By localizing the equation with the help of Caarelli-Silvestre extension, we solve the problem by a delicate Carleman inequality in the (n + 1)-dimensional half space

FRACTIONAL LAPLACIAN WITH A DRIFT TERM

PU-ZHAO KOW AND JENN-NAN WANG

Abstract. In this paper, we study a Landis-type conjecture for the fractional Schrödinger equation with a drift term. The Landis-type conjecture is a question of unique continuation property from the innity. More precisely, we would like to prove the following: if any solution of the fractional Schrödinger equation with a drift term decays at a certain exponential rate, then such solution must be trivial. By localizing the equation with the help of Caarelli-Silvestre extension, we solve the problem by a delicate Carleman inequality in the (n + 1)-dimensional half space.

1. Introduction

In this work, we consider the unique continuation from the innity for the fractional Schrödinger equation with drift term

(1.1) ((−∆)^s+ b(x)x · ∇ + q(x))u = 0 in Rⁿ,

where s ∈ (0, 1) and b, q are scalar-valued functions. Precisely, we are interested in investigating the decay rate of u at innity that implies the solution u is trivial. This problem is closely related to the Landis conjecture [KL88]. For s = 1, b = 0, Landis conjectured that if if |q(x)| ≤ 1 and |u(x)| ≤ C0 satises |u(x)| ≤ exp(−C|x|¹⁺), then u ≡ 0. The Landis conjecture was disproved by Meshkov [Me91], who constructed a complex-valued potential q and a nontrivial complex-valued u with |u(x)| ≤ C exp(−C|x|⁴³) such that u is a solution of the Schrödinger equation with potential q. He also showed that if |u(x)| ≤ C exp(−C|x|⁴³⁺), then u ≡ 0.

In view of Meshkov's counterexample, Kenig [Ke06] rened the Landis conjecture and asked whether this conjecture is true for real-valued potentials and solutions. This real version Landis conjecture was conrmed partially in [KSW15] where n = 2 and q ≥ 0.

This result was later extended to the more general situation with ∆ being replaced by any second order elliptic operator [DKW17]. The Landis conjecture in the real case with n = 1 was studied in [Ro18]. Recently, the real version Landis conjecture in the plane case was resolved by Logunov, Malinnikova, Nadirashvili, Nazarov [LMNN20].

A Landis-type conjecture was considered in [RW19] for the fractional Laplacian with b = 0. Both qualitative and quantitative estimates were proved in [RW19]. For example, when q is dierentiable and satises

|x · ∇q(x)| ≤ 1,

if u satises the following decay behavior: ∃ α > 1 such that Z

Rⁿ

e^|x|α|u|²dx < ∞,

2020 Mathematics Subject Classication. 35R11; 35B40.

Key words and phrases. Fractional Laplacian, Landis-type conjecture, Caarelli-Silvestre extension.

1

then u ≡ 0. On the other hand, for a non-dierentiable potential q, if s ∈ (1/4, 1), kqk_L^∞_(Rⁿ₎ ≤ 1, and u satises the decay behavior: ∃ α > _4s−1^4s such that

Z

Rⁿ

e^|x|α|u|²dx < ∞, then u ≡ 0.

The main theme of this work is to extend the results in [RW19] to the fractional Schrödinger equation with a drift term (1.1). The unique continuation property established in [GSU20] states that if u ∈ H^−r(Rⁿ) for some r ∈ R and u = (−∆)^su = 0 in some open set W ⊂ Rⁿ, then u ≡ 0. It follows from this property that the unique continuation property holds for (1.1). However, to our best knowledge, the strong unique continuation property for the fractional Laplacian with a drift term remains open. Our work is the rst attempt to study the unique continuation property for any solution of the fractional Laplacian with a drift term that satises some decaying condition. Motivated by the result of [Ro18], we consider the drift term only involving the radial derivative. Inspired by the ideas in [Me89] and [RW19], we show that if both b and q are dierentiable, then any non-trivial solution of the fractional Schrödinger equation does not decay super-exponentially at innity. The detailed statement is described in the following theorem.

Theorem 1.1. Let n = 1, 2, 3 and (1.2)

(s ∈ (¹₂, 1) when n = 1 or 2, s ∈ (³₄, 1) when n = 3.

We also assume that there exists a constant λ such that

(1.3) |q(x)| ≤ λ and 0 ≤ b(x) ≤ λ|x|^−β for all x ∈ Rⁿ and the radial derivatives of q, b satisfy

(1.4) |x · ∇q(x)| ≤ λ and |x · ∇b(x)| ≤ λ|x|^−β for all x ∈ Rⁿ for some β > 1. If u ∈ H^s(Rⁿ) is a solution to (1.1) such that

(1.5) Z

Rⁿ

e^|x|β



|u(x)|²+ |∇u(x)|²



dx ≤ λ, then u ≡ 0.

For non-dierentiable potential q, we also can prove the following result.

Theorem 1.2. Let n = 1, 2, 3 and s given in (1.2). Let β > _4s−1^4s , and both b, q satisfy (1.3). Here we assume the radial derivative of b satises

|x · ∇b(x)| ≤ λ|x|^−β for all x ∈ Rⁿ. If u ∈ H^s(Rⁿ) is a solution to (1.1) satisfying (1.5), then u ≡ 0.

Remark 1.3. Since we treat the drift term as a lower order addition, it is reasonable to expect that s > 1/2 in Theorem 1.1 and 1.2.

Like several existing results, we want to prove Theorem1.1and1.2by an appropriate Carleman estimate. Since such estimate is a local estimate, we rst localize the equation (1.1) by the Caarelli-Silvestre extension [CS07] and derive the Carleman estimate for a degenerate elliptic equation in the (n + 1)-dimensional upper half-space Rⁿ⁺¹+ .

Our strategy in proving both theorems is similar to that of [RW19]. We will mainly handle the CS extended solution ˜u in Rⁿ⁺¹+ . The condition (1.5) can be considered as a boundary decay for ˜u. We then pass the boundary decay to the bulk decay of ˜u in Rⁿ⁺¹+ . We would like to point out that unlike the pure potential case considered in [RW19], here, in order to guarantee the bulk decay of ˜u, we also need the boundary decay of ∇u due to the addition of the drift term.

We now comment on the form of drift coecient in (1.1). Such choice is due to the limitation in the Carleman estimate for ˜u in Rⁿ⁺¹+ . Since the boundary term contains the rst derivatives of u, to bound this boundary term, we need to include the second derivatives of ˜u in the Carleman estimate in view of the trace inequality (LemmaA.3).

However, the parameter appears in the rst derivatives of u is τ, while the parameter in the second derivatives of ˜u is τ⁻¹ (see (4.4)). Without further restrictions, this boundary term can not be removed. We also remark that the operator

(1.6) − ((−∆)^s+ λx · ∇) (where λ is a positive constant)

is related to a 2s-stable Ornstein-Uhlenbeck process in R^d for s ∈ (0, 1), see [Jak08, Sect. 2]. Precisely, let Xt be the 2s-stable Ornstein-Uhlenbeck process in R^d, given in the following stochastic integral:

X_t= e^−λtX₀+ Z t

0

e^−λ(t−s)d ˆX_s, Xˆ₀ = 0,

where the integral is in the Stieltjes sense, and ( ˆXt, P^x) is the isotropic 2s-stable Lévy process in R^d with index of stability s ∈ (0, 1) and characteristic function

E⁰e^{i ˆXt·ξ} = e^−t|ξ|2s for all ξ ∈ R^d and for all t ≥ 0.

Here, E^x denotes the expectation with respect to the distribution P^x of the process starting from x ∈ R^d. Indeed, the innitesimal genrator of Xt is equal to (1.6).

This paper is organized as follows. In Section 2, we introduce some notations and state the Caarelli-Silvestre extension. In Section 3, we discuss how the boundary decay of u implies the bulk decay of ˜u in Rⁿ⁺¹+ . The derivation of the needed Carleman estimate is discussed in great detail in Section4. Section 5 is devoted to the proofs of main results.

2. The Caffarelli-Silvestre extension

Basically, we shall follow the denitions and notations in [RW19]. We now restate what we need. Let Rⁿ⁺¹+ := Rⁿ × R+ =  x = (x⁰, x_n+1) x⁰ ∈ Rⁿ, x_n+1 > 0

and x₀ = (x⁰, 0) ∈ Rⁿ× {0}. For r, R > 0, we denote

B_r⁺(x₀) := x ∈ Rⁿ⁺¹+ |x − x₀| ≤ r , B_r⁰(x₀) := x ∈ Rⁿ× {0} |x − x0| ≤ r ,

B⁺_r := B_r⁺(0), B_r⁰ := B_r⁰(0), A⁺_r,R := x ∈ Rⁿ⁺¹+ r ≤ |x| ≤ R , A⁰_r,R := x ∈ Rⁿ× {0} r ≤ |x| ≤ R .

For Ω ⊂ Rⁿ⁺¹+ , we dene H˙¹(Ω, x^1−2s_n+1) :=



v : Ω → R Z

Ω

x^1−2s_n+1|∇v|²dx < ∞

 , H¹(Ω, x^1−2s_n+1) :=



v : Ω → R Z

Ω

x^1−2s_n+1(|v|²+ |∇v|²) dx < ∞

 .

Write x = rθ, where r > 0 and θ ∈ S+ⁿ, i.e., θ = (θ⁰, θn+1) ∈ Sⁿ with θn+1 > 0. We also denote

H¹(S₊ⁿ, θ_n+1^1−2s) :=



v : S₊ⁿ → R Z

S₊ⁿ

θ^1−2s_n+1 (|v|²+ |∇Sⁿv|²) dθ < ∞

 , where ∇^Sn = (Θ₁, · · · , Θ_n+1). Here, Θk are vector elds on Sⁿ.

For s ∈ (¹₂, 1)and u ∈ H^s(Rⁿ), let ˜u ∈ ˙H¹(Rⁿ⁺¹+ , x^1−2s_n+1)be a solution to the degenerate elliptic equation

(2.1)

(∇ · x^1−2s_n+1∇˜u = 0 in Rⁿ⁺¹+ ,

˜

u = u on Rⁿ× {0},

where ∇ = (∇⁰, ∂n+1) = (∂1, · · · , ∂n, ∂n+1). It was established in [CS07] that

−(−∆)^su(x) = c_n,s lim

xn+1→0x^1−2s_n+1∂_n+1u(x)˜

for some constant cn,s > 0. In view of this observation, (1.1) can be reformulated as the local, degenerate elliptic equation

(2.2)











∇ · x^1−2s_n+1∇˜u = 0 in Rⁿ⁺¹+ ,

˜

u = u on Rⁿ× {0},

cn,s lim

xn+1→0x^1−2s_n+1 ∂n+1u = bx˜ ⁰· ∇⁰u + qu on Rⁿ× {0}.

3. Boundary decay implies bulk decay

In this section, we will show that the boundary decay (1.5) implies the bulk decay of

˜

uin Rⁿ⁺¹+ .

Proposition 3.1. Let s be given in (1.2) and ˜u ∈ H¹(Rⁿ⁺¹+ , x^1−2s_n+1) be a solution to (2.2). Assume that there exists a constant λ > 0 such that

(3.1) |q(x⁰)| ≤ λ and |b(x⁰)||x⁰| ≤ λ for all x⁰ ∈ Rⁿ.

If there exist constants C, β > 1 such that (1.5) holds, then there exist constants C1, c₁ >

0 such that

(3.2) |˜u(x)| ≤ C₁e^−c1|x|β for all x ∈ Rⁿ⁺¹+ . The following lemma can be found in [RW19, Equ.(19)].

Lemma 3.2. Let s ∈ (¹₂, 1) and ˜u ∈ H¹(Rⁿ⁺¹+ , x^1−2s_n+1) be a solution to (2.1). If x0 ∈ Rⁿ× {0}, then there exist α = α(n, s) ∈ (0, 1) and c = c(n, s) ∈ (0, 1) such that

kx

1−2s 2

n+1uk˜ _L2(B⁺cr(x0))

≤ C

 kx

1−2s 2

n+1 uk˜ _L2(B_16r⁺ (x0))+ r^1−skuk_L²_(B⁰

16r(x0))

α

×

 r^s+1

xn+1lim→0x^1−2s_n+1∂_n+1u˜

L²(B⁰_16r(x0))

+ r^1−skuk_L²_(B⁰

16r(x0))

1−α

(3.3)

for some positive constant C.

Combining Lemma 3.2 and LemmaA.2 implies the following lemma.

Lemma 3.3. Let s be given by (1.2) and ˜u ∈ H¹(Rⁿ⁺¹+ , x^1−2s_n+1) be a solution to (2.2).

If q ∈ L^∞(Rⁿ) and b(x⁰)x⁰ ∈ (L^∞(Rⁿ))ⁿ, then there exist α = α(n, s) ∈ (0, 1) and c = c(n, s) ∈ (0, 1) such that

k˜uk_L∞(B⁺cr 4)

≤ Cr⁻ⁿ²



r^s−1kx_n+1^1−2s2 uk˜ _L2(B⁺_16r)+ kuk_L²_(B⁰

16r)

α

×



r^2skbx⁰ · ∇⁰u + quk_L²_(B⁰

16r)+ kuk_L²_(B⁰

16r)

1−α

+ r^2skbx⁰· ∇⁰u + quk_L²_(B⁰

16r)

 (3.4)

for all r > 1, where the constant C is independent to r.

Proof. Given any r > 1, let ˜v(x) = ˜u(rx) in Rⁿ⁺¹+ and v(x⁰) = u(rx⁰)on Rⁿ× {0}. Note that

c_n,s lim

xn+1→0x^1−2s_n+1∂_n+1v(x) = c˜ _n,s lim

xn+1→0x^1−2s_n+1 ∂_n+1 ˜u(rx)

= rcn,s lim

xn+1→0x^1−2s_n+1∂n+1u(rx)˜

= rc_n,s lim

rxn+1→0r^−1+2s(rx_n+1)^1−2s∂_n+1u(rx)˜

= r^2sc_n,s lim

rxn+1→0(rx_n+1)^1−2s∂_n+1u(rx)˜

= r^2sb(rx⁰)rx⁰· ∇⁰u(rx) + q(rx)u(rx)

(By (2.2))

= r^2sb(rx⁰)x⁰ · ∇⁰u(rx) + r^2sq(rx)u(rx)

= r^2sb(rx⁰)x⁰ · ∇⁰v(x) + r^2sq(rx)v(x), that is,

(3.5)











∇ · x^1−2s_n+1∇˜v = 0 in Rⁿ⁺¹+ ,

˜

v = v on Rⁿ× {0},

cn,s lim

xn+1→0x^1−2s_n+1∂n+1v = b˜ rx⁰ · ∇⁰v + qrv on Rⁿ× {0},

where br(x⁰) := r^2sb(rx⁰) and qr(x⁰) := r^2sq(rx). Applying Lemma A.2 to (3.5) with p = 2 (since (1.2)) a1 = 0 and a2 = b_rx⁰· ∇⁰v + q_rv, we obtain that

(3.6) k˜vk_L∞(B_1/4⁺ )≤ C



kx_n+1^1−2s2 ˜vk_L2(B₁⁺)+ kb_rx⁰ · ∇⁰v + q_rvk_L²_(B⁰

1)

 , where C = C(n, s). Since ˜v(x) = ˜u(rx), we have

k˜vk_L∞(B⁺_1/4) = sup

x∈Rⁿ⁺¹₊ ,|x|≤¹₄

|˜v(x)| = sup

x∈Rⁿ⁺¹₊ ,|rx|≤^r₄

|˜u(rx)| = k˜uk_L∞(B_r/4⁺ ). On the other hand, we can derive

kx

1−2s 2

n+1vk˜ ²_L2(B⁺₁) = Z

x∈Rⁿ⁺¹₊ ,|x|≤1

x^1−2s_n+1|˜v(x)|²dx = r^2s−2−nkx

1−2s 2

n+1uk˜ ²_L2(B⁺r)

and

kb_rx⁰ · ∇⁰v + q_rvk²_L2(B₁⁰)= Z

|x⁰|<1

|b_r(x⁰)x⁰· ∇⁰v(x⁰) + q_r(x⁰)v(x⁰)|²dx⁰

= r^4s−nkbx⁰ · ∇⁰u + quk²_L2(B_r⁰). Thus, (3.6) implies

k˜uk_L∞(B⁺_r/4) ≤ Cr⁻ⁿ²



r^s−1kx

1−2s 2

n+1uk˜ _L2(Br⁺)+ r^2skbx⁰· ∇⁰u + quk_L²_(B⁰_r)

 .

Here, r > 1 is arbitrary and C is independent of r. Replacing r by cr, where c is the constant given in Lemma3.2, we have

k˜uk_L∞(B_cr/4⁺ ) ≤ Cr⁻ⁿ²



r^s−1kx_n+1^1−2s2 uk˜ _L2(B⁺cr)+ r^2skbx⁰· ∇⁰u + quk_L²_(B⁰

cr)

 .

Here, C is another constant independent of r. Combining this inequality with (3.3) and using

c_n,s lim

xn+1→0x^1−2s_n+1∂_n+1u = bx˜ ⁰· ∇⁰u + qu,

we obtain our desired result. 

We are ready to prove the main result of this section.

Proof of Proposition3.1. Let R > 1 be suciently large and x0 ∈ Rⁿ× {0}with |x0| = 32R. From (1.5), it follows that

λ ≥ Z

B_16R⁰ (x0)

e^|x0|β



|u(x⁰)|²+ |∇⁰u(x⁰)|²

 dx⁰

≥ e^(16R)β Z

B_16R⁰ (x0)



|u(x⁰)|²+ |∇⁰u(x⁰)|²

 dx⁰

= e^(16R)β



kuk²_L2(B_16R⁰ (x0))+ k∇⁰uk²_L2(B_16R⁰ (x0))

 , that is,

(3.7) kuk_L²_(B⁰

16R(x0))+ k∇⁰uk_L²_(B⁰

16R(x0)) ≤ ˜Ce^−˜cRβ

for some constants ˜C and ˜c. Note that the Caarelli-Silvestre extension ˜u satises kx

1−2s 2

n+1∇˜uk_L2(Rⁿ⁺¹+ ) ≤ C and k˜u(•, xn+1)k_L²_(Rⁿ₎ ≤ kuk_L²_(Rⁿ₎

(see [St10, page 48-49]). Thus, we have

kx_n+1^1−2s2 uk˜ ²_L2(B+

16R(x0))≤ Z 16R

0

x^1−2s_n+1k˜u(•, x_n+1)k²_L2(Rⁿ)dx_n+1

≤ Z 16R

0

x^1−2s_n+1kuk²_L2(Rⁿ)dx_n+1

= (16R)^2−2s

2 − 2s kuk²_L2(Rⁿ)

≤ CR^2−2s.

Plugging cn,slim_xn+1→0x^1−2s_n+1∂_n+1u = bx˜ ⁰· ∇⁰u + qu into (3.3) (with r = R) and using (3.1) yields

kx

1−2s 2

n+1uk˜ _L2(B⁺_cR(x0)) ≤ ˜Ce^−˜cRβ.

Next, choosing r = ^cR₁₆ in (3.4) and putting together (3.1) and (1.5), we have k˜uk_L∞(B⁺

c2R 64

(x0)) ≤ ˜Ce^−˜cRβ for |x0| = 32R, which implies

k˜uk_L∞(B⁺

c ˜R(x0))≤ ˜Ce^{−˜c ˜Rβ} for all large ˜R  1.

The decay estimate (3.2) then follows from the chain-of-balls argument described in

[RW19, Proposition 2.2, Step 2]. 

4. Carleman estimates

This section is mainly devoted to the derivations of Carleman estimates. We will discuss the estimates corresponding to both dierentiable and non-dierentiable potentials.

4.1. Carleman estimate with dierentiable potential. The proof of the Carleman estimate below follows from the argument in [RW19].

Theorem 4.1. Let s ∈ (¹₂, 1) and let ˜u ∈ H¹(Rⁿ⁺¹+ , x^1−2s_n+1) with supp(˜u) ⊂ Rⁿ⁺¹+ \ B₁⁺ be a solution to











∇ · x^1−2s_n+1∇˜u = f in Rⁿ⁺¹+ ,

˜

u = u on Rⁿ× {0},

xn+1lim→0x^1−2s_n+1∂_n+1u = W x˜ ⁰ · ∇⁰u + V u + g on Rⁿ× {0},

where f ∈ L²(Rⁿ⁺¹+ , x^2s−1_n+1) is compactly supported in Rⁿ⁺¹+ , g ∈ L²(Rⁿ) is compactly supported in Rⁿ, and x⁰ · ∇⁰W, x⁰· ∇⁰V exist. Let α > 1 be a real constant and dene φ(x) := |x|^α. If

W (x⁰) ≥ 0 for all x⁰ ∈ Rⁿ\ B₁⁰.

Then there exists a real number τ0 > 1 such that τ³ke^{τ φ}|x|^{3α2 −1}x

1−2s 2

n+1uk˜ ²_L2

(Rⁿ⁺¹₊ )+ τ ke^{τ φ}|x|^α2x

1−2s 2

n+1∇˜uk²_L2

(Rⁿ⁺¹₊ )

+ τ⁻¹ke^{τ φ}|x|^−α2x

1−2s 2

n+1∇(x · ∇˜u)k²_L2

(Rⁿ⁺¹₊ )

≤ C



ke^{τ φ}|x|x

2s−1 2

n+1f k²_L2(Rⁿ⁺¹₊ )

+ τ ke^{τ φ}|x|^α2|V |¹²uk²_L2(Rⁿ×{0})+ τ ke^{τ φ}|x|^α2|x⁰ · ∇⁰V |¹²uk²_L2(Rⁿ×{0})

+ τ²ke^{τ φ}|x|^α|W |¹²uk˜ ²_L2(Rⁿ×{0})+ τ²ke^{τ φ}|x|^α|x⁰· ∇⁰W |¹²uk²_L2(Rⁿ×{0})

+ τ²ke^{τ φ}|x|^α2uk²_L2(Rⁿ×{0})

+ τ⁻¹ke^{τ φ}|x|^−α2|V |¹²(x⁰· ∇⁰u)k²_L2(Rⁿ×{0})

+ τ⁻¹ke^{τ φ}|x|^−α2|x⁰ · ∇⁰V |¹²(x⁰ · ∇⁰u)k²_L2(Rⁿ×{0})

+ ke^{τ φ}|W |¹²(x⁰· ∇⁰u)k²_L2(Rⁿ×{0})+ ke^{τ φ}|x⁰· ∇⁰W |¹²(x⁰· ∇⁰u)k²_L2(Rⁿ×{0})

+ ke^{τ φ}|x|^−α2(x⁰· ∇⁰u)k²_L2(Rⁿ×{0})

+ τ²ke^{τ φ}|x|^32α|x|^1+2s2 gk²_L2(Rⁿ×{0})+ ke^{τ φ}|x|^−α2|x|^s(x⁰· ∇⁰g)k²_L2(Rⁿ×{0})

 (4.1)

for all τ ≥ τ0. Here, the positive constant C is independent of τ.

Proof. Step 1: Pass to conformal polar coordinates. Let x = e^tθ with t ∈ R and θ ∈ S₊ⁿ. Set u := e^{n−2s2 t}u˜, we have

(4.2)

















θ^1−2s_n+1∂²_t + ∇Sⁿ· θ_n+1^1−2s∇Sⁿ − θ^1−2s_n+1 (n − 2s)² 4



u = ˜f in S+ⁿ× R,

θn+1lim→0θ^1−2s_n+1ν · ∇Sⁿu =



V −˜ n − 2s 2

W˜



u + ˜W ∂_tu + ˜g on ∂S+ⁿ × R, where ˜f = e^{n+2+2s2 t}f, ν = (0, · · · , 0, 1), ˜V = e^2stV, ˜W = e^2stW, and ˜g = e^{n+2s2 t}g.

Next, by setting ˜∆_Sⁿ := θ

2s−1 2

n+1 ∇_Sⁿ·θ^1−2s_n+1∇_Sⁿθ

2s−1 2

n+1, ϕ(t) = φ(e^tθ) = e^αt, v = e^{τ ϕ}θ

1−2s 2

n+1 u, f = e^{τ ϕ}f˜, and g = e^{τ ϕ}g˜, (4.2) can be written as











L_ϕv := (S + A)v = θ

2s−1 2

n+1 f in S+ⁿ× R,

lim

θn+1→0θ^1−2s_n+1ν · ∇Sⁿθ

2s−1 2

n+1 v = I + ˜W θ

2s−1 2

n+1∂_tv + g on ∂S+ⁿ× R, where

S := ∂_t²+ ˜∆Sⁿ −(n − 2s)²

4 + τ²|ϕ⁰|², A := −2τ ϕ⁰∂_t− τ ϕ⁰⁰,

I :=



V −˜ n − 2s 2

W˜

 θ

2s−1 2

n+1v − τ ϕ⁰W θ˜

2s−1 2

n+1 v.

To shortened the notations, we denote the norm and the scalar product in the bulk and the boundary space by

(4.3)

(k • k := k • k_L²_(Sⁿ

+×R), h•, •i := h•, •i_L²_(Sⁿ

+×R)

k • k₀ := k • k_L²_(∂Sⁿ

+×R), h•, •i₀ := h•, •i_L²_(∂Sⁿ

+×R)

and we omit the notation limθn+1→0 in k • k0 and h•, •i0.

Since S, A are only symmetric and anti-symmetric up to boundary contributions, we can see that

kL_ϕvk² = kSvk² + kAvk²+ h[S, A]v, vi + 4τ hθ^1−2s_n+1ν · ∇_Sⁿθ

2s−1 2

n+1 v, ϕ⁰θ

2s−1 2

n+1∂_tvi₀ (4.4)

+ 2τ hθ^1−2s_n+1ν · ∇Sⁿθ

2s−1 2

n+1 v, ϕ⁰⁰θ

2s−1 2

n+1 vi₀. From ˜W ≥ 0, it follows that h ˜W θ

2s−1 2

n+1 ∂_tv, ϕ⁰θ

2s−1 2

n+1∂_tvi₀ ≥ 0 and kL_ϕvk² ≥ kSvk²+ kAvk²+ h[S, A]v, vi

+ 4τ hI, ϕ⁰θ

2s−1 2

n+1 ∂_tvi₀+ 4τ hg, ϕ⁰θ

2s−1 2

n+1 ∂_tvi₀ (4.5)

+ 2τ hθ_n+1^1−2sν · ∇Sⁿθ

2s−1 2

n+1v, ϕ⁰⁰θ

2s−1 2

n+1vi₀.

Step 2: Estimate the commutator term h[S, A]v, vi. The commutator term can be computed explicitly

[S, A]v = 4τ³|ϕ⁰|²ϕ⁰⁰v − 4τ ϕ⁰⁰∂_t²v − 4τ ϕ⁰⁰⁰∂_tv − τ ϕ⁰⁰⁰⁰v.

Since supp(˜u) ⊂ Rⁿ⁺¹+ \ B⁺₁, we can see that supp(v) ⊂  (θ, t) ∈ S+ⁿ× R t ≥ 0 Recall that ϕ(t) = e^αt for α > 1. We thus have |ϕ⁰|²ϕ⁰⁰ ≥ ϕ⁰⁰⁰⁰ in supp(v). By this. inequality, we can estimate

h[S, A]v, vi ≥ 2τ³kϕ⁰(ϕ⁰⁰)¹²vk²+ 4τ k(ϕ⁰⁰)¹²∂_tvk². Combining this inequality with (4.5) yields

kL_ϕvk² ≥ kSvk² + kAvk²+ 2τ³kϕ⁰(ϕ⁰⁰)¹²vk²+ 4τ k(ϕ⁰⁰)¹²∂_tvk² + 4τ hI, ϕ⁰θ

2s−1 2

n+1 ∂_tvi₀ + 4τ hg, ϕ⁰θ

2s−1 2

n+1 ∂_tvi₀ + 2τ hθ_n+1^1−2sν · ∇Sⁿθ

2s−1 2

n+1 v, ϕ⁰⁰θ

2s−1 2

n+1 vi0. (4.6)

Step 3: Derive the full gradient inequality from the symmetric part kSvk². Let c0 be a positive constant to be determined later. Note that

− ˜∆Sⁿ = −S + ∂_t²+ τ²(ϕ⁰)²− (n − 2s)²

4 .

Hence, we have

c₀τ k(ϕ⁰)¹²θ

1−2s 2

n+1 ∇Sⁿθ

2s−1 2

n+1 vk²+ c₀τ hθ^1−2s_n+1 ν · ∇Sⁿθ

2s−1 2

n+1 v, ϕ⁰θ

2s−1 2

n+1 vi₀

= −c₀τ hϕ⁰v, ˜∆Sⁿvi

= c₀τ



− hϕ⁰v, Svi − hϕ⁰⁰v, ∂_tvi + k(ϕ⁰)¹²∂_tvk²+ τ²k(ϕ⁰)³²vk²

− (n − 2s)²

4 k(ϕ⁰)¹²vk²



≤ −c₀τ hϕ⁰v, Svi − c₀τ hϕ⁰⁰v, ∂_tvi + c₀τ k(ϕ⁰)¹²∂_tvk²+ 2c₀τ³k(ϕ⁰)³²vk² (4.7)

for all τ  1. Combining (4.6) and (4.7) gives

kSvk²+ kAvk²+ 2τ³kϕ⁰(ϕ⁰⁰)¹²vk²+ 4τ k(ϕ⁰⁰)¹²∂tvk² + c₀τ k(ϕ⁰)¹²θ

1−2s 2

n+1 ∇Sⁿθ

2s−1 2

n+1vk² + 4τ hI, ϕ⁰θ

2s−1 2

n+1∂_tvi₀+ 4τ hg, ϕ⁰θ

2s−1 2

n+1 ∂_tvi₀ + 2τ hθ^1−2s_n+1 ν · ∇Sⁿθ

2s−1 2

n+1v, ϕ⁰⁰θ

2s−1 2

n+1 vi₀ + c₀τ hθ^1−2s_n+1 ν · ∇Sⁿθ

2s−1 2

n+1 v, ϕ⁰θ

2s−1 2

n+1 vi₀

≤ kL_ϕvk²− c₀τ hϕ⁰v, Svi − c₀τ hϕ⁰⁰v, ∂_tvi + c₀τ k(ϕ⁰)¹²∂_tvk²+ 2c₀τ³k(ϕ⁰)³²vk².

Using ϕ⁰⁰ = αϕ⁰ and choosing suciently small c0 > 0, we can derive kSvk²+ kAvk²+ τ³kϕ⁰(ϕ⁰⁰)¹²vk²

+ τ k(ϕ⁰⁰)¹²∂_tvk²+ τ k(ϕ⁰)¹²θ

1−2s 2

n+1 ∇Sⁿθ

2s−1 2

n+1vk²

≤ C



kLϕvk²+ τ |hI, ϕ⁰θ

2s−1 2

n+1∂tvi0| + τ |hg, ϕ⁰θ

2s−1 2

n+1 ∂tvi0| + τ |hθ_n+1^1−2sν · ∇Sⁿθ

2s−1 2

n+1 v, ϕ⁰⁰θ

2s−1 2

n+1 vi₀|

 (4.8)

for some positive constant C, which is independent of τ.

Step 4: Estimate the second derivatives from the symmetric part kSvk². Let c1 be a positive constant to be determined later. In view of ϕ⁰ ≥ 1 in supp(v), we obtain

kSvk² ≥ c1τ⁻¹k(ϕ⁰)⁻¹²Svk² ≥ c1τ⁻¹k(ϕ⁰)⁻¹²S⁰vk²− Cτ⁻¹k(ϕ⁰)⁻¹²vk², where S⁰ := ∂_t²+ ˜∆Sⁿ+ τ²(ϕ⁰)². It is clear that

k(ϕ⁰)⁻¹²S⁰vk² = k(ϕ⁰)⁻¹²∂_t²vk²+ k(ϕ⁰)⁻¹²∆˜Sⁿvk²+ τ⁴k(ϕ⁰)³²vk²

+ 2h(ϕ⁰)⁻¹∂_t²v, ˜∆Sⁿvi + 2τ²h∂_t²v, ϕ⁰vi + 2τ²h ˜∆Sⁿv, ϕ⁰vi.

Repeating the integration by parts gives

h∂_t²v, ϕ⁰vi = −h∂tv, ϕ⁰⁰vi − k(ϕ⁰)¹²∂tvk², h ˜∆Sⁿv, ϕ⁰vi = −hθ_n+1^1−2sν · ∇Sⁿθ

2s−1 2

n+1v, ϕ⁰θ

2s−1 2

n+1 vi0− k(ϕ⁰)¹²θ

1−2s 2

n+1 ∇Sⁿθ

2s−1 2

n+1 vk²,

and

h(ϕ⁰)⁻¹∂_t²v, ˜∆Sⁿvi = − hθ^1−2s_n+1ν · ∇Sⁿθ

2s−1 2

n+1 v, (ϕ⁰)⁻¹θ

2s−1 2

n+1 ∂_t²vi0

− 1

2kϕ⁻¹(ϕ⁰)¹²θ

1−2s 2

n+1 ∇Sⁿθ

2s−1 2

n+1 vk² + k(ϕ⁰)⁻¹²θ

1−2s 2

n+1 ∇Sⁿθ

2s−1 2

n+1∂_tvk². Therefore, we can derive

kSvk² ≥ c1τ⁻¹k(ϕ⁰)⁻¹²Svk²

≥ c₁τ⁻¹k(ϕ⁰)⁻¹²∂_t²vk²+ 2c₁τ⁻¹k(ϕ⁰)⁻¹²θ

1−2s 2

n+1 ∇Sⁿθ

2s−1 2

n+1 ∂_tvk² + c₁τ³k(ϕ⁰)³²vk²− c₁τ⁻¹kϕ⁻¹(ϕ⁰)¹²θ

1−2s 2

n+1 ∇_Sⁿθ

2s−1 2

n+1 vk²

− c₁τ h∂_tv, ϕ⁰⁰vi − c₁τ k(ϕ⁰)¹²∂_tvk²

− 2c₁τ⁻¹hθ_n+1^1−2sν · ∇Sⁿθ

2s−1 2

n+1v, (ϕ⁰)⁻¹θ

2s−1 2

n+1 ∂_t²vi₀

− 2c₁τ hθ_n+1^1−2sν · ∇Sⁿθ

2s−1 2

n+1v, ϕ⁰θ

2s−1 2

n+1 vi₀. (4.9)

Putting this inequality and (4.8) together and choosing suciently small c1 > 0, we have

τ³kϕ⁰(ϕ⁰⁰)¹²vk²+ τ k(ϕ⁰⁰)¹²∂_tvk²+ τ k(ϕ⁰)¹²θ

1−2s 2

n+1 ∇Sⁿθ

2s−1 2

n+1 vk² + τ⁻¹k(ϕ⁰)⁻¹²∂_t²vk² + τ⁻¹k(ϕ⁰)⁻¹²θ

1−2s 2

n+1 ∇Sⁿθ

2s−1 2

n+1∂tvk²

≤ C



kL_ϕvk² + τ |hI, ϕ⁰θ

2s−1 2

n+1∂_tvi₀| + τ |hg, ϕ⁰θ

2s−1 2

n+1 ∂_tvi₀| + τ |hθ_n+1^1−2sν · ∇Sⁿθ

2s−1 2

n+1v, ϕ⁰⁰θ

2s−1 2

n+1vi₀| + τ⁻¹|hθ_n+1^1−2sν · ∇_Sⁿθ

2s−1 2

n+1v, (ϕ⁰)⁻¹θ

2s−1 2

n+1 ∂_t²vi₀|

 (4.10) .

Step 5.1: Estimate the term τ|hI, ϕ⁰θ

2s−1 2

n+1∂tvi0| + τ |hg, ϕ⁰θ

2s−1 2

n+1∂tvi0|. Write ˜U :=

V −˜ ^n−2s₂ W˜. Performing integration by parts leads to

|τ h ˜U θ

2s−1 2

n+1 v, ϕ⁰θ

2s−1 2

n+1∂_tvi₀|

= τ    

− 1 2h ˜U θ

2s−1 2

n+1 v, ϕ⁰⁰θ

2s−1 2

n+1 vi₀− 1 2h∂_tU θ˜

2s−1 2

n+1 v, ϕ⁰θ

2s−1 2

n+1 vi₀    

≤ τ

2k| ˜U |¹²|ϕ⁰⁰|¹²θ

2s−1 2

n+1vk²₀+ τ

2k|∂_tU |˜ ¹²|ϕ⁰|¹²θ

2s−1 2

n+1 vk²₀

≤ C



τ k| ˜V |¹²|ϕ⁰⁰|¹²θ

2s−1 2

n+1vk²₀+ τ k|∂_tV |˜ ¹²|ϕ⁰|¹²θ

2s−1 2

n+1vk²₀ + τ k| ˜W |¹²|ϕ⁰⁰|¹²θ

2s−1 2

n+1vk²₀+ τ k|∂_tW |˜ ¹²|ϕ⁰|¹²θ

2s−1 2

n+1vk²₀