = O(f(r)/r) and u = 0 on ∆1, we obtain fromthe divergence theorem and (1.4) the following identity

rd−2D(r) =∫∂Br∩Ω

u∂u

∂νdσ =

∫∂Br∩Ω

µu∂u

∂rdσ +

∫∂Br∩Ω

u∇u.α dσ

=∫∂Br∩Ω

µu∂u

∂rdσ − 1/2

∫∂Br∩Ω

u2divα dσ

=∫∂Br∩Ω

µu∂u

∂rdσ +O(f(r)/r)rd−1H(r).

Thus,

H ′(r) =2rD(r) +O(f(r)/r)H(r)(1.7)

rd

dr(logH(r)) = 2N(r) +O(f(r)).(1.8)

Proceeding in the standard way, we will show that there exists r0 > 0 dependingon λ, d, K, and the Lipschitz character of Ω and C > 0 depending on the sameparameters and on f(1) such that the frequency function Z(r) = N(r) + 1 verifies

(1.9) Z ′(r) ≥ −C(1 + f(r)/r)Z(r) for all 0 < r < r0,

and this is well known to imply theorem 8.Integrating over Γr the Rellich-Necas identity

div(βA∇u.∇u) = 2 div(β.∇uA∇u) + div(β)A∇u.∇u− 2DiβkaijDjuDku− 2β.∇u div(A∇u) + βkDkaijDiuDju.


we obtain from (1.5) and the identities

β. Q|Q| = r, β.∇uA∇u. Q|Q| =r

µ

(∂u

∂ν

)2

on ∂Br

β.NA∇u.∇u = β.∇uA∇u.N =AQ.NAN.N

µ

(∂u

∂N

)2

on ∆r

the following formula(1.10)∫

∂Br∩Ω

A∇u.∇u dσ = 2∫∂Br∩Ω

1µ

(∂u

∂ν

)2

dσ +1r

∫∆r

AQ.NAN.N

µ

(∂u

∂N

)2

dσ

+(d− 2r

+O(f(r)/r))∫

Γr

A∇u.∇u dX − 2r

∫Γr

β.∇u div(A∇u) dX.

Then, setting

B(r) =2

rd−2

∫∂Br∩Ω

1µ

(∂u

∂ν

)2

dσ +1

rd−1

∫∆r

AQ.NAN.N

µ

(∂u

∂N

)2

dσ,

I1(r) =1

rd−2

∫Γr

u div(A∇u) dX, I2(r) =2

rd−2

∫Γr

β.∇u div(A∇u) dX,

I3(r) = − 1rd−3

∫∂Br∩Ω

u div(A∇u) dσ

and differentiating D(r) = A(r) + I1(r) we get from (1.10) the identity

(1.11) D′(r) = B(r)− 1r[(d− 2)I1(r) + I2(r) + I3(r) +O(f(r))A(r)].

From lemma 2, (1.4), (1.5) and the equation satisfied by u we have

(1.12) |I1(r)|+ |I2(r)| ≤ C1r[A(r) +H(r)] for all 0 < r < 1,

where C1 depends on λ, d, K and the Lipschitz character of Ω . Also, from theidentity (1.10), lemma 2 and Holder’s inequality we obtain

(1.13)1

rd−3

∫∂Br∩Ω

|∇u|2 dσ ≤ C2[rB(r) +A(r) +H(r)] for all 0 < r < 1,

where C2 depends on λ, d, K, f(1) and the Lipschitz character of Ω. Finally, (1.13)and Holder’s inequality imply

(1.14) |I3(r)| ≤ C3r[H(r) +A(r) +√rB(r)H(r)] for all 0 < r < 1.

It follows from (1.12) and the identity D(r) = A(r) + I1(r), that we can findr0 > 0 depending on C1 such that for all 0 < r < r0 the following holds

(1.15)A(r) ≤ 2D(r) +H(r)

|D(r)| ≤ 2[A(r) +H(r)].

10 VILHELM ADOLFSSON

and from (1.11), (1.12) and (1.14) we find that with constants depending on λ, d,K,f(1) and the Lipschitz character of Ω

(1.16) B(r) ≤ 2D′(r) +O(1r)(H(r) +A(r)) for all 0 < r < 1.

From (1.15), the function Z(r) = N(r) + 1 is nonnegative for 0 < r < r0.Therefore, if Z ′(r) ≥ 0, (1.9) holds and there is nothing to prove. Otherwise, wehave D′(r)H(r) ≤ H ′(r)D(r), and multiplying (1.16) by rH(r) we get that for0 < r < 1

rB(r)H(r) ≤ 2rD′(r)H(r) + C[H(r) +A(r)]2 ≤ 2rH ′(r)D(r) + C(H(r) +A(r))2.

Replacing the identity (1.7) on the right hand side of the last inequality we getfrom the second inequality in (1.15)√

rB(r)H(r) ≤ C[A(r) +H(r)] for all 0 < r < r0,

where C depends on λ, d, K, f(1) and the Lipschitz character of Ω. . This inequalityand (1.14) imply

|I3(r)| ≤ Cr[A(r) +H(r)],

and from the last inequality, (1.12) and (1.11) we obtain

D′(r) ≥ B(r)− C(1 + f(r)/r)[A(r) +H(r)].

Then, the above inequality, (1.7) and (1.15) imply

(1.17) Z ′(r) ≥ rB(r)H(r)− 2D(r)2

rH(r)2− C(1 + f(r)/r)Z(r) for all 0 < r < r0.

On the other hand, by definition we have

r2d−3[rB(r)H(r)− 2D(r)2] =∫

∆r

AQ.NAN.N

µ

(∂u

∂N

)2

dσ

∫∂Br∩Ω

µu2dσ

+2r∫

∂Br∩Ω

1µ

(∂u

∂ν

)2

dσ

∫∂Br∩Ω

µu2dσ −(∫

∂Br∩Ω

u∂u

∂νdσ

)2and from Holder’s inequality and the assumption that A(Q)Q.N(Q) ≥ 0 on ∆1 itfollows that the first term on the right hand side of (1.17) is nonnegative, whichproves the theorem.

Remark. The above proof when the boundary of Ω is empty is somehow a newproof of the main result in [6], where they proof the doubling property for solutionsto divergence form operators with Lipschitz coefficients, and which avoids intro-ducing normal coordinates to obtain the identity (1.10), (in this case the term in(1.10) which involves integration over ∆r does nor appear) and replacing them bya suitable choice of the vector field β in the Rellich-Necas identity.


Theorem 9. Let Ω be a Lipschitz domain in Rd with 0 ∈ ∂Ω and u be a non zerosolution to Lu = 0 in Γ1 whose conormal derivative vanishes almost everywhere on∆1, and where the operator L is as in (1.3) and satisfies the following assumptions:

(i) A(0) = I and A(Q)Q.N(Q) = 0 for almost every Q ∈ ∆1.(ii) There is a nondecreasing function f : (0,+∞) −→ (0,+∞) such that for

all X in Rd

|∇A(X)| ≤ f(|X|)/|X| and |A(X)−A(0)| ≤ f(|X|),

and ∫ 2

0

f(r)r

dr <∞

Then, there exist r0 depending on λ, d, K and the Lipschitz character of Ω anda constant C depending on λ, d, K, the value of the above integral, the Lipschitzcharacter of Ω and u, such that the following holds∫

Γ2r

u2 dX ≤ C

∫Γr

u2 dX for all 0 < r < r0.

Moreover, if the coefficient c of the zero order term of the operator L is identicallyzero, both u and u− l satisfy the doubling property with a constant C independentof l ∈ R.

Proof of Theorem 9. In this proof we let µ and β be the function and vector fielddefined in the proof of theorem 8. For u and L as in theorem 8 we define thefunctions H(r), A(r), D(r) and N(r) as in theorem 8. Now, since A(Q)Q.N(Q) = 0on ∆1 we obtain from the divergence theorem the following identity

H(r) = r1−d∫

Γr

div(A(X)X|X| u2

)dX

and

H ′(r) = −d− 1r

H(r) + 2r1−d∫∂Br∩Ω

u∂u

∂νdσ + r1−d

∫∂Br∩Ω

div(A(X)X|X|

)u2 dσ.

Since Lu = 0 in Γ1 it follows from the above identity, the divergence theorem andthe assumptions on the coefficients matrix A that

H ′(r) =2rD(r) +O(f(r)/r)H(r)(1.18)

rd

dr(logH(r)) = 2N(r) +O(f(r)).(1.19)

As in the proof of theorem 8 we integrate over Γr the Rellich-Necas identity

div(βA∇u.∇u) = 2 div(β.∇uA∇u) + div(β)A∇u.∇u− 2DiβkaijDjuDku− 2β.∇u div(A∇u) + βkDkaijDiuDju


and from (1.5) and the identities

β. Q|Q| = r, β.∇uA∇u. Q|Q| =r

µ

(∂u

∂ν

)2

on ∂Br

β.NA∇u.∇u =AQ.N

µA∇u.∇u = 0 and β.∇uA∇u.N = 0 on ∆r

we obtain the following formula

(1.20)

∫∂Br∩Ω

A∇u.∇u dσ = 2∫∂Br∩Ω

1µ

(∂u

∂ν

)2

dσ

+(d− 2r

+O(f(r)/r))∫

Γr

A∇u.∇u dX − 2r

∫Γr

β.∇u div(A∇u) dX.

Then, setting

B(r) =2

rd−2

∫∂Br∩Ω

1µ

(∂u

∂ν

)2

dσ I1(r) =1

rd−2

∫Γr

u div(A∇u) dX

I2(r) =2

rd−2

∫Γr

β.∇u div(A∇u) dX. I3(r) = − 1rd−3

∫∂Br∩Ω

u div(A∇u) dσ

and differentiating D(r) = A(r) + I1(r) we get from (1.20) the identity

(1.21) D′(r) = B(r)− 1r[(d− 2)I1(r) + I2(r) + I3(r) +O(f(r))A(r)].

At this point, from (1.18) and (1.21) the argument proceeds as in the proof oftheorem 8, obtaining that the function Z(r) = N(r)+1 is nonnegative for 0 < r < r0and it either satisfies Z ′(r) ≥ 0 or

(1.22) Z ′(r) ≥ rB(r)H(r)− 2D(r)2

rH(r)2− C(1 + f(r)/r)Z(r).

On the other hand, by definition

r2d−3[rB(r)H(r)− 2D(r)2] =

2r∫

∂Br∩Ω

1µ

(∂u

∂ν

)2

dσ

∫∂Br∩Ω

µu2dσ −(∫

∂Br∩Ω

u∂u

∂νdσ

)2and from Holder’s inequality it follows that the first term on the right hand side of(1.22) is nonnegative. Thus, in any case we obtain

Z ′(r) ≥ −C(1 + f(r)/r)Z(r) for all 0 < r < r0,

which proves the first claim in theorem 9.When the coefficient c of the zero order term in the operator L is identically zero,

both u and u − l are solutions to Lw = 0 whose conormal derivative vanishes on∆1 for all l ∈ R, and if Hl(r), Dl(r) and Nl(r) denote the corresponding functionsdefined in (1.6) and associated to u− l, we observe that to prove the second claimin theorem 9 suffices to show that supl∈RNl(r0) is finite. This clearly holds whenu is constant, otherwise, from (1.15) we have

Nl(r0) ≤ 2 + 2A(r0)Hl(r0)

for all l ∈ R,

and since the operator L has the interior unique continuation property [3], it followsthat inf l∈RHl(r0) > 0, which proves the theorem.


2. Zero Dirichlet boundary condition

Proof of theorem 2. Let D and u be as in theorem 2. We fix a point Q ∈ ∆3(Q0).Then, after a translation we may assume Q = 0, and that u is harmonic on Γ1,where Γ1 is the set of points Y in the unit ball of Rd, Y = (y, yd), such thatyd > ϕ(y), where ϕ is a Lipschitz function in Rd−1 verifying ϕ(0) = 0,

|∇ϕ(y)−∇ϕ(0) | ≤ θ(|y|) for all y ∈ Rd−1,

and where θ is a Dini function. From the mean value theorem we get

|y∇ϕ(y)− ϕ(y)| ≤ 2|y|θ(|y|) for all y ∈ Rd−1,

and in particular

(2.1) y∇ϕ(y)− ϕ(y) ≥ −2|y|θ(|y|) for all y ∈ Rd−1.

We define

θ(r) =1

log2 2

∫ 2r

r

1t

∫ 2t

t

θ(s)s

dsdt

and consider the change of variables Y = Ψ(X) = (x, xd + 3|X|θ(|X|)). Fromcalculus we have

(2.2)

∂yi∂xj

= δij for all i = 1, . . . , d− 1 and j = 1, . . . , d

∂yn∂xj

= δjd +xjα(|X|)|X|

for all j = 1, . . . , d,

and where α(r) = 3 ddr (rθ(r)). If JΨ denotes the Jacobian of the transformation Ψ

we have

det JΨ(X) =∂yd∂xd

= 1 +xdα(|X|)|X|

≥ 1− α(|X|),

and since |α(r)| ≤ 3θ(4r) and θ(r) tends to zero as r tends to zero, the map-ping Ψ defines a C1-diffeomorphism from a neighborhood of zero in the X-spaceonto a neighborhood of zero in the Y -space. In particular there exist r1 and r0depending on θ such that Ψ : Br1 −→ Br0 defines a C1-diffeomorphism, and1/2 ≤ det JΨ(X) ≤ 2 for all X ∈ Br1 . Defining w(X) = u Ψ(X) and

Ω = X ∈ Rd : xd > ϕ(x)− 3|X|θ(|X|) ,we have that w is a W 1,2-solution to Lw = div(A∇w) = 0 on Γr1 = Ω∩Br1 whichvanishes continuously on the associated set ∆r1 . Moreover, the coefficients matrixA(X) is given by

A(X) = det (JΨ(X))JΨ−t(X)JΨ−1(X).

From (2.2) and simple algebra we obtain writing A(X) = (aij(X))

(2.3)

aij(X) = δij [1 +xdα(|X|)|X|

] for i, j = 1, . . . , d− 1.

ajd(X) = adj(X) = −xjα(|X|)|X|

for j = 1, . . . , d− 1.

add(X) =1 +

|x|2α(|X|)2

|X|2

1 +xdα(|X|)|X|


We will show that at scale r1, the function w, the domain Ω and the operatorL, satisfy the conditions in theorem 8, which implies that w satisfies the doublingproperty. Then, undoing the change of variables defined by Ψ , we will obtain thatour original harmonic function u satisfies the corresponding doubling property.

From (2.3) we have that the matrix A verifies the following properties: First, Ais symmetric, A(0) = I, and taking r1 sufficiently small we have that A satisfiesthe ellipticity condition

1/2|ξ|2 ≤∑i,j=1

aij(X)ξiξj ≤ 2|ξ|2 for all X ∈ Br1 and all ξ ∈ Rd.

and for some constant C depending on d

|∇A(X)| ≤ Cθ(4|X|)/|X|, |A(X)−A(0)| ≤ Cθ(4|X|) for all X ∈ Rd

Moreover, we claim that A(Q)Q.N(Q) ≥ 0 on the bottom boundary of Ω. To seethis, we observe that from our assumptions on θ and the implicit function theorem,Ω near 0 ∈ ∂Ω is the region above a C1-graph defined implicitly by a function φsatisfying the equation

(2.4)

φ(x) = ϕ(x)− 3|Q|θ(|Q|)φ(0) = 0

where Q = (x, φ(x)).

Differentiating (2.4) implicitly we have

(2.5) ∇φ(x) =∇ϕ(x)− α(|Q|)

|Q|x

1 +φ(x)α(|Q|)

|Q|

for |x| < r1.

Now, the exterior unit normal to ∂Ω is given almost everywhere by

N(Q) =(∇φ(x),−1)√1 + |∇φ(x)|2

,

and (2.3), (2.5) and elementary algebra show that for Q ∈ ∂Ω we have

(2.6) A(Q)Q.N(Q) =x∇ϕ(x)− φ(x)

det JΨ(Q)√

1 + |∇φ(x)|2.

But from (2.4), (2.1) and observing that θ ≤ θ we obtain

(2.7) x∇ϕ(x)−φ(x) = x∇ϕ(x)−ϕ(x)+3|Q|θ(|Q|) ≥ |Q|θ(|Q|) ≥ 0 for |x| < r1.

Therefore, A(Q)Q.N(Q) ≥ 0 for all Q ∈ ∆r1 , which proves the claim and theorem2.

Remarks.


1. The reader will realize that we can relax the condition on the domain D toobtain the doubling property. In fact, we could just assume that ∂D is given locallyas the graph of Lipschitz functions ϕ satisfying the following lower side condition:for almost every x ∈ Rd−1 and all y ∈ Rd−1

(2.8) (x− y)∇ϕ(x)− (ϕ(x)− ϕ(y)) ≥ −2|x− y|θ(|x− y|),

and where θ is a Dini function. If we set P = (x, ϕ(x)) and Q = (y, ϕ(y)), we canrewrite the above condition as

(2.9) (P −Q).N(P ) ≥ −|P −Q|θ(|P −Q|) for a.e. P ∈ ∂D and all Q ∈ ∂D,

where N(P) denotes the exterior unit normal vector to D at P ∈ ∂D. We observethat condition (2.9) is invariant under linear transformations. To see this, we let Adenote an invertible linear transformation in Rd, and consider the domain

A−1(D) = X ∈ Rd : AX ∈ D .

Clearly, A−1(D) is a Lipschitz domain and any two boundary points P ′ and Q′ canbe respectively written as A−1P and A−1Q, for some P and Q in ∂D. Moreover,N(P ′) = AtN(P )

|AtN(P )| and from this we have

(P ′ −Q′).N(P ′) = (P −Q).N(P )

|AtN(P )|≥ −|P ′ −Q′|θ(|P ′ −Q′|),

where θ(r) = Cθ(Cr) for some C depending on the linear transformation A.In this sense, we can generalize the result for convex domains in [2] to M -convex

domains, where by an M -convex domain we understand a Lipschitz domain D givenlocally as the graph of functions ϕ verifying that ϕ +M |x|2 is a convex function,since in this case, (2.8) holds with θ(r) = (M/2)r.

2. Theorem 2 also holds under the same hypothesis when u is a solution to adivergence form operator L with Lipschitz coefficients and bounded coefficients inthe lower order terms as in (1.3), but in this case, the constants r0 and C alsodepend on K, the bound on the L∞ norm of the gradient of the coefficients matrixand on λ. To see this, we could assume in the previous proof that instead of beingu harmonic in D, u is a solution to Lu = 0 on D, where L is as in (1.3). Aftera linear change of coordinates we could assume that A(0) = I, then the functionw = u Ψ would be a solution in Ω to

Lw = div(A∇w) + b∇w + c w = 0

with w = 0 on ∆r1 , and where the coefficient matrix A(X) = (aij(X)) and thecoefficients b = (b1, . . . , bd) and c are given by

A(X) = det JΨ(X)JΨ−t(X)A Ψ(X)JΨ−1(X)

b(X) = det JΨ(X)JΨ−1(X)b Ψ(X)

c(X) = det JΨ(X)c Ψ(X).


Then, it is clear that the operator L satisfies the condition (ii) in theorem 8with f(r) = C(r + θ(4r)), and where C depends on d, λ and the L∞ norm of thegradient of the matrix A. Moreover, from (2.6) and (2.7), we see that for Q ∈ ∆r1

and taking r1 smaller if necessary, we have

A(Q)Q.N(Q) ≥ 12|Q|θ(|Q|) + det JΨ(Q)JΨ(Q)−t(A Ψ(Q)− I)JΨ(X)−1

Q.N(Q)

=12|Q|θ(|Q|) +O(|Q|2),

which would remain larger than 0. Thus, w, the operator L and Ω verify the as-sumptions in theorem 8, implying the doubling property for w and as a consequencefor u.

3. From the previous remarks, the results in [2] for convex domains can beextended to solutions to divergence form operators with Lipschitz coefficients andbounded coefficients in the lower order terms, showing that on and M -convex do-main D, a nonconstant solution u to one of these operators which vanishes con-tinuously on an open set V contained in ∂D satisfies the doubling property withrespect to balls centered at points Q ∈ V , and obtaining that it can not vanish toinfinite order at those points.

4. To prove theorem 3 we will need a more detailed definition of the frequencyfunction that we have cooked up to prove the doubling property. In fact, if u andD are as in the last proof and Q = (y, ϕ(y)) ∈ ∆3(Q0), undoing the change ofvariables Ψ and defining

(2.10)

A(Q, r) = r2−d∫Sr(Q)∩D

|∇u|2 dX

H(Q, r) = r1−d∫∂Br(Q+3rθ(r)ed)∩D

ru2

r + 3(xd − ϕ(y)− 3rθ(r))(rθ(r))′dσ

N(Q, r) =A(Q, r)H(Q, r)

,

where ed = (0, . . . , 0, 1) and Sr =⋃

0<s<r∂Bs(Q+ 3sθ(s)ed), we have from theorem

8 and standard arguments

(2.11) eβ(r)N(Q, r), where β(r) = C

∫ r

0

θ(4t)t

dt

is a nondecreasing function for 0 < r < r0, and where r0 is as in theorem 2 and Cdepends on d, and the value of the integral from 0 to 1 of θ(r)/r.

Proof of Theorem 4. Let D, u, V and Q ∈ V be as in theorem 4. Without loss ofgenerality we may assume that V = ∆6(Q0), Q = 0 and that the boundary of Dbelow Γ1 is given as the graph of a C1 function ϕ in Rd−1 verifying ϕ(0) = 0. Fromthe continuity of ∇ϕ we find that for all ε > 0 we can find r = r(ε) > 0 such that

|∇ϕ(y)−∇ϕ(0)| ≤ ε whenever |y| ≤ r(ε),

and in particular

(2.12) y∇ϕ(y)− ϕ(y) ≥ −2ε|y| for |y| ≤ r(ε).


Proceeding as in the proof of theorem 2, we consider the change of variables

Ψ(X) = (x, xd + 3ε|X|)

where now θ(r) = ε for all r. The same calculations as before and with α(r) = 3εfor all r give,

det JΨ(X) = 1 +xdα(|X|)|X|

= 1 +3εxd|X|

.

When 3ε < 1 the transformation Ψ defines a bilipschitz mapping from Rd ontoitself whose inverse can be explicitly found. Defining as before w = u Ψ and

Ω = X ∈ Rd : xd > ϕ(x)− 3ε|X| ,

we have that w is an W 1,2-solution to Lw = div(A∇w) = 0 on Γr1 = Ω∩Br1 , whichvanishes continuously on the corresponding set ∆r1 . Moreover, the coefficientsmatrix A(X) is as in the proof of theorem 2 given by the formula (2.3) and withα(r) = 3ε for all r. It is then easy to check that defining A(0) = I, we havethat w and the operator L satisfy the condition (ii) in theorem 8 with f(r) = Cε.Moreover, since ∂yd

∂xd> 0 when X 6= 0 we have that Ω is the region above a Lipschitz

graph φ defined implicitly by the equationφ(x) = ϕ(x)− 3ε|Q|φ(0) = 0

where Q = (x, φ(x)).

Proceeding from here as in the proof of theorem 2 we find that for Q ∈ ∂Ω wehave

A(Q)Q.N(Q) =x∇ϕ(x)− ϕ(x) + 3ε|Q|

det JΨ(Q)√

1 + |∇φ(x)|2,

and from this identity and (2.12) we obtain that A(Q)Q.N(Q) ≥ 0 on ∆r(ε). Atthis point, the argument in the proof of theorem 8 implies that the function H(r)and the frequency function N(r) associated to w, L and Ω satisfy the differentialinequalities

N ′(r) ≥ −CεrN(r)

rd

drlogH(r) ≤ 2N(r) + Cε

for all r ≤ r(ε),

where C depends only on d. Integrating these inequalities we obtain

H(r) ≥ H(r(ε))(

r

r(ε)

)Cεexp

(− 2Cε

(r(ε)r

)CεN(r(ε))

)for all r < r(ε).

But assuming that (0.3) in theorem 4 holds for u, implies that

H(r) ≤ Cexp

(− 1

rβ2

)for all r < r(ε),


and comparing the last two inequalities, we see that taking ε > 0 such that Cε < β2

implies that H(r(ε)) must be equal to zero. Thus, u must be identically zero on anopen neighborhood of zero, which proves the theorem.

Remark. With the obvious changes in the above proof, theorem 4 holds when wereplace the Laplace operator for an elliptic selfadjoint operator with Lipschitz co-efficients and bounded coefficients in the lower order terms.

Proof of Theorem 3. The first claim in theorem 3 follows from the doubling propertyand theorem 1. To prove the second claim we fix m > 0 and a Dini function θ.We let F denote the set of all mappings ϕ : Rd−1 −→ R verifying the followingconditions

(1) ‖∇ϕ‖L∞(Rd−1) ≤ m and ϕ(0) = 0.(2) |∇ϕ(x)−∇ϕ(y) | ≤ θ(|x− y|) for all x, y ∈ Rd−1.For each ϕ ∈ F we define D(ϕ) = X ∈ Rd : |x| < 3 and ϕ(x) < xd <

ϕ(x) + 3m+ 3 .Now, if ϕ ∈ F and u is a harmonic function on D(ϕ) vanishing continuously on

the bottom boundary of D(ϕ), the change of coordinates

(2.13)yi = xi

yd = xd + ϕ(x)

transforms u into a function u defined on the set Ω = X ∈ Rd : |x| < 3, 0 < xd <3m+ 3 and such that if Aϕ(X) = (aij(X)) is the matrix given by

(2.14)

aij(X) = δij for i, j = 1, . . . , d− 1

ajd(X) = adj(X) = Djϕ(x) for j = 1, . . . , d− 1

add(X) = 1 + |∇ϕ(x)|2

and Lϕ denotes the divergence form operator Lϕ = div(Aϕ∇.), then u is a solutionto Lϕw = 0 on Ω vanishing continuously on the bottom boundary of Ω. Next, welet L denote the set formed by all the subsets E of the open unit ball B1 in Rd−1

verifying that there exists a function ϕ ∈ F and a nonconstant harmonic functionu defined on D(ϕ) vanishing continuously on the bottom boundary of D(ϕ), suchthat

E = x ∈ B1 : |∇u(x, 0)| = 0 = x ∈ B1 : |∇u(x, ϕ(x))| = 0 .

From [14], it follows that if ϕ ∈ F and u is harmonic function on Dϕ vanishingcontinuously on the bottom boundary of Dϕ, there is a constant C depending ond, m and the value of the integral from 0 to 1 of θ(r)/r, such that(2.15)

‖∇u‖L∞(B+2 (0)) ≤ C

(∫Ω

u 2 dX

)1/2

|∇u(X)−∇u(X ′)| ≤ C

∫ |X−X′|

0

θ(r)r

dr

(∫Ω

u 2 dX

)1/2

for all X,Y ∈ B+2 (0).

From this, it is clear that for E ∈ L, E is a relatively closed subset of B1. Wesay Ei → E for a sequence of Ei ⊂ L and E ∈ L, if for each ε > 0 there is a i(ε)such that

B1−ε ∩ Ei ⊂ x ∈ B1 : d(E, x) < ε for all i ≥ i(ε).


Now we want to verify the following properties of L.(P1) (Closure under appropriate scaling and translation). If |y| ≤ 1−λ, 0 < λ <

1, and if E ∈ L, then Ey,λ ∩B1 ∈ L. Here, Ey,λ = λ−1(E − y).This is because the fact that if u is a nonconstant harmonic function on a domain

D(ϕ) vanishing on the bottom boundary of D(ϕ), setting Y = (y, ϕ(y)), uy,λ(X) =u(λX + Y ) and ϕy,λ(x) = (ϕ(λx+ y)− ϕ(y)) /λ, then ϕy,λ ∈ F and uy,λ is anonconstant harmonic function on D(ϕy,λ) vanishing on the bottom boundary ofD(ϕy,λ) for |y| ≤ 1− λ and 0 < λ < 1. Then, it is easy to see that

Ey,λ ∩B1 = x ∈ B1 : |∇uy,λ(x, ϕy,λ(x))| = 0 .

Thus, Ey,λ ∩B1 ∈ L.(P2) (Existence of homogeneous degree zero tangent set). If |y| < 1, if λk,

0 < λk < 1, decreases to zero, and if E ∈ L, then there is a subsequence λk′ andF ∈ L such that Ey,λk′ ∩B1 → F and F0,λ ∩B1 = F for each 0 < λ < 1.

To show (P2), we let ϕ ∈ F and u be a nonconstant harmonic function on D(ϕ)vanishing continuously on the bottom boundary of D(ϕ), and such that

E = x ∈ B1 : |∇u(x, ϕ(x))| = 0 .

For |y| < 1, and a decreasing sequence λk converging to zero, we set as before Y =(y, ϕ(y)), uy,λk

(X) = u(λkX + Y ), ϕy,λk(x) = (ϕ(λkx+ y)− ϕ(y)) /λ. Without

loss of generality, we may assume ∇u(Y ) = 0, for otherwise, by the continuity of∇u (or ∇u) up to the bottom boundary of D(ϕ), it is clear that Ey,λk

∩ B1 = ∅for all k sufficiently large. Then, we simply take F = ∅, which is associated to theharmonic function u(X) = xd and the domain D(0) = X ∈ Rd : xd > 0 , whichshows that (P2) holds in this case.

Since u is nonconstant, we have from (2.11)

(2.16) eβ(r)N(Y, r) ≤ eβ(r0)N(Y, r0) < +∞ for all 0 < r < r0,

where N(Y, r) is the frequency function defined in (2.10) and associated to u, D(ϕ),θ and Q = Y . Now, for each λk ≤ (r0(1− |y|))/2, we define

vk =uy,λk( ∫

∂B1(0)∩D(ϕy,λk)

u2y,λk

dσ

)1/2.

We observe that if Nk(r) denotes the frequency function defined in (2.10) andassociated to vk, D(ϕy,λk

), Q = 0 and θk(r) = θ(λkr) we have

(2.17) βk(r) = β(λkr), Nk(r) = N(Y, λkr) for all 0 < r < 2.

This implies that the squares of the harmonic functions vk satisfy on the correspond-ing domains of definition the doubling property with respect to balls centered atzero and with a constant independent of k. Through the change of coordinates(2.13) associated to ϕy,λk

and the uniform doubling property, we obtain that thereis a constant C depending on d, m, the value of the integral from 0 to 3 of θ(r)/rand u, such that for k large enough

C−1 ≤∫

Ω

v 2k dX ≤ C


Since Lϕy,λkvk = 0 on Ω, Aϕy,λk

(X) converges to Aψ(X), where ψ(x) = ∇ϕ(y).x,the above inequalities and (2.15), we can find a subsequence of k’s such that vk′converges in C1(Ω) to a function v. Then Lψ v = 0 on Ω, v is nonconstant andvanishes continuously on the bottom boundary of Ω. Clearly, the pull back of vunder the change of variables (2.13) corresponding to ψ is a harmonic function onD(ψ) and setting

F = x ∈ B1 : |∇v(x, ψ(x))| = 0 ∈ L

we have, Ey,λk′ ∩ B1 → F . Moreover, one observes from the formulae (2.10) thatdefining

D0(r) = r2−d∫Br∩D(ψ)

|∇v|2 dX, H0(r) = r1−d∫∂Br∩D(ψ)

v2 dX

N0(r) =D0(r)H0(r)

we have limk′→∞

Nk′(r) = N0(r) for all 0 < r < 1 and

K = limr→0

eβ(r)N(Y, r) ≤ eβ(λkr)N(Y, λkr) = eβk(r)Nk(r) ≤ eβ(λk)N(Y, λk)

for all 0 < r < 1. Hence the frequency function N0(r) for v satisfies, N0(r) = Kfor all 0 < r < 1. This implies that v(λX) = λKv(X) for all 0 < λ < 1. Hence, vis a homogeneous harmonic function of degree K and F0,λ ∩B1 = F for 0 < λ < 1,proving (P2).

Now, by [12, Appendix A] we have the following

Theorem. Let L be a collection of relatively closed proper subsets of the unit openball in Rd−1 which satisfies (P1) and (P2). Then, the Hausdorff dimension of eachof the elements in L is less or equal than d− 2.

This finishes the proof of theorem 3.

Remarks.1. Theorem 3 also holds when we replace the Laplace operator for an operator

L as in (1.3) whose coefficient matrix is Lipschitz. In this case, the second claim intheorem 3 follows from the doubling property and the obvious changes in the lastproof. The first claim follows from the doubling property, the arguments used in[2] to prove theorem 1 and the following analog of the inequality (1.2).

Lemma 3. Let D be a Lipschitz domain in Rd, L an operator as in (1.3) whosecoefficient matrix A is Lipschitz and satisfies ‖∇A‖L∞(Rd) ≤ m for some m >

0. Then, for each ε > 0 there exists a constant C(ε) depending on the Lipschitzcharacter of D, m, λ, K and ε, such that if Q ∈ ∂D, 0 < r < 1 and u is a solutionto Lu = 0 on Γ2r(Q) vanishing continuously on ∆2r(Q) the following holds∫

Γr(Q)

|u| dX ≤ C(ε)r2∫

∆2r(Q)

| ∂u∂N

| dσ + ε

∫Γ2r(Q)

|u| dX.

Proof. We let let F denote the set of all Lipschitz maps ϕ : Rd−1 −→ R satisfyingϕ(0) = 0 and ‖∇ϕ‖L∞(Rd−1) ≤ m for some m > 0, and L denote the set of


all the operators L as in (1.3) whose coefficient matrix A satisfies A(0) = I and‖∇A‖L∞(Rd) ≤ m. For each ϕ ∈ F we define D(ϕ) = X ∈ Rd : xd > ϕ(x) .

To prove this lemma suffices to show the following:For each ε > 0 there exists a constant C(ε) depending on λ, ε, K and m, such

that if ϕ ∈ F , L ∈ L and u is a solution to Lu = 0 on B2 ∩ D(ϕ) vanishingcontinuously on B2 ∩ ∂D(ϕ) the following holds∫

B1∩D(ϕ)

|u| dX ≤ C(ε)∫B2∩∂D(ϕ)

| ∂u∂N

| dσ + ε

∫B2∩D(ϕ)

|u| dX.

To prove it we proceed by contradiction. Otherwise, there exist ε > 0 andsequences ϕn in F , Ln in L with

Lnw = div(An∇w) +Bn.∇w + Cnw,

and a sequence un of functions verifying Lnun = 0 on B2 ∩ D(ϕn) ,un = 0 inB2 ∩ ∂D(ϕn),

(2.18)∫B1∩D(ϕn)

|un| dX = 1

and

(2.19) ε

∫B2∩D(ϕn)

|un| dX + n

∫B2∩∂D(ϕn)

|∂un∂N

| dσ ≤ 1.

For each n we let vn denote the zero extension of un to the whole ball B2

outside of the domain D(ϕn). Since all the above sequences are compact in theproper topologies, we can find subsequences which we may assume to be the wholesequences, such that ϕn → ϕ ∈ L and An → A uniformly over compact sets,Bn → B and Cn → C weakly in L2(B3/2), where the operator Lw = div(A∇w) +B.∇w + Cw lies in L, and vn → v uniformly on B3/2, weakly in W 1,2(B3/2), and∇vn converges uniformly to ∇v over compact sets contained in B3/2 \ ∂D(ϕ), [7]. Moreover, the divergence theorem and (2.19) imply that there is a constant Cdepending on λ, such that for all ψ ∈ C∞0 (B3/2) and n ≥ 1 we have

(2.20)∣∣∣∣∫ An∇vn∇ψ −Bn.∇vnψ − Cnvnψ dX

∣∣∣∣ ≤ C

n‖ψ‖L∞(B3/2)

.

Passing to the limit in (2.18) and (2.20) we find a function v ∈ W 1,2(B3/2)satisfying Lv = 0 on B3/2,

(2.21)∫B1

|v| dX = 1,

and v is identically zero on B1 \ D(ϕ). The Lipschitz character of ϕ implies thatwe can find a nonempty open set U contained in B1 \D(ϕ) where v is identicallyzero, but an operator L ∈ L has the interior unique continuation property [3], thusv = 0 on B3/2.This is in contradiction with (2.21) and proves the lemma.

2. The previous lemma can also be used to show that lemma 1 holds whenreplace the Laplace operator for an operator L as in (1.3) whose coefficients matrixis Lipschitz.


3. Zero Neumann Boundary Condition

Proof of Theorem 5. To prove theorem 5 we need the following analog of the in-equality (1.2) and lemma 3

Lemma 4. Let D be a Lipschitz domain in Rd. Then, for each ε > 0 there existsa constant C(ε) depending on ε, d and the Lipschitz character of D, such that ifQ ∈ ∂D and u is a harmonic function on Γ2r(Q) whose normal derivative vanishesalmost everywhere on ∆2r(Q) and∫

∆r(Q)

u dσ = 0

the following holds∫Γr(Q)

|u| dX ≤ C(ε)r2∫

∆2r(Q)

|∇u| dσ + ε

∫Γ2r(Q)

|u| dX.

Proof. As in lemma 3 we let F denote the set of Lipschitz mappings ϕ on Rd−1

verifying ϕ(0) = 0 and ‖∇ϕ‖L∞(Rd−1) ≤ m for some m > 0. Then, if lemma 4 werefalse we would find ε > 0 and sequences ϕn in F and un verifying that un isharmonic on B2 ∩D(ϕn), ∂un

∂N = 0 on B2 ∩ ∂D(ϕn),

(3.1)∫B1∩∂D(ϕn)

un dσ = 0,∫B1∩D(ϕn)

|un| dX = 1

and

(3.2) ε

∫B2∩D(ϕn)

|un| dX + n

∫B2∩∂D(ϕn)

|∇u| dσ ≤ 1.

For each n we let vn and fn = (f1n, . . . , f

dn) denote the zero extensions to the

whole ball B2 outside of the domain D(ϕn) of un and ∇un respectively. From (3.2)and standard interior estimates [7], there is a constant C depending on m, d and εsuch that

‖vn‖L2(B3/2)+ ‖fn‖L2(B3/2)

≤ C for all n ≥ 1.

Since the above sequences are compact in the proper topologies, we can findsubsequences which we may assume to be the whole sequences, such that ϕn →ϕ ∈ F uniformly over compact sets, vn → v, fn → f = (f1, . . . , fd) weakly inL2(B3/2) and uniformly over compact sets contained in B3/2 \ ∂D(ϕ). From (3.2),the divergence theorem and Poincare inequality on ∆2 we find that there is aconstant C depending on m and d such that for all ψ ∈ C∞0 (B3/2) and 1 ≤ j ≤ d

(3.3)∣∣∣∣∫ vnDjψ + f jnψ dX

∣∣∣∣+ ∣∣∣∣∫ fn.∇ψ dX∣∣∣∣ ≤ C

n‖ψ‖L∞(Rd).

Taking limits in (3.1) and (3.3) we find that the limit v satisfies: v ∈ W 1,2(B3/2),v is harmonic on B3/2,

(3.4)∫B1∩D(ϕ)

|v| dX = 1,


and v vanishes on B1 \D(ϕ). As in lemma 3 these imply that v is identically zeroon B3/2, which contradicts (3.4) and proves the lemma.

Now, if D and u are as in theorem 5, r > 0 and Q ∈ ∂D we obtain from thedoubling property (0.4) and choosing ε > 0 sufficiently small in lemma 4 that thereis a constant C depending on M , d and the Lipschitz character of D verifying thefollowing

(3.5)∫

Γr/2(Q)

|u− l| dX ≤ Cr2∫

∆r(Q)

|∇u| dσ

where l denotes the average of u over ∆r/2(Q). Now, from the Rellich identity,the doubling property (0.4) and standard estimates for harmonic functions whosenormal derivative vanishes almost everywhere on the boundary [7] we obtain

(3.6)

(∫∆r(Q)

|∇u|2 dσ

)1/2

≤ Cr−d+32

∫Γr/2(Q)

|u− l| dX,

and (3.5), (3.6) imply the inequality (0.5), proving the theorem.

Proof of theorem 6. Let D and u be as in theorem 6. We fix a point Q ∈ ∆3(Q0).Then after a translation we may assume Q = 0, and that u is harmonic in Γ1, whereΓ1 is the set of points Y = (y, yd) in the unit ball of Rd such that yd > ϕ(y), whereϕ is a C1,1 function in Rd−1 verifying ϕ(0) = 0. For j = 1, . . . , d − 1 we considerthe functions

uj(x, xd) = xj − xdΘxd∗Djϕ(x)

and define

ψj(x, xd) =uj(x, xd) for xd ≥ 04uj(x,−xd/2)− 3uj(x,−xd) for xd < 0,

where Θ is a smooth compactly supported regularization of the identity in Rd−1

and ∗ denotes convolution. Next, we consider the C1,1-function

Ψ(X) = (ψ1(x, xd), . . . , ψd−1(x, xd), xd + ϕ(x))

whose Jacobian at points (x, 0), x ∈ Rd−1, is given by the matrix

JΨ(x, 0) =

1 0 0 . . . 0 D1ϕ0 1 0 . . . 0 D2ϕ0 0 1 . . . 0 D3ϕ...

......

. . ....

...0 0 0 . . . 1 Dd−1ϕ

−D1ϕ −D2ϕ −D3ϕ . . . −Dd−1ϕ 1

and det JΨ(x, 0) = 1 + |∇ϕ(x)|2. Setting JΨ−1(x, 0) = (aij(x)) we have

add(X) =1

1 + |∇ϕ|2


andaij(x) = δij −

DiϕDjϕ

1 + |∇ϕ|2

ajd(X) = − Djϕ

1 + |∇ϕ|2

adj(X) =Djϕ

1 + |∇ϕ|2

for i, j = 1, . . . , d− 1.

From the inverse function theorem, Ψ defines a C1-diffeomorphism from a neigh-borhood of zero in the X-space onto a neighborhood of zero in the Y -space. Inparticular, there exist r1 and r0 depending on ϕ such that Ψ : Br1 −→ Br0 definesa C1-diffeomorphism. Defining w(X) = u Ψ(X), w solves Lw = div(A∇w) = 0on Br1 ∩ Ω, where

A(X) = det JΨ(X)JΨ−t(X)JΨ−1(X).

and Ω = X ∈ Rd : xd+ϕ(x)−ϕ(ψ1(x, xd), . . . , ψd−1(x, xd)) > 0 . Since Ψ(x, 0) =(x, ϕ(x)) and

∂

∂xd[xd + ϕ(x)− ϕ(ψ1(x, xd), . . . , ψd−1(x, xd))] = 1 + |∇ϕ(x)|2 +O(|xd|)

we have that for sufficiently small r1, Br1 ∩ Ω = X ∈ Br1 : xd > 0 .A calculation shows that for x ∈ Rd−1, A(x, 0) = I, and since we are assuming

that ∂u∂N = 0 on ∆1 and ∇w = JΨ∇u Ψ, the conormal derivative of w associated

to the operator L on ∂Ω ∩Br1 is given by

A(x, 0)∇w(x, 0).(−ed) = −Ddw(x, 0) = Dju(x, ϕ(x))Djϕ(x)−Ddu(x, ϕ(x)) = 0.

These imply that w, L and Ω verify the conditions in theorem 9 with f(r) = Cr,where C depends on d and the C1,1-norm of ϕ. Thus, proving theorem 6.

Proof of theorem 7. The first claim in theorem 7 follows from the doubling propertyand theorem 5. The second claim is proved with an argument similar to the onegiven in the proof of theorem 3. Actually, the same proof we gave there will workjust by considering harmonic functions whose normal derivative vanishes at thebottom boundary of the domains D(ϕ) instead of harmonic functions vanishingcontinuously on the bottom boundary.

Remark. Theorems 5, 6, 7 and lemma 4 also hold when we replace the Laplaceoperator by an operator L as in (1.3) whose coefficient matrix is Lipschitz and wherethe coefficient c of the zero order term is identically zero, and when we considersolutions u to Lu = 0 whose conormal derivative vanishes on the correspondingboundary portion of the domain. In fact, it is well known that these results arefalse when the coefficient c of the zero order term is not zero, and that a non zerosolution to ∆u+ cu = 0 could be constant on a ball. For instance, the function

u(X) =

1 for |X| ≤ 1

exp(

11−|X|2

)+ 1 for |X| ≥ 1

satisfies ∆u+ cu = 0 with c = −∆u/u ∈ L∞(Rd).


To proof theorem 6 in this general setting, we consider an operator L as in (1.3)with Lu = div(A∇u) + b.∇u, A(X) = (aij(X)) and assume that u is a solutionto Lu = 0 on Γ1 whose conormal derivative vanishes on ∆1, where Γ1 is the setof points Y = (y, yd) in the unit ball of Rd such that yd > ϕ(y), ϕ(0) = 0,ϕ ∈ C1,1(Rd−1), A(0) = I, and ‖∇A‖L∞(Rd) < +∞.

Then, for each i = 1, . . . , d we consider the functions

βi(x) =d−1∑j=1

aij(x, ϕ(x))Djϕ(x)− aid(x, ϕ(x))

and define

uj(x, xd) = xj − xdΘxd∗ βj(x) for j = 1, . . . , d− 1,

ud(x, xd) = ϕ(x)− xdΘxd∗ βd(x),

ψj(x, xd) =uj(x, xd) for xd ≥ 04uj(x,−xd/2)− 3uj(x,−xd) for xd < 0.

for j = 1, . . . , d,

where Θ is a smooth compactly supported regularization of the identity on Rd−1.Next, we consider the C1,1-function

Ψ(X) = (ψ1(x, xd), . . . , ψd−1(x, xd), ψd(x, xd))

whose Jacobian at points (x, 0), x ∈ Rd−1, is given by the matrix

JΨ(x, 0) =

1 0 0 . . . 0 D1ϕ0 1 0 . . . 0 D2ϕ0 0 1 . . . 0 D3ϕ...

......

. . ....

...0 0 0 . . . 1 Dd−1ϕ

−β1 −β2 −β3 . . . −βd−1 −βd

and det JΨ(x, 0) = −βd(x) +

d−1∑j=1

βjDjϕ(x) = 1 + |∇ϕ(x)|2 +O(|x|).

Setting JΨ−1(x, 0) = (cij(x)) and β = (β1, . . . , βd−1) we have

cdd(X) =1

β.∇ϕ− βd,

cij(x) = δij −Diϕβj

β.∇ϕ− βd

cjd(X) = − Djϕ

β.∇ϕ− βd

cdj(X) =βj

β.∇ϕ− βd

for i, j = 1, . . . , d− 1.

The inverse function theorem implies as before that Ψ is a C1-diffeomorphismfrom a neighborhood of zero in the X-space onto a neighborhood of zero in theY -space. Defining w(X) = u Ψ(X), w solves

Lw = div(A∇w) + b.∇w = 0


on Br1 ∩ Ω, where Ω = X ∈ Rd : xd > 0 and

A(X) = det JΨ(X)JΨ−t(X)A Ψ(X)JΨ−1(X)

b(X) = det JΨ(X)JΨ−1(X)b Ψ(X)

A calculation shows that the operator L, w and Ω satisfy the conditions intheorem 9 with f(r) = Cr, where C depends on λ, d, the C1,1-norm of ϕ and‖∇A‖L∞(Rd) , and these prove this generalization of theorem 6.

4. The parabolic case

Here we show that a similar result holds for caloric functions vanishing contin-uously on the lateral boundary of a cylinder D × (0,∞), where D is as in theorem2. In particular we have the following theorem.

Theorem 10. Let D be a Dini domain in Rd and u(X, t) satisfy∆u− ∂u

∂t = 0 on D × (0,∞)u(X, 0) = f(X) for X ∈ Du(Q, t) = 0 for Q ∈ ∂D and t > 0

for some f in a suitable class. Assume, that the set

E = (Q, t) ∈ ∂D × (0,∞) :∂u

∂N(Q, t) = 0

has positive surface measure on ∂D × (0,∞). Then, u must be identically zero.

Proof. In the first paragraph in [2, section 3] the following was claimed:“The reader will observe in the next proof, that in general, the same result

can be obtained when D is just a Lipschitz domain and the corresponding uniquecontinuation property holds at points (Q, 0) with Q ∈ ∂D, for harmonic functionsdefined on W ∩D×(0,∞) and vanishing continuously on W ∩∂(D×(0,∞)), whereW is an open set in Rd+1 containing the boundary point (Q, 0)”.

In fact, this is not precise and it should say that the corresponding doublingproperty should be known to hold at points (Q, τ) ∈ W ∩ ∂p(D × (0,∞)), where∂p denotes the parabolic boundary. Hence, from [2, Theorem 4] to prove the abovetheorem suffices to show that if u(X, y) is harmonic on Ω = D× (0, 3) and vanishescontinuously on ∂pΩ, there are constants C and r0 depending on u, d and the Dinicharacter of D, such that for all (Q, τ) ∈ ∂p(D×(0, 1)) and 0 < r < r0 the followingholds

(4.1)∫B2r(Q,τ)∩Ω

u2 dXdy ≤ C

∫Br(Q,τ)∩Ω

u2 dXdy.

Now, it is an easy exercise to show that when D satisfies (2.8), there exists aconstant C depending on the Lipschitz character of D such that for almost every(P, t) ∈ ∂pΩ and all (Q, τ) ∈ ∂pΩ

[(P, t)− (Q, τ)].N(P, t) ≥ −C√|P −Q|2 + (t− τ)2θ(

√|P −Q|2 + (t− τ)2),


where N(P, t) denotes the exterior unit normal vector to Ω at the point (P, t).Therefore, the corresponding boundary portion of Ω verifies (2.9) and from thefirst remark after the proof of theorem 2 we obtain (4.1), proving the theorem.

Remark. The above theorem also holds when we replace caloric functions for func-tions u satisfying

div(A(X)∇u) + b(X).∇u+ c(X)u− ∂u∂t = 0 on D × (0,∞)

u(X, 0) = f(X) for X ∈ Du(Q, t) = 0 for (Q, t) ∈ ∂D × (0,∞)

where the coefficient matrix A(X) is Lipschitz and

(4.2) ‖∇A‖L∞(Rd) + ‖b‖L∞(Rd) + ‖c‖L∞(Rd) ≤ K.

This follows from the obvious modifications in the proof of theorem 4 in [2] andthe following lemma replacing lemma 2 in [2] (See also the arguments in [10]), whoseproof can be carried out by contradiction as lemma 3.

Lemma 5. Let D denote a Lipschitz domain in Rd and L be a parabolic operator asabove satisfying (4.2). Then, for each ε > 0 there exists a constant C(ε) dependingon the Lipschitz character of D, d, K and ε, such that if τ ∈ R, Q ∈ ∂D, 0 < r < 1,and u is a solution to Lu = 0 on Γ2r(Q) × (τ − 4r2, τ ] vanishing continuously on∆2r(Q)× (τ − 4r2, τ ] the following holds∫ τ

τ−r2

∫Γr(Q)

|u| dXdt ≤ C(ε)r2∫ τ

τ−4r2

∫∆2r(Q)

| ∂u∂N

| dσdt

+ε∫ τ

τ−4r2

∫Γ2r(Q)

|u| dXdt.

In fact, the constants C(ε) appearing in lemmas 3, 4 and 5 can be estimatedusing suitable Carleman inequalities, obtaining that C(ε) = ε−θ for some positiveθ depending on the Lipschitz character of D, dimension, the ellipticity constantsof the operator involved and the L∞-norm of the gradient of the coefficients of thesecond order terms.

References

[1] Almgren F.J., Minimal Submanifolds and Geodesics (ed. by M. Obata), North Holland, 1979,

pp. 1–6.[2] Adolfsson V., Escauriaza L. and Kenig C.E., Convex domains and unique continuation at

the boundary, Rev. Mat. Iberoamericana 11, 3 (1995), 513–525.

[3] Aronszajn N., Krzywicki A. and Szarski J., A unique continuation theorem for exteriordifferential forms on Riemannian manifolds, Ark. Mat. 4 (1962), 417–453.

[4] Bourgain J. and Wolff T., A remark on gradients of harmonic functions in dimension d ≥ 3,Colloquium Mathematicum LX/LXI (1990), 253–260.

[5] Dahlberg B.E.J., Estimates for harmonic measure, Archive for Rational Mechanics and Anal-ysis 65 (1977), 275–285.

[6] Garofalo N. and Lin F.H., Monotonicity properties of variational integrals, Ap-weights and

unique continuation, Indiana Univ. Math. J. 35 (1986), 245–267.

[7] Gilbarg D. and Trudinger N.S., Elliptic Partial Differential Equations of Second Order,Springer–Verlag, 1983.


[8] Kenig C.E., Spring School ’94: Potential Theory and Analysis. Paseky nad Jizerou (Czech

Republic). Lectures by Carlos E. Kenig. Notes by Michael Korey, preprint.[9] Kukavica I., Dini domains and unique continuation at the boundary, preprint (1996).

[10] Lin F.H., A uniqueness theorem for parabolic equations, Indiana Univ. Math. J. XLIII(1990), 127–136.

[11] Lin F.H., Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure and Appl.

Math. XLIV (1991), 287–308.[12] Simon L., Lectures on Geometric Measure Theory, Proc. C.M.A., Aust. Nat. Univ., 1983.

[13] Wang W., A remark on gradients of harmonic functions, Rev. Mat. Iberoamericana (to

appear).[14] Widman K.O., Inequalities for the Green function and boundary continuity of the gradient

of solutions of elliptic differential equations, Math. Scand. 21 (1967), 17–37.

[15] Zhiren J., A unique continuation property on the boundary for solutions of elliptic equations,Trans. Amer. Math. Soc. 336, 2 (1993), 639–653.

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