Gradient Estimates on Dirichlet and Neumann Eigenfunctions

Gradient Estimates on Dirichlet and Neumann Eigenfunctions

Marc Arnaudon¹, Anton Thalmaier², Feng-Yu Wang³

1Institut de Math´ematiques de Bordeaux, Universit´e de Bordeaux, 351 Cours de la Lib´eration, F–33405 Talence Cedex, France

[email protected]

2Mathematics Research Unit, FSTC, University of Luxembourg, Maison du Nombre, L–4364 Esch-sur-Alzette, Grand Duchy of Luxembourg

[email protected]

3Center for Applied Mathematics, Tianjin University, Tianjin 300072, China, and Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, United Kingdom

[email protected]

August 9, 2018

Abstract

By methods of stochastic analysis on Riemannian manifolds, we derive explicit constants c₁(D) andc₂(D) for ad-dimensional compact Riemannian manifoldD with boundary such that

c1(D)√

λkφk_∞6k∇φk_∞6c2(D)√ λkφk_∞

holds for any Dirichlet eigenfunctionφof−∆ with eigenvalueλ. In particular, whenDis convex with non-negative Ricci curvature, the estimate holds for

c₁(D) = 1

de, c₂(D) =√ e

√2

√π+

√π 4√

2

! .

Corresponding two-sided gradient estimates for Neumann eigenfunctions are derived in the sec- ond part of the paper.

AMS subject Classification: 35P20, 60H30, 58J65

Keywords: Eigenfunction, gradient estimate, diffusion process, curvature, second fundamental form.

This work was supported by Fonds National de la Recherche Luxembourg (Grant No. O14/7628746 GEOMREV) and the University of Luxembourg (Grant No. IRP R-AGR-0517-10/AGSDE. Feng-Yu Wang acknowledges support in part by NNSFC (11771326, 11431014).

1 Introduction

Let D be a d-dimensional compact Riemannian manifold with boundary ∂D. We write (φ, λ) ∈ Eig(∆) if φis a Dirichlet eigenfunction of −∆ inDwith eigenvalue λ >0. According to [7], there exist two constantsc1(D), c2(D)>0 such that

(1.1) c1(D)

√

λkφk_∞6k∇φk_∞6c2(D)

√

λkφk_∞, (φ, λ)∈Eig(∆).

An analogous statement for Neumann eigenfunctions has been derived in [5].

Concerning Dirichlet eigenfunctions, an explicit upper constant c2(D) can be derived from the uniform gradient estimate of the Dirichlet semigroup in an earlier paper [10] of the third named author. More precisely, letK, θ>0 be two constants such that

(1.2) RicD >−K, H∂D >−θ,

where RicD is the Ricci curvature on Dand H∂D the mean curvature of ∂D. Let

(1.3) α0= 1

2max θ,p

(d−1)K .

Consider the semigroup P_t = e^t∆ for the Dirichlet Laplacian ∆. According to [10, Theorem 1.1]

wherec= 2α0, for any nontrivial f ∈Bb(D) andt >0, the following estimate holds:

k∇P_tfk_∞

kfk∞ 69.5α₀+2√

α0(1 + 4^2/3)^1/4(1 + 5×2^−1/3)

(tπ)^1/4 +

p1 + 2^1/3(1 + 4^2/3) 2√

tπ =:c(t).

Consequently, for any (φ, λ)∈Eig(∆),

k∇φk∞6kφk∞inf

t>0c(t)e^λt. In particular, when Ric_D >0, H_∂D >0,

(1.4) k∇φk_∞6

pe (1 + 2^1/3) (1 + 4^2/3)

√2π

√

λkφk_∞, (φ, λ)∈Eig(∆).

In this paper, by using stochastic analysis of the Brownian motion on D, we develop two-sided gradient estimates; the upper bound given below in (1.8) improves the one in (1.4). Our result will also be valid for α0 ∈R satisfying

(1.5) 1

2∆ρ∂D 6α0 outside the focal set,

whereρ_∂D is the distance to the boundary. The caseα₀<0 appears naturally in many situations, for instance when D is a closed ball with convex distance to the origin. Note that by [10, Lemma 2.3], if under (1.2) we defineα0 by (1.3) then condition (1.5) holds as a consequence.

For x>0, in what follows in the limiting case x= 0 we use the convention 1

1 +x 1/x

:= lim

r↓0

1 1 +r

1/r

= 1 e.

Theorem 1.1. LetK, θ >0 be two constants such that (1.2) holds and letα0 be given by (1.3) or more generally satisfy (1.5). Then, for any nontrivial (φ, λ)∈Eig(∆),

(1.6) λ

pde(λ+K) 6 λ pd(λ+K)

λ λ+K

λ/(2K)

6 k∇φk_∞

kφk_∞

and

(1.7) k∇φk_∞

kφk_∞ 6

(pe(λ+K) if √

λ+K >2A

√e A+ ^λ+K_4A

if √

λ+K 62A, where

A:= 2α⁺₀ +

p2(λ+K)

√π exp

− α₀² 2(λ+K)

. In particular, when Ric_D >0, H_∂D >0,

(1.8)

√

√λ

de 6 k∇φk_∞ kφk_∞ 6

√ λ

√

√2e π +

√πe 4√

2

!

, (φ, λ)∈Eig(∆).

Proof. This result follows from Theorem 2.1 and Theorem 2.2 below in the special caseV = 0. In this case, Ric^V_D = Ric_D >−K is equivalent to (2.1) with n=d. More sophisticated upper bounds are given below in Theorem 2.2.

By (1.8), if Dis convex with non-negative Ricci curvature then (1.1) holds with c₁(D) = 1

√

de, c₂(D) =

√2e

√π +

√πe 4√

2.

To give explicit values of c₁(D) and c₂(D) for positive K or θ, let λ₁ > 0 be the first Dirichlet eigenvalue of −∆ onD. Then Theorem 1.1 implies that (1.1) holds for

c1(D) =

√λ1

pde(λ₁+K), c₂(D) =

pe(λ1+K)

√λ₁ 1_{B>2A}+

√e

√λ₁ 2α⁺₀ +

r2(λ1+K)

π + λ1+K

4 2α⁺₀ +p

2(λ1+K)/π

!

1_{B62A}

with

B =p

λ₁+K and A= 2α⁺₀ +

r2(λ₁+K)

π .

This is due to the fact that the expression forc1(D) is an increasing function ofλand the expression forc₂(D) a decreasing function ofλ. Since there exist explicit lower bound estimates onλ₁ (see [9]

and references within), this gives explicit lower bounds ofc₁(D) and explicit upper bounds ofc₂(D).

The lower bound for k∇φk_∞ will be derived by using Itˆo’s formula for |∇φ|²(Xt) where Xt is a Brownian motion (with drift) on D, see Subsection 2.1 for details. To derive the upper bound estimate, we will construct some martingales to reduce k∇φk_∞ to k∇φk_∂D,∞ := sup_∂D|∇φ|, and to estimate the latter in terms of kφk∞, see Subsection 2.2 for details.

Next, we consider the Neumann problem. Let Eig_N(∆) be the set of non-trivial eigenpairs (φ, λ) for the Neumann eigenproblem, i.e. φ is non-constant, ∆φ =−λφ withN φ|_∂D = 0 for the unit inward normal vector fieldN of∂D. LetI∂D be the second fundamental form of∂D,

I∂D(X, Y) =−h∇_XN, Yi, X, Y ∈T_x∂M, x∈∂M.

With a concrete choice of the function f, the next theorem implies (1.1) for (φ, λ) ∈ Eig_N(∆) together with explicit constantsc₁(D), c₂(D).

Theorem 1.2. LetK, δ∈R be constants such that

(1.9) Ric_D >−K, I∂D >−δ.

Forf ∈C_b²( ¯D) with inf

D f = 1 andNlogf|_∂D >δ, let c_ε(f) = sup

D

4ε|∇logf|²

1−ε +K−2∆ logf

, ε∈(0,1), K(f) = sup

D

2|∇logf|²+K−∆ logf .

Then for any non-trivial (φ, λ)∈Eig_N(∆), we haveλ+c_ε(f)>0 and sup

ε∈(0,1)

ελ²

de(λ+c_ε(f))kfk²_∞ 6 sup

ε∈(0,1)

ελ²

d(λ+c_ε(f))kfk²_∞

λ λ+c_ε(f)

λ/cε(f)

6 k∇φk²_∞

kφk²_∞ 6 2kfk²_∞(λ+K(f)) π

1 +K(f) λ

λ/K(f)

62ekfk²_∞λ+K(f)

π .

Proof. Under the conditions (1.2), Theorem 3.3 below applies with L= ∆, K_V = K and n = d.

The desired estimates are immediate consequences.

When ∂D is convex, i.e. I∂D > 0, we may take f ≡ 1 in Theorem 1.2 to derive the following result. According to Theorem 3.2 below, this result also holds for∂D =∅where Eig(∆) is the set of eigenpairs for the closed eigenproblem.

Corollary 1.3. Let ∂D be convex or empty. If Ric^V_D > −K for some constant K, then for any non-trivial (φ, λ)∈Eig_N(∆), we haveλ+K >0 and

λ²

de(λ+K⁺) 6 λ² d(λ+K)

λ λ+K

λ/K

6 k∇φk²_∞

kφk²_∞ 6 2(λ+K) π

1 +K

λ λ/K

6 2e(λ+K⁺)

π .

2 Proof of Theorem 1.1

In general, we will consider Dirichlet eigenfunctions for the symmetric operatorL:= ∆ +∇V onD whereV ∈C²(D). We denote by Eig(L) the set of pairs (φ, λ) where φis a Dirichlet eigenfunction of −L on Dwith eigenvalue λ.

In the following two subsections, we consider the lower bound and upper bound estimates respectively.

2.1 Lower bound estimate

In this subsection we will estimatek∇φk_∞from below using the following Bakry- ´Emery curvature- dimension condition:

(2.1) 1

2L|∇f|²− h∇Lf,∇fi>−K|∇f|²+(Lf)²

n , f ∈C^∞(D),

where K ∈R, n>d are two constants. When V = 0, this condition with n =d is equivalent to Ric_D >−K.

Theorem 2.1 (Lower bound estimate). Assume that (2.1) holds. Then (2.2) k∇φk²_∞>kφk²_∞sup

t>0

λ²(e^Kt−1)

nKe^(λ+K)+t, (φ, λ)∈Eig(L).

Consequently, for K⁺:= max{0, K} there holds (2.3) k∇φk²_∞> λ²kφk²_∞

n(λ+K⁺) λ

λ+K⁺ λ/K⁺

> λ²kφk²_∞

ne(λ+K⁺), (φ, λ)∈Eig(L).

Proof. Let X_t be the diffusion process generated by ¹₂LinD, and let τ_D := inf{t>0 :X_t∈∂D}.

By Itˆo’s formula, we have

(2.4) d|∇φ|²(X_t) = 1

2L|∇φ|²(X_t) dt+ dM_t, t6τ_D,

for some martingale Mt. By the curvature dimension condition (2.1) andLφ=−λφ, we obtain

(2.5) 1

2L|∇φ|²= 1

2L|∇φ|²− h∇Lφ,∇φi −λ|∇φ|² >−(K+λ)|∇φ|²+λ² nφ². Therefore, (2.4) gives

d|∇φ|²(Xt)>

λ²

nφ²−(K+λ)|∇φ|²

(Xt) dt+ dMt, t6τD. Hence, for anyt >0,

e^(K+λ)+tk∇φk²_∞>E

h|∇φ|²(Xt∧τ_D)e^{(K+λ)(t∧τD)}i

> λ² nE

Z t∧τ_D 0

e^(K+λ)sφ(Xs)²ds

= λ² nE

Z t 0

1_{s<τD}e^(K+λ)sφ(X_s)²ds

. Since φ|_∂D= 0 and Lφ=−λφ, by Jensen’s inequality we have

E

1_{s<τD}φ(X_s)²

> E[φ(Xs∧τ_D)]2

= e^−λsφ(x)²,

where x = X₀ ∈ D is the starting point of X_t. Then, by taking x such that φ(x)² = kφk²_∞, we arrive at

e^(K+λ)+tk∇φk²_∞> λ² n

Z t 0

e^(K+λ)se^−λsφ(x)²ds

= λ²kφk²_∞ n

Z t 0

e^Ksds= λ²(e^Kt−1) nK kφk²_∞. This completes the proof of (2.2).

Since (2.1) holds for K⁺ replacing K, we may and do assume that K > 0. By taking the optimal choice t= _K¹ log(1 + ^K_λ) (by convention t=λ⁻¹ ifK = 0) in (2.2), we obtain

k∇φk²_∞> λ²kφk²_∞ λ+K

λ λ+K

λ/K

> λ²kφk²_∞ ne(λ+K). Hence (2.3) holds.

2.2 Upper bound estimate

Let Ric^V_D = Ric_D −Hess_V. For K₀, θ>0 such that Ric_D >−K₀ and H_∂D>−θ, let

(2.6) α= 1

2

max θ,p

(d−1)K₀ +k∇Vk_∞ We note that ¹₂Lρ∂D6α by [10, Lemma 2.3].

Theorem 2.2 (Upper bound estimate). LetKV, θ>0 be constants such that Ric^V_D >−K_V, H_∂D>−θ.

Letα∈Rbe such that

(2.7) 1

2Lρ_∂D6α.

1. Assume α>0. Then, for any nontrivial (φ, λ)∈Eig(L),

(2.8) k∇φk_∞

kφk_∞ 6

(pe(λ+K_V) if √

λ+K_V >2A

√e

A+^λ+K_4A^V

if √

λ+K_V 62A, where

(2.9) A:=α+

p2(λ+K_V)

√π exp

− α² 2(λ+KV)

+|α| ∧

√ 2α² pπ(λ+K_V). In particular, (2.8) holds with Areplaced by

(2.10) A⁰:= 2α+

p2(λ+KV)

√π exp

− α² 2(λ+K_V)

.

We also have

(2.11) k∇φk_∞

kφk_∞ 6√

e 2α+p

2(λ+K_V)

√π +λ+K_V 4

√π 2α+p

2(λ+K_V)

! .

2. Assume α60. Then, for any nontrivial (φ, λ)∈Eig(L),

(2.12) k∇φk_∞

kφk_∞ 6

(pe(λ+K_V) if √

λ+K_V >2A^∗

√e

A^∗+^λ+K_4A∗^V

if √

λ+K_V 62A^∗, where

(2.13) A^∗ :=

p2(λ+K_V)

√π exp

− α² 2(λ+K_V)

.

In particular,

(2.14) k∇φk_∞

kφk_∞ 6p

λ+K_V r2

π +1 4

rπ 2

!√ e.

In addition, the following estimate holds:

(2.15) k∇φk∞

kφk_∞ 6

(pe(λ+KV) if √

λ+KV >2√ e ˆA e ˆA+^λ+KV

4 ˆA if √

λ+K_V <2√ e ˆA, where

(2.16) Aˆ:=α+

√

√2λ π e^−α

2

2λ +|α| ∧

√ 2α²

√πλ.

The strategy to prove Theorem 2.2 will be to first estimate k∇φk_∞ in terms of kφk_∞ and k∇φk_∂D,∞ (see estimate (2.24) below) where kfk_∂D,∞ := k1_∂Dfk_∞ for a function f on D. The this end we construct appropriate martingales in terms of φand ∇φ.

We start by recalling the necessary facts about the diffusion process generated by ¹₂L, see for instance [1, 3]. For anyx∈D, the diffusionXt solves the SDE

(2.17) dX_t= 1

2∇V(X_t) dt+u_t◦dB_t, X₀=x, t6τ_D,

whereBtis ad-dimensional Brownian motion,utis the horizontal lift of Xtonto the orthonormal frame bundle O(D) with initial valueu₀∈O_x(D),and

τD := inf{t>0 :Xt∈∂D}

is the hitting time ofXt to the boundary∂D. SettingZ :=∇V, we have

(2.18) du_t= 1

2Z^∗(u_t) dt+

d

X

i=1

H_i(u_t)◦dBⁱ_t

where Z^∗(u) := hu(Z_π(u)) and Hi(u) := hu(uei) are defined by means of the horizontal lift hu: T_π(u)D → TuO(D) at u ∈ O(D). Note that formally hut(ut◦dBt) = P

ihut(utei)◦dB_tⁱ = P

iH_i(u_t)◦dB_tⁱ.

For f ∈C^∞(D), leta:= df ∈Γ(T^∗D). Settingmt:=u⁻¹_t a(Xt), we see by Itˆo’s formula that

(2.19) dmt

=m 1

2u⁻¹_t (a+∇_Za)(Xt) dt

wherea= tr∇²adenotes the so-called connection (or rough) Laplacian on 1-forms and= equality^m modulo the differential of a local martingale.

Denote by Q_t:T_xD → T_XtD the solution, along the paths of X_t, to the covariant ordinary differential equation

DQt=−1

2(Ric^V_D)^]Qtdt, Q0 = id_TxD, t6τ_D, whereD:=u_tdu⁻¹_t and where by definition

(Ric^V_D)^]v= Ric_D^V(·, v)^], v∈TxD.

Thus, condition Ric^V_D >−K_V implies

(2.20) |Q_tv|6e^{KV2 t}|v|, t6τD.

Finally, note that for any smooth function f on D, we have by the Weitzenb¨ock formula:

d ∆ +Z

f = d −d^∗df + (df)Z

= ∆⁽¹⁾df+∇_Zdf+h∇.Z,∇fi

= (+∇_Z)(df)−Ric^V_D(·,∇f)

= −Ric^V_D +∇_Z (df) (2.21)

where ∆⁽¹⁾ denotes the Hodge-deRham Laplacian on 1-forms.

Now let (φ, λ)∈Eig(L), i.e. Lφ=−λφ, where L= ∆ +Z. Forv∈TxD, consider the process nt(v) := (dφ)(Qtv).

Then

n_t(v) =h∇φ(X_t), Q_tvi=hu⁻¹_t (∇φ)(X_t), u⁻¹_t Q_tvi.

Using (2.19), we see by Itˆo’s formula and formula (2.21) that dn_t(v)=^m 1

2(dφ+∇_Zdφ)(X_t)Q_tvdt+ dφ(X_t)(DQ_tv) dt=−λ

2n_t(v) dt.

It follows that

(2.22) e^λt/2n_t(v) = e^λt/2h∇φ(X_t), Q_tvi, t6τ_D, is a martingale.

Lemma 2.3. Let (φ, λ) ∈ Eig(L). We keep the notation from above. Then, for any function h∈C¹([0,∞);R), the process

Nt(v) :=hte^λt/2h∇φ(X_t), Qtvi −e^λt/2φ(Xt) Z t

0

hh˙sQsv, usdBsi, t6τD, (2.23)

is a martingale. In particular, for fixed t >0 and h ∈C¹([0, t]; [0,1]) monotone such that h₀ = 1 and ht= 0, we have

k∇φk_∞6 k∇φk_∂D,∞P{t > τ_D}e^(λ+KV)+t/2 +kφk_∞e^λt/2P{t6τ_D}^1/2

Z t 0

|h˙_s|²e^KVsds 1/2

. (2.24)

Proof. Indeed, from (2.22) we deduce that h_te^λt/2h∇φ(X_t), Q_tvi −

Z t 0

h˙_se^λs/2h∇φ(X_s), Q_svids, t6τ_D, is a martingale as well. By the formula

e^λt/2φ(Xt) =φ(X0) + Z t

0

e^λs/2h∇φ(X_s), usdBsi

we see then thatN_t(v) is a martingale. To check inequality (2.24), we deduce from the martingale property of {Ns∧τ_D(v)}_s∈[0,t] that

k∇φk_∞6 k∇φk_∂D,∞E h

1{t>τ_D}e^λτD/2|h_τD| |Q_τD|i +kφk_∞e^λt/2E

"

1_{t6τD} sup

|v|61

Z t 0

hh˙_sQ_sv, u_sdB_si 2#1/2

.

The claim follows by using (2.20).

To estimate the boundary norm k∇φk_∂D,∞, we shall compareφ(x) and ψ(t, x) :=P(τ_D^x > t), t >0,

for small ρ_∂D(x) := dist(x, ∂D). LetP_t^D be the Dirichlet semigroup generated by ¹₂L. Then ψ(t, x) =P_t^D1D(x),

so that

(2.25) ∂_tψ(t, x) = 1

2Lψ(t,·)(x), t >0.

Lemma 2.4. For any (φ, λ)∈Eig(L),

(2.26) k∇φk_∂D,∞6kφk_∞ inf

t>0e^λt/2k∇ψ(t,·)k_∂D,∞.

Proof. To prove (2.26), we fix x∈∂D. For small ε >0, let x^ε= exp_x(εN), where N is the inward unit normal vector field of∂D. Sinceφ|_∂D= 0 and ψ(t,·)|_∂D= 0, we have

(2.27) |∇φ(x)|=|N φ(x)|= lim

ε→0

|φ(x^ε)|

ε , |∇ψ(t,·)(x)|= lim

ε→0

|ψ(t, x^ε)|

ε . LetX_t^ε be the L-diffusion starting atx^ε and τ_D^ε its first hitting time of∂D. Note that

Nt:=φ(X_t∧τ^{ε ε}

D) e^{λ(t∧τDε)/2}, t>0, is a martingale. Thus, for each fixed t >0, we can estimate as follows:

|∇φ(x)|= lim

ε→0

|φ(x^ε)|

ε

= lim

ε→0

E[φ(X_t^ε) 1_{t<τ^ε

D}] e^{λ(t∧τDε)/2} ε

6kφk∞e^λt/2lim

ε→0

E[1{t<τ_D^ε}] ε 6kφk_∞e^λt/2lim

ε→0

ψ(t, x^ε) ε

=kφk_∞e^λt/2|∇ψ(t,·)|(x).

Taking the infimum over tgives the claim.

We now work out an explicit estimate for k∇ψ(t,·)k_∂D,∞. Let cut(D) be the cut-locus of ∂D, which is a zero-volume closed subset ofDsuch that ρ_∂D := dist(·, ∂D) is smooth inD\cut(D).

Proposition 2.5. Letα∈Rsuch that

(2.28) 1

2Lρ_∂D 6α.

Then

k∇ψ(t,·)k_∂D,∞6α+

√2

√πt+ Z t

0

1−e^−α

2s

√ 2

2πs³ ds

6α+

√2

√πte^−α

2t 2 + min

|α|, α²√

√ 2t π

, (2.29)

and

(2.30) k∇ψ(t,·)k_∂D,∞6

√

√2

πt +α+

√

√ t 2πα²

Notice that by [10, Lemma 2.3] the condition ¹₂Lρ_∂D6α holds for α defined by (2.6).

Proof. Letx∈Dand letXtsolve SDE (2.17). As shown in [6], (ρ_∂D(Xt))_t6τD is a semimartingale satisfying

(2.31) ρ_∂D(X_t) =ρ_∂D(x) +b_t+ 1 2

Z t 0

Lρ_∂D(X_s) ds−l_t, t6τ_D,

where b_t is a real-valued Brownian motion starting at 0, and l_t a non-decreasing process which increases only when X_t^x ∈ cut(D). Setting ε = ρ_∂D(x), we deduce from (2.31) together with

1

2Lρ_∂D6α, that

(2.32) ρ_∂D(Xt(x))6Y_t^α(ε) :=ε+bt+αt, t6τD. Consequently, lettingT^α(ε) be the first hitting time of 0 by Y_t^α(ε), we obtain

(2.33) ψ(t, x)6P(t < T^α(ε)).

On the other hand, since ψ(t,·) vanishes on the boundary and is positive in D, we have for all y∈∂D

(2.34) |∇ψ(t, y)|= lim

x∈D, x→y

ψ(t, x) ρ_∂D(x).

Hence, by (2.33), to prove the first inequality in (2.29) it is enough to establish that

(2.35) lim sup

ε↓0

P(t < T^α(ε))

ε 6α+

√2

√πt+ Z t

0

1−e^−α

2s

√ 2

2πs³ ds.

It is well known that the (sub-probability) density fα,ε ofT^α(ε) is (2.36) fα,ε(s) = εexp −(ε+αs)²/(2s)

√

2πs³ ,

which can be obtained by the reflection principle for α= 0 and the Girsanov transform forα 6= 0.

Thus

P(t>T^α(ε)) =ε Z t

0

exp −(ε+αs)²/(2s)

√

2πs³ ds

=εexp(−αε) Z t

0

e^−α2s/2

√

2πs³ exp

−ε² 2s

ds

= exp(−αε) Z 2t/ε²

0

e^−1/r

√

πr³ exp

−α²ε²r 4

dr, (2.37)

where we have made the change of variable r = 2s/ε². With the change of variable v = 1/r we easily check that

(2.38)

Z ∞ 0

r^−3/2e^−1/rdr= Γ(1/2) =√ π,

and this allows to write

(2.39) P(t>T^α(ε)) = exp(−αε) 1− Z ∞

2t/ε²

e^−1/r

√

πr³ dr− Z 2t/ε²

0

e^−1/r

√ πr³

1−e^−α2ε2r/4

dr

! .

Asε→0,

Z ∞ 2t/ε²

e^−1/r

√

r³ dr= Z ∞

2t/ε²

√1

r³dr+ o(ε) = ε√

√2

t + o(ε), and with change of variables= ¹₂ε²r

Z 2t/ε² 0

e^−1/r

√ πr³

1−e^−α

2ε2r 4

dr =ε Z t

0

e^−ε

2

√ 2s

2πs³

1−e^−α

2s 2

ds

=ε Z t

0

1−e^−α

2s

√ 2

2πs³ ds+ o(ε)

by monotone convergence. Combining these with e^−αε= 1−αε+ o(ε), we deduce from (2.39) that (2.40) P(t>T^α(ε)) = 1−ε



α+

√2

√πt+ Z t

0

1−e^−α

2s

√ 2

2πs³ ds



+ o(ε) which yields (2.35).

Next, an integration by parts yields (2.41)

Z t 0

1−e^−α

2s

√ 2

2πs³ ds= α²

√2π Z t

0

√1 ue^−α

2u 2 du−

√2

√πt

1−e^−α

2t 2

.

With the change of variable s=|α|

ru

t in the first term in the right we obtain

(2.42) α²

√2π Z t

0

√1 ue^−α

2u

2 du=|α|

r2t π

Z |α|

0

e^−s

2t 2 ds.

We arrive at

(2.43) f(α) :=α+

√

√2 πt+

Z t 0

1−e^−α

2s

√ 2

2πs³ ds=

√

√2 πte^−α

2t

2 +α+|α|

r2t π

Z |α|

0

e^−s

2t 2 ds.

Bounding r2t

π Z |α|

0

e^−s

2t 2 ds by

r2t π

Z ∞ 0

e^−s

2t

2 ds = 1, respectively bounding e^−s

2t

2 by 1 in the integral, yields (2.29).

The function

f(α) =

√2

√πte^−α

2t

2 +α+|α|

r2t π

Z |α|

0

e^−s

2t 2 ds

is smooth and an easy computation shows that

(2.44) f(0) =

√2

√πt, f⁰(0) = 1, f⁰⁰(α) =

√2t

√πe^−α

2t 2

Using the fact that f(α)−α is even, we also get

(2.45) f(α) =

√

√2

πt +α+ Z |α|

0

√

√2t πe^−s

2t 2 s ds6

√

√2

πt+α+

√t

√2πα². which yields (2.30).

Remark 2.6. One could use estimate (2.24) (optimizing the right-hand side with respect to t) together with Lemma 2.4 (again optimizing with respect to t) to estimate k∇φk_∞ in terms of kφk_∞. We prefer to combine the two steps.

Lemma 2.7. Assume Ric^V_D >−K_V for some constant K_V ∈R. Letα be determined by (2.28).

(a) Ifα >0, then for any (φ, λ)∈Eig(L), k∇φk∞6inf

t>0 max

ε∈[0,1]e

(λ+K+ V)t 2

(

ε α+

√2

√πte^−α

2t 2 + min

|α|, α²√

√ 2t π

! +

r1−ε t

) kφk_∞,

as well as

k∇φk_∞6inf

t>0 max

ε∈[0,1]e^(λ+KV+)t/2 (

ε α+ r2

πt+

√t

√ 2πα²

! +

r1−ε t

) kφk_∞

and

k∇φk_∞6inf

t>0 max

ε∈[0,1]e^(λ+KV+)t/2 (

ε 2α+ r 2

πt

! +

r1−ε t

) kφk_∞.

(b) Ifα 60, then

k∇φk_∞6inf

t>0 max

ε∈[0,1]e^(λ+KV+)t/2 (

ε r 2

πte^−α

2t

2 +

r1−ε t

) kφk_∞.

In particular,

k∇φk_∞6inf

t>0 max

ε∈[0,1]e^(λ+KV+)t/2 (

ε r 2

πt +

r1−ε t

) kφk_∞.

Proof. For fixedt >0 in (2.23), we takeh∈C¹([0, t]; [0,1]) such thath0 = 1 andht= 0. Then, by the martingale property of {N_s∧τD(v)}_s∈[0,t], we obtain

|∇_vφ|(x) =|N₀(v)|=|ENt∧τ_D(v)|

= E

1_{t>τD}e^λτD/2h_τDh∇φ(X_τD), Q_τDvi −1_{t6τD}e^λt/2φ(X_t) Z t

0

hh˙_sQ_sv, u_sdB_si

. (2.46)

Note that using (2.20) along with Lemma 2.4 we may estimate

E

h

1{t>τ_D}e^λτD/2hτDh∇φ(X_τD), QτDvii

6E h

1_{t>τD}e^λτD/2|h_τD| k∇φk_∂D,∞e^KVτD/2|v|i

6E

h

1_{t>τD}e^λτD/2|h_τD| kφk_∞k∇ψ(t−τ_D,·)k_∂D,∞e^{λ(t−τD)/2}e^KVτD/2|v|i

=E h

1_{t>τD}|h_τD| kφk_∞k∇ψ(t−τD,·)k_∂D,∞e^λt/2e^KVτD/2|v|i 6e^(λ+K+V)t/2kφk_∞E

1_{t>τD}|h_τD| k∇ψ(t−τ_D,·)k_∂D,∞|v|

, as well as

E

1_{t6τD}e^λt/2φ(Xt) Z t

0

hh˙sQsv, usdBsi

6e^λt/2kφk_∞P{t6τD}^1/2 Z t

0

|h˙s|²e^KVsds 1/2

.

Taking

hs= t−s

t , s∈[0, t], we obtain thus from (2.46)

|∇φ(x)|6 e^(λ+KV+)t/2

t kφk_∞E

1{t>τ_D}(t−τD)k∇ψ(t−τD,·)k_∂D,∞

+ e^λt/2kφk_∞P{t6τ_D}^1/21 t

e^KV+t−1 K_V⁺

1/2

.

Note that

e^KV+t−1

K_V⁺ 6te^KV+t. (i) By (2.29), assuming thatα>0, we have on {t > τ_D}:

t−τD

t k∇ψ(t−τD,·)k_∂D,∞6αt−τD

t +

√

√2 π

√t−τD

t + t−τD

t

Z t−τD

0

1−e^−α

2s

√ 2

2πs³ ds

6α+

√

√2 πt+

Z t 0

1−e^−α

2s

√ 2

2πs³ ds

6α+

√2

√πte^−α

2t

2 + min

α, α²√

√ 2t π

.

Thus, letting ε=P(t > τ_D), we obtain

|∇φ(x)|6e^(λ+KV+)t/2kφk_∞

"

ε α+

√2

√πte^−α

2t 2 + min

α, α²√

√ 2t π

! +

r1−ε t

# .

(ii) Still under the assumptionα>0, this time using estimate (2.30), we have on{t > τ_D}:

k∇ψ(t−τD,·)k_∂D,∞6

√ 2

pπ(t−τ_D) +α+

√t√−τD

2π α², and thus letting ε=P(t > τD), we get

|∇φ(x)|6 e^(λ+KV+)t/2

t kφk_∞E

"

1_{t>τD} r2

π

√t−τD+α(t−τD) +(t−τD)^3/2

√

2π α²

!#

+ e^λt/2kφk_∞P{t6τ_D}^1/21 t

e^KV+t−1 K_V⁺

1/2

6e^(λ+KV+)t/2kφk_∞

"

ε r2

πt+α+

√t

√2πα²

! +

r1−ε t

# .

(iii) In the caseα60, we get from (2.29) in a similar way:

|∇φ(x)|6e^(λ+KV+)t/2kφk_∞ (

ε

√

√2 πte^−α

2t

2 +

r1−ε t

) .

This concludes the proof of Lemma 2.7.

Proposition 2.8. We keep the assumptions of Lemma 2.7.

(a) If α>0, then for any (φ, λ)∈Eig(L), k∇φk_∞6√

e max

ε∈[0,1]





 ε



α+ q

2(λ+K_V⁺)

√π exp

− α² 2(λ+K_V⁺)

+ min

|α|,

√2α² q

π(λ+K_V⁺)





+√ 1−ε

q

(λ+K_V⁺)

kφk_∞,

as well as k∇φk∞6√

e max

ε∈[0,1]





 ε



α+ q

2(λ+K_V⁺)

√π + α²

q

2π(λ+K_V⁺)



+√ 1−ε

q

(λ+K_V⁺)





 kφk_∞

and

k∇φk_∞6√ e max

ε∈[0,1]





 ε



2α+ q

2(λ+K_V⁺)

√π



+√ 1−ε

q

(λ+K_V⁺)





 kφk_∞

(b) If α60, then k∇φk∞6√

e max

ε∈[0,1]





 ε

q

2(λ+K_V⁺)

√π exp

− α² 2(λ+K_V⁺)

+√

1−ε q

(λ+K_V⁺)





 kφk_∞.

Proof. Take t= 1/(λ+K_V⁺) in Lemma 2.7.

We are now ready to complete the proof of Theorem 2.2.

Proof of Theorem 2.2. The claims of Theorem 2.2 (with the exception of estimate (2.15)) follow directly from the inequalities in Proposition 2.8 together with the fact that for any A, B>0,

(2.47) max

ε∈[0,1]

εA+√

1−εB =B1_{B>2A}+

A+B² 4A

1_{B62A}. Finally, to check (2.15) we may go back to (2.24) from where we have

k∇φk_∞6εe^(λ+KV)+t/2k∇φk_∂D,∞+√

1−εe^λt/2kφk_∞ Z t

0

|h˙s|²e^KVsds 1/2

.

Taking

h_s= e^−KVt−e^−KVs

e^−KVt−1 , s∈[0, t], we obtain

k∇φk_∞6inf

t>0 max

ε∈[0,1]

(

εe^(λ+KV)+t/2k∇φk_∂D,∞+kφk_∞e^λt/2√ 1−ε

K_V 1−e^−KVt

1/2) .

Noting that

K_V

1−e^−KVt 6 K_V⁺

1−e^−KV+t 6t⁻¹e^KV+t, and takingt= (K_V⁺+λ)⁻¹ we obtain

k∇φk_∞6√ e max

ε∈[0,1]

εk∇φk_∂D,∞+ q

(1−ε)(λ+K_V⁺)kφk_∞

.

Applying Lemma 2.4 and Proposition 2.5 witht= 1/λ, we arrive at k∇φk_∞6kφk_∞ max

ε∈[0,1]

(

eε α+

√

√2λ π e^−α

2

2λ +|α| ∧α²√

√ 2 πλ

! +

q

e(1−ε)(λ+K_V⁺) )

.

The proof is then finished as above with observation (2.47).

3 Proof of Theorem 1.2

As in Section 2, we consider L = ∆ +∇V and let Eig_N(L) be the set of the corresponding non- trivial eigenpairs for the Neumann problem of L. We also allow ∂D = ∅, then we consider the eigenproblem without boundary. We first consider the convex case, then extend to the general situation. In this section, Pt denotes the (Neumann if ∂D 6=∅) semigroup generated by L/2 on D. LetX_tbe the corresponding (reflecting) diffusion process which solves the SDE

(3.1) dXt=ut◦dBt+1

2∇V(Xt) dt+N(Xt) d`t,

where B_t is a d-dimensional Euclidean Brownian motion, u_t the horizontal lift of X_t onto the orthonormal frame bundle, and `tthe local time of Xton ∂D.

We will apply the following Bismut type formula for the Neumann semigroup Pt, see [15, Theorem 3.2.1], where the multiplicative functional process Q_s was introduced in [4].

Theorem 3.1 ([15]). Let Ric^V_D >−K_V and I∂D >−δ for some K_V ∈C( ¯D) and δ ∈C(∂D). Then there exists a R^d⊗R^d-valued adapted continuous processQs with

(3.2) kQ_tk6exp

1 2

Z t 0

KV(Xs)ds+ Z t

0

δ(Xs)d`s

, s>0, such that for anyt >0 andh∈C¹([0, t]) with h(0) = 0, h(t) = 1, there holds

(3.3) ∇P_tf =E

f(Xt)

Z t 0

h⁰(s)QsdBs

, f ∈Bb(D).

3.1 The case with convex or empty boundary

In this part we assume that∂D is either convex or empty. When∂D is empty,Dis a Riemannian manifold without boundary and Eig_N(L) denotes the set of eigenpairs for the eigenproblem without boundary. In this case, if Ric^V > K_V for some constant K_V ∈ R, then λ+K_V > 0 for (φ, λ) ∈ Eig_N(L), see for instance [8].

Theorem 3.2. Assume that∂D is either convex or empty.

(1) If the curvature-dimension condition (2.1) holds, then for any (φ, λ)∈Eig_N(L), k∇φk²_∞> λ²kφk²_∞

n(λ+K) λ

λ+K λ/K

> λ²kφk²_∞ ne(λ+K⁺). (2) If Ric^V_D >−K_V for some constantKV ∈R, then for any (φ, λ)∈Eig_N(L),

k∇φk²_∞

kφk²_∞ 6 2(λ+K_V) π

1 +K_V

λ λ/K_V

6 2e(λ+K_V⁺)

π .

Proof. (a) We start by establishing the lower bound estimate. By Itˆo’s formula, for any (φ, λ) ∈ Eig_N(L) we have

(3.4) d|∇φ|²(Xt) = 1

2L|∇φ|²(Xt) dt+ 2I∂D(∇φ,∇φ)(X_t) d`t+ dMt, t>0,

where `_t is the local time of X_t at ∂D, which is an increasing process. Since I∂D >0, and since (2.1) and Lφ=−λφ imply

1

2L|∇φ|² >−(K+λ)|∇φ|²+λ² nφ², we obtain

d|∇φ|²(X_t)>λ²

nφ²−(λ+K)|∇φ|²

(X_t) dt+ dM_t, t>0.

Noting that for X0 =x∈D we have

E[φ(Xs)²]>(E[φ(Xs)])² = e^−λsφ(x)², we arrive at

e^(λ+K)tk∇φk²_∞>e^(λ+K)tE[|∇φ|²(X_t)]> λ² n

Z t 0

e^(λ+K)sE[φ²(X_s)] ds

> λ² n

Z t 0

e^Ksφ(x)²ds= λ²(e^Kt−1) nK φ(x)².

Multiplying by e^−(λ+K)t, choosing t= _K¹ log(1 +^K_λ) (noting thatλ+K >0, in case λ+K = 0 takingt→ ∞), and taking the supremum over x∈D, we finish the proof of (1).

(b) Let ∂D be convex and Ric^V_D >−K_V for some constant K_V. Then Theorem 3.1 holds for δ= 0, so that

σt:=

E

Z t 0

|h⁰(s)|²kQ_sk²ds 1/2

6 Z t

0

|h⁰(s)|²e^KVsds 1/2

.

Taking

h(s) = Rs

0 e^−KVrdr Rt

0e^−KVrdr we obtain

σ_t6 K_V 1−e^−KVt

1/2

. Therefore,

k∇P_tfk_∞6kfk_∞E

Z t 0

h⁰(s)Q_sdB_s 6kfk∞

√ 2 2π σt

Z ∞ 0

sexp

− s² 2σ²_t

ds

=kfk_∞σ_t√

√ 2

π , t >0, f ∈Bb(D).

(3.5)

Applying this to (φ, λ)∈Eig_N(L), we obtain e^−λt/2|∇φ|6kφk_∞σ_t√

√ 2

π 6kφk_∞

2K_V π(1−e^−2KVt)

1/2

, t >0.

Consequently, λ+K_V >0. Takingt= _K¹

V log(1 +^K_λ^V) as above, we arrive at k∇φk²_∞

kφk²_∞ 6 2(λ+K_V) π

1 +K_V

λ λ/KV

.

3.2 The non-convex case

When ∂D is non-convex, a conformal change of metric may be performed to make ∂M convex under the new metric; this strategy has been used in [2, 12, 13, 14] for the study of functional inequalities on non-convex manifolds. According to [15, Theorem 1.2.5], for a strictly positive function f ∈ C^∞( ¯D) with I∂D+Nlogf|_∂D > 0, the boundary ∂D is convex under the metric f⁻²h·,·i. For simplicity, we will assume thatf >1. Hence, we take as class of reference functions

D:=

f ∈C²( ¯D) : inff = 1, I∂D+Nlogf >0 .

Assume (2.1) and Ric^V_D > −K_V for some constants n > d and K, K_V ∈ R. For any f ∈ D and ε∈(0,1), define

c_ε(f) := sup

D

4ε|∇logf|²

1−ε +εK+ (1−ε)K_V −2Llogf

.

We let λ^N₁ be the smallest non-trivial Neumann eigenvalue of −L. The following result implies λ₁>−c_ε(f).

Theorem 3.3. Letf ∈D.

(1) If (2.1) and Ric^V_D > −K_V hold for some constants n > d and K, K_V ∈ R. Then for any non-trivial (φ, λ)∈Eig_N(L), we haveλ+c_ε(f)>0 and

kfk²_∞k∇φk²_∞

kφk²_∞ > sup

ε∈(0,1)

ελ² n(λ+c_ε(f))

λ λ+c_ε(f)

λ/cε(f)

> sup

ε∈(0,1)

ελ² ne(λ+c_ε(f)⁺).