Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space II

www.imstat.org/aihp 2011, Vol. 47, No. 3, 699–724

DOI:10.1214/10-AIHP381

© Association des Publications de l’Institut Henri Poincaré, 2011

Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space II

Viorel Barbu

, Giuseppe Da Prato

and Luciano Tubaro

aUniversity Al. I. Cuza and Institute of Mathematics Octav Mayer, Blvd. Carol, 11, 700506 Iasi, Romania. E-mail:vb41@uaic.ro bScuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy. E-mail:daprato@sns.it

cDepartment of Mathematics, University of Trento, Via Sommarive, 14, 38050 Povo (Trento), Italy. E-mail:tubaro@science.unitn.it Received 20 January 2010; revised 2 July 2010; accepted 8 July 2010

Abstract. This work is concerned with the existence and regularity of solutions to the Neumann problem associated with a Ornstein–Uhlenbeck operator on a bounded and smooth convex setKof a Hilbert spaceH. This problem is related to the re- flection problem associated with a stochastic differential equation inK.

Résumé. Dans cet article nous étudions l’existence et la régularité des solutions d’un problème de Neumann associé à un opérateur de Ornstein–Uhlenbeck défini sur un domaine convexeK, borné et régulier dans un espace de HilbertH. Le problème est lié à un problème de réflexion associé à une équation différentielle stochastique dans le domaineK.

MSC:60J60; 47D07; 15A63; 31C25

Keywords:Neumann problem; Ornstein–Uhlenbeck operator; Kolmogorov operator; Reflection problem; Infinite-dimensional analysis

1. Introduction

We are given a non-degenerate Gaussian measure μ=NQwith mean 0 and covariance operatorQin a separable Hilbert space H (with scalar product·,·and norm | · |). We fix α∈ [0,1] and consider the following Neumann problem on a regular convex subsetKofH,

λϕ−L_αϕ=f inK,

∂ϕ

∂n=0 onΣ, (1.1)

whereλ >0,Σis the boundary ofK,f:H→Ris a given function onHandLis the Ornstein–Uhlenbeck operator Lαϕ:=1

2Tr

Q^1−αD²ϕ

−1 2

x, Q^−αDϕ

. (1.2)

We shall denote by Athe self-adjoint operatorA:=Q⁻¹. Sinceμ is not degenerate, there existsδ >0 such that Ax, x ≥δ|x|²,∀x∈D(A)for someδ >0. Of course we have also that TrA⁻¹<∞.

ConcerningK, we shall assume that

1This work is supported by CNCSIS project PN II IDEI ID-70/2008.

2Partially supported by the Italian National Project MURST “Equazioni di Kolmogorov.”

Hypothesis 1.1. There exists a convexC^∞-functiong:H → [0,∞)withg(0)=0, g(0)=0 and D²g positively defined,i.e.,D²g(x)h, h ≥κ|h|²,∀h∈H, x∈H,whereκ >0,such that

K=

x∈H:g(x)≤1

, Σ=

x∈H:g(x)=1 .

Moreover,we also suppose thatD²gis bounded onKand thatgand all its derivatives grow at infinity at the most polynomially.

We denote byμ_Σthe surface measure induced byμonΣ(see [5,11,12]) and byn(y)the inner normal toKaty, that is

n(y)= Dg(y)

|Dg(y)| ∀y∈Σ. (1.3)

By Hypothesis1.1it follows that

Lemma 1.2. Kis convex,closed and bounded.Moreover there areγ,ρ,δ >0such that Dg(x), x

≥γ|x|² ∀x∈H, Dg(x) ≤δ ∀x∈K, (1.4)

g(x)≥γ

2|x|² ∀x∈H, (1.5)

Dg(x) ≥ρ ∀x∈Σ. (1.6)

Proof. We have Dg(x)=

₁

0

D²g(tx)xdt ∀x∈H.

Therefore Dg(x), x

= ₁

0

D²g(tx)x, x

dt≥κ|x|² ∀x∈H,

which implies the first estimate in (1.4) and also thatDgis bounded onK.

Similarly by g(x)=

₁

0

Dg(tx), x

dt ∀x∈H

and (1.4) it follows (1.5). This implies thatKis bounded and 0∈K, where^◦ K^◦ is the interior ofK. Finally by (1.4) it follows (1.6) otherwise there is{x_n} ⊂Σsuch thatDg(x_n)→0. Taking into account that 0< g(x)≤ Dg(x), xand that{xn}is bounded the latter implies that 1=g(xn)→0 which is of course absurd.

It is easy to see thatμis the unique invariant measure of the Ornstein–Uhlenbeck process inH, dX(t )+¹₂A^αX(t )dt=A^(α−1)/2dW (t ),

X(0)=x∈H, (1.7)

whereW is a cylindrical Wiener process in a filtered probability space Ω,F,P,{Ft}t≥0

of the form

W (t ), z

= ∞ k=1

βk(t)z, ek, t≥0∀z∈H.

Here{β_k}is a sequence of mutually independent real Brownian motions on(Ω,F,{Ft}t≥0,P)(see, e.g., [9]) and {e_k}is an orthonormal basis inHwhich will be taken as a system of eigen-functions forAfor simplicity, i.e.,

Ae_k=a_ke_k ∀k∈N, where obviouslyak≥δ.

Let us describe the results of the paper. First we consider the symmetric Dirichlet form a(ϕ, ψ )=

K

A^(α−1)/2Dϕ, A^(α−1)/2Dψ

dν ∀ϕ, ψ∈C¹(K), (1.8)

whereν=_μ(K)¹ μand show thata is closable (equivalently continuous) in the spaceW^1,2

A^α−1(K, ν)(see Section2).

We notice that forα=0 this space reduces to the Malliavin spaceD^1,2(K, ν). Here we use a recent result about an integration by parts formula onKproved in [4].

Then we define a weak solution of the Neumann problem (1.1) in the usual way as a solutionϕ∈W^1,2

A^α−1(K, ν)of the equation

λ

H

ϕψdμ+1

2a(ϕ, ψ )=

H

f ψdν ∀ψ∈W^1,2

A^α−1(K, ν), (1.9)

wheref∈L²(K, ν).

If we denote byNthe Kolmogorov operator corresponding to the Dirichlet form (1.8) then (1.9) can be equivalently written asλϕ−N ϕ=f. The second-order regularity ofϕas well as the proof that it satisfies the Neumann boundary condition onΣ in the sense of trace is one of the main results of this work (Theorem3.5). In the previous work [4]

this result was proved in the caseα=1. It should be emphasized that, though the treatment closely follows [4], there are, however, some notable differences which will be mentioned later on. The nice feature of problem (1.1) is that for allαthe corresponding Ornstein–Uhlenbeck operators (1.7) have the same invariant measureμ=N_Qand this allows a unified treatment. Moreover, since the trace assumption onA^−α is weaker than that onA⁻¹we can treat into this general functional setting reflection problem not treatable forα=1.

We note that in specific situationsAis a linear elliptic operator with suitable boundary conditions on a bounded and open subsetOofR^d. (See Section5below.)

The second part of the paper is devoted to the construction of a processX(t, x)such that the semigroupP_tgenerated byN is expressed asP_tϕ(x)=E[ϕ(X(t, x))]whereXis formally the solution to the following stochastic variational inequality

dX+¹₂A^αXdt+A^α−1N_K(X)dtA^(α−1)/2dWt,

X(0)=x, (1.10)

whereN_Kis the normal cone toK, i.e., N_K(x)=∅ ifx∈K,^◦

N_K(x)=

λn(x), λ≥0

ifx∈Σ.

Whenα=1 this problem is known in literature as the stochastic reflection problem on convex setK and was studied in finite-dimensional spaces H by [2,3,6,8]. If H is infinite-dimensional, however, no results concerning existence and uniqueness of strong solutions with the notable exception of the 1992 work of Nualart and Pardoux [14]

which treats this problem inH=L²(0,1)and forK= {y∈L²(0,1): y≥0 a.e. in(0,1)}.

The transition semigroup (Ptϕ)(x)=E

ϕ

X(t, x)

∀ϕ∈Cb(K), t≥0 (1.11)

formally relates the Neumann problem (1.1) and Eq. (1.10) but no rigorous proof of this conjecture exists except the cases mentioned above (see also [16]). However, in [1] this is proven forα=1 via some sharp arguments involving

theory of Langrangian flows. In particular, it is proven the existence and uniqueness of a martingale solution in sense of Stroock and Varadhan.

Whenα∈ [0,1)the operatorA^α−1N_Kis not monotone inH, so no existence results in the literature for Eq. (1.10) seems to be available. The second part of the paper is concerned with representation of semigroupP_t as a transition Markov semigroup in the special case whereKis a ball and Tr[A^2δ−1]<∞for someδ >0. The proof of existence of the process is constructive and relies on some sharpBV-estimates on solutions to approximating equation associated with (1.10) and the Skorohod theorem.

2. Notations and preliminary results

Everywhere in the followingDϕ is the derivative of a functionϕ:H →R. ByD²ϕ:H→L(H, H )we shall de- note the second derivative ofϕ. We shall denote also byC_b(H )andC_b^k(H ), k∈N,the spaces of all continuous and bounded functions onH and, respectively, ofk-times differentiable functions with continuous and bounded deriva- tives. The spaceC^k(K), k∈N,is defined as the space of restrictions of functions ofC_b^k(H )to the subsetK. Also we refer to [7,9] for notations and basic results on infinite-dimensional processes.

We denote by{e_k}the orthonormal basis inH of eigenfunctions ofQ, i.e.

Qe_k=λ_ke_k ∀k∈N, (2.1)

whereλ_k=_a¹_k with{a_k, k∈N}the eigenvalues ofA, byD_kthe derivative in the directione_kand setx_k= x, e_kfor allx∈H, k∈N. We denote byE(H )the linear span of all exponential functions{e^x,eh, h∈N}.

Then we recall a basic integration by parts formula inH.

H

D_kϕdμ= 1 λk

H

x_kϕdμ ∀k∈N, ϕ∈C_b¹(H ). (2.2)

We denote byM_α: C_b¹(H )⊂L²(H, μ)→L²(H, μ;H ) Mαϕ:=A^(α−1)/2Dϕ, ϕ∈C_b¹(H ).

HereM₀is the Malliavin derivative [12]. It is well known (and easy to show thanks to (2.2)) thatM_α is closable. We shall denote its closure byMα and also byA^(α−1)/2D.

The domain of the closure ofM_α will be denoted byW^1,2

A^α−1(H, μ). It is a Hilbert space with the inner product ϕ, ψ_W^1,2

Aα−1(H,μ)=

H

ϕψdμ+

H

A^(α−1)/2Dϕ, A^(α−1)/2Dϕ dμ

=

H

ϕψdμ+ ∞ k=1

H

λ^α_k⁻¹D_kϕD_kψdμ.

Denote byL²(H, μ)andL²(K, ν)the space ofμ-square integrable functions (ν-square integrable functions) onH andK, respectively.

In a similar way we define the spaceW^2,2

A^α−1(H, ν). The corresponding inner product is defined by (see [4,7,10]) ϕ, ψ_W^2,2

Aα−1(H,μ)= ϕ, ψ_W^1,2

Aα−1(H,μ)+

H

Tr

A^2(α−1)D²ϕD²ψ dμ

= ϕ, ψ_W^1,2

Aα−1(H,μ)+ ∞ h,k=1

H

λ¹_h^−αλ¹_k^−αD²_h,kϕD_h,k² ψdμ.

2.1. The integration by parts formula onK

The following result is proved in [4]. For reader’s convenience we recall it here, deferring to theAppendixfor a proof (TheoremA.2).

Lemma 2.1. LetK= {x∈H:g(x)≤1}whereg∈C²(H )is convex and|Dg(x)|⁻¹∈L^p(H, μ)for allp≥1.Then

K

D_hϕ(x)μ(dx)= 1 μ(K)

Σ

n_h(y)ϕ(y)μ_Σ(dy) + 1

λ_h

K

x_hϕ(x)μ(dx) ∀h∈H, ϕ∈C_b¹(H ), (2.3) wherenh(y)= n(y), eh.

With the help of this result we can define the spacesW^1,2

A^α−1(K, ν)andW^2,2

A^α−1(K, ν)as in [4].

Moreover, we can define the trace of a functionϕ∈W^1,2

A^α−1(K, ν)thanks to the following result.

Proposition 2.2. For anyϕ∈C¹_b(H )we have

Σ

Q^1/2n(y) ²ϕ²(y)μ_Σ(dy)

≤C

K

ϕ²(x)μ(dx)+

K

Q^1/2Dϕ(x) ²μ(dx)

. (2.4)

Proof. Letϕ∈C_b¹(H )andh∈N. Replacing in (2.3)ϕwithλ_hD_hgϕ²and thenD_hϕwith 2λhD_hgϕD_hϕ+λ_hD²_hgϕ², yields

2

K

λ_hD_hgϕD_hϕdμ+

K

λ_hD_h²gϕ²dμ

= 1 μ(K)

Σ

λ_hn_h(y)D_hgϕ²dμΣ+

K

x_hD_hgϕ²dμ.

Summing up onhyields 2

K

QDϕ, Dgϕdμ+

K

Tr QD²g

ϕ²dμ

= 1 μ(K)

Σ

Qn(y), Dg

ϕ²dμΣ+

K

x, Dgϕ²dμ.

But, taking into account (1.3), (1.6) we have Qn(y), Dg(y)

= Dg(y) Qn(y),n(y)

≥ρ

Qn(y),n(y)

∀y∈Σ.

Substituting in the previous identity yields 1

ρμ(K)

Σ

Qn(y),n(y)

ϕ²dμΣ+

Kx, Dgϕ²dμ

≤2

K

QDϕ, Dgϕdμ+

K

Tr QD²g

ϕ²dμ.

Taking into account thatKis bounded and thatDg,D²gare bounded onK, the conclusion follows.

We can now define the trace of a functionϕ∈W^1,2

A^α−1(K, ν). Let{ϕ_j} ⊂C_b¹(K)be such that limn→∞ϕ_j=ϕ inL²(K, ν),

limn→∞A^(α−1)/2Dϕ_j=A^(α−1)/2Dϕ inL²(K, ν).

Then by (2.4) it follows that the sequence{|Q^1/2n(y)|γ0(ϕ_j)}, whereγ0(ϕ_j)denotes the trace ofϕ_j, is convergent in L²(Σ, μ_Σ)to a functionψ∈L²(Σ, μ_Σ). Then we define the traceγ₀(ϕ)ofϕas

γ₀(ϕ)= ψ

|Q^1/2n(y)|. 2.2. Trace of the normal derivative Proposition 2.3. Assume thatϕ∈W^2,2

A^α−1(K, ν).Then the following estimate holds,

Σ

Q^1/2n(y) ² A^(α−1)/2Dϕ ²(y)μΣ(dy)

≤C

K

A^(α−1)/2Dϕ(x) ²μ(dx)+

K

Tr

A^α−1D²ϕ(x)2 μ(dx)

. (2.5)

Proof. Letϕ∈W_A^2,2

α−1(K, ν)and let{ϕ_j} ⊂C²(K)be convergent toϕ inW_A^2,2

α−1(K, ν). Fori∈Nwe apply (2.3) to a_i^(α−1)/2D_iϕ_j. We have

Σ

Q^1/2n(y) ² a^(α−1)/2

i D_iϕ_j ²(y)μ_Σ(dy)

≤Ca^(α_i ^−1)/2

K

D_iϕ_j(x) ²μ(dx)+a^(α_i ^−1)/2

K

A^(α−1)/2DD_iϕ_j(x) ²μ(dx)

. Summing up oniyields

Σ

Q^1/2n(y) ² A^(α−1)/2Dϕ_j ²(y)μ_Σ(dy)

≤C

K

A^(α−1)/2Diϕj(x) ²μ(dx)+

K

Tr

A^α−1D²ϕj(x)2 μ(dx)

.

Now the conclusion follows lettingj→ ∞.

3. The penalized problem

We are here concerned for anyε >0 with the penalized equation dXε(t)+₁

2A^αX_ε(t)+A^α−1β_ε X_ε(t)

dt=A^(α−1)/2dWt,

Xε(0)=x, (3.1)

where

β_ε(x)=1 ε

x−Π_K(x)

∀x∈H.

Sinceβ_εis Lipschitz continuous, it is easily seen that Eq. (3.1) which can be equivalently be written as X_ε(t)=e^{−t Aα/2}x−

t 0

A^α−1e^{−Aα(t−s)/2}β_ε X_ε(s)

ds+ t

0

e^{−Aα(t−s)/2}A^(α−1)/2dWs

has a unique mild solution X_ε(·, x)∈L²

Ω, C

[0,+∞);H .

Moreover, it is easy to see that there is a unique invariant probability measureνεforXεgiven by

ν_ε(dx)=Z⁻_ε¹e^−dK2(x)/ε, (3.2)

wheredK is the distance toKand Zε=

H

e^−dK2(y)/εμ(dy). (3.3)

The corresponding Kolmogorov operator reads as follows, Nεϕ=Lϕ−

A^α−1βε(x), Dϕ

, ϕ∈E(H )∀ε >0, (3.4)

whereLis the Ornstein–Uhlenbeck operator Lϕ=1

2Tr

A^α−1D²ϕ

−1 2

x, A^αDϕ

∀ϕ∈E(H ).

One can easily check thatν_ε(as defined in (3.2) and (3.3)) is an invariant measure forN_εand that

H

N_εϕψdνε= −1 2

H

A^α−1Dϕ, Dψ

dνε ∀ϕ, ψ∈E(H ). (3.5)

Moreover, sinceβ_εis Lipschitz continuous, the operatorN_εis essentiallym-dissipative inL²(H, ν_ε)(we still denote byN_ε its closure) andE(H )is a core forN_ε, see [7].

Section3.1below is devoted to several estimates for(λI−N_ε)⁻¹f wheref ∈L²(H, ν_ε). Then these estimates are used in Section3.2to prove that(λI−N_ε)⁻¹f converges to(λI−N )⁻¹f asε→0, whereN is the self-adjoint operator corresponding to the Dirichlet form (1.8) (see (3.32) below), for anyf ∈L²(K, ν). Moreover, we shall end up the section by proving a few sharp properties of the domainD(N )ofN.

3.1. Estimates for(λI−Nε)⁻¹f Letλ >0, ε >0,ϕ∈E(H ). We set

f_ε=λϕ−N_εϕ. (3.6)

We are going to prove for later use a few estimates of the first and second derivatives ofϕ. To this purpose, sinceβ_ε is not differentiable, we need a further approximationβ_ε,ηofβ_ε.

More precisely, for anyε >0, η >0 we consider the penalized equation dXε,η(t)+₁

2A^αX_ε,η(t)+A^α−1β_ε,η

X_ε,η(t)

dt=A^{−(1−α)/2}dWt,

X_ε,η(0)=x, (3.7)

whereβ_ε,ηis the regularization ofβ_εgiven by the infinite-dimensional mollifier βε,η(x)=e^−ηA

H

βε

e^−ηAx+y

μη(dy), x∈H, η >0. (3.8)

Hereμ_ηis the Gaussian measure onH with mean 0 and covariance operator Q_η:=1

2A⁻¹

1−e^−2ηA .

Notice thatβ_ε,ηis of classC^∞and its derivatives of all order are bounded. Moreover,β_ε,ηis a monotone mapping inH and

ηlim→∞βε,η(x)=βε(x) inH∀ε >0, x∈H. (3.9)

Sinceβε,ηis Lipschitz, Eq. (3.7) has a unique mild solutionXε,η(t, x). Moreover, it is easy to see that there is a unique invariant probability measureν_ε,ηfor (3.7) given by

ν_ε,η(dx)=Z⁻_ε,η¹e^{−dK,η2 (x)/ε}, (3.10)

where Zε,η=

H

e^{−dK,η2 (y)/ε}μ(dy). (3.11)

1

2εd_K,η² is the potential associated withβε,η, that is 1

2εDd_K,η² (x)=β_ε,η(x) ∀x∈H, (3.12)

equivalently 1

2εd_K,η² (x)= ₁

0

β_ε,η(tx), x

dt ∀x∈H.

The corresponding Kolmogorov operator reads as follows, N_ε,ηϕ=Lϕ−

A^α−1β_ε,η(x), Dϕ

, ϕ∈E(H ), ε >0, (3.13)

whereLis the Ornstein–Uhlenbeck operator introduced before. Thenν_ε,ηis an invariant measure forN_ε,ηand

H

N_ε,ηϕψdνε,η= −1 2

H

A^α−1Dϕ, Dψ

dνε,η ∀ϕ, ψ∈E(H ). (3.14)

Moreover, sinceβ_ε,ηis Lipschitz continuous, the operatorN_ε,ηis essentiallym-dissipative inL²(H, ν_ε,η)andE(H ) is a core forN_ε,η(see [10]). We shall denote again byN_ε,ηthe closure ofN_ε,ηinL²(H, ν_ε,η). Moreover, we have

ηlim→0

X_ε,η(t, x)−X_ε(t, x) =0 ∀t≥0, x∈H,P-a.s. (3.15) Indeed by (3.1) and (3.7) we have for allt≥0, ε >0, η >0,

X_ε,η(t, x)−X_ε(t, x)

= − _t

0

A^1−αe^{−Aα(t−s)/2} β_ε,η

X_ε,η(t, x)

−β_ε

X_ε(t, x)

ds P-a.s.

and this yields

X_ε,η(t, x)−X_ε(t, x) ≤C _t

0

β_ε,η

X_ε,η(t, x)

−β_ε

X_ε(t, x) ds +C

_t

0

X_ε,η(t, x)−X_ε(t, x) ds ∀t≥0, ε, η >0P-a.s.,

because

βε,ηLip≤1

ε ∀η >0.

Since

ηlim→0βε,η

Xε(t, x)

=βε

Xε(t, x) ,

we obtain by Gronwall’s lemma that (3.15) holds.

Lemma 3.1. Letλ >0, ε >0, η >0,ϕ∈E(H )and set

f_ε,η=λϕ−N_ε,ηϕ. (3.16)

Then the following estimates hold

H

ϕ²dνε,η≤ 1 λ²

H

f_ε,η² dνε,η, (3.17)

H

A^(α−1)/2Dϕ ²dνε,η≤ 2 λ

H

f_ε,η² dνε,η, (3.18)

λ

H

A^(α−1)/2Dϕ ²dνε,η+1 2

H

Tr

A^α−1D²ϕ2 dνε,η

+1 2

H

A^α/2Dϕ ²dνε,η≤4

H

f_ε,η² dνε,η. (3.19)

Proof. Multiplying both sides of (3.16) byϕ, taking into account (3.14) and integrating inν_ε,ηoverH, yields λ

H

ϕ²dνε,η+1 2

H

A^(α−1)/2Dϕ ²dνε,η=

H

ϕf_ε,ηdνε,η. (3.20)

Now (3.17) and (3.18) follow easily from the Hölder inequality. To prove (3.19) we differentiate both sides of (3.16) in the direction ofekand obtain that

λDkϕ−Nε,ηDkϕ+1

2akDkϕ+ ∞ h=1

Dkβε,ηeh, ekDhϕ=Dkfε.

Next we multiply both sides of latter equation bya_k^α−1D_kϕ. Taking into account (3.14), integrating in ν_ε,ηoverH and summing up overk, yields

λ

H

A^(α−1)/2Dϕ ²dνε,η+1 2

H

Tr

A^α−1D²ϕ2 dνε,η

+1 2

H

A^α/2Dϕ ²dνε,η+

K^c

Dβ_ε,ηA^(α−1)/2Dϕ, A^(α−1)/2Dϕ dνε,η

=

H

A^(α−1)/2Dϕ, A^(α−1)/2Df_ε,η

dνε,η. (3.21)

Noting finally that, again in view of (3.14),

H

A^(α−1)/2Dϕ, A^(α−1)/2Df_ε,η dνε,η

=2

H

f_ε²dνε,η−2λ

H

fε,ηϕdνε,η≤4

H

f_ε,η² dνε,η,

the conclusion follows.

Taking into account (3.15) and that

ηlim→0N_ε,ηϕ(x)=N_εϕ(x) ∀ε >0, lettingη→0 we obtain the following result.

Corollary 3.2. Letλ >0, ε >0, ϕ∈E(H )and let

fε=λϕ−Nεϕ. (3.22)

Then the following estimates hold

H

ϕ²dνε≤ 1 λ²

H

f_ε²dνε, (3.23)

H

A^(α−1)/2Dϕ ²dνε≤ 2 λ

H

f_ε²dνε, (3.24)

λ

H

A^(α−1)/2Dϕ ²dνε+1 2

H

Tr

A^α−1D²ϕ2 dνε

+1 2

H

A^α/2Dϕ ²dνε≤4

H

f_ε²dνε. (3.25)

Now we are able to prove.

Proposition 3.3. Letλ >0,f ∈L²(H, νε)and letϕεbe the solution of the equation

λϕ_ε−N_εϕ_ε=f. (3.26)

Thenϕ_ε∈W^2,2

A^α−1(H, ν_ε),A^α/2Dϕ_ε∈L²(H, ν_ε)and the following estimates hold

H

ϕ_ε²dνε≤ 1 λ²

H

f²dνε, (3.27)

H

A^(α−1)/2Dϕε ²dνε≤2 λ

H

f²dνε, (3.28)

λ

H

A^(α−1)/2Dϕ_ε ²dνε+1 2

H

Tr

A^α−1D²ϕ_ε2 dνε

+1 2

H

A^α/2Dϕε ²dνε≤4

H

f²dνε. (3.29)

Proof. Inequality (3.27) is obvious since by (3.5),N_ε is dissipative in L²(H, ν_ε). Let us prove (3.28). Letλ >0, f∈L²(H, ν_ε)and letϕ_ε,be the solution to Eq. (3.26). SinceE(H )is a core forN_εthere exists a sequence{ϕ_ε,n}n∈N⊂ E(H )such that

nlim→∞ϕ_ε,n→ϕ_ε, lim

n→∞N_εϕ_ε,n→N_εϕ_ε inL²(H, ν_ε).

We setfε,n=λϕε,n−Nεϕε,n. Clearly,fε,n→f inL²(H, νε)asn→ ∞. We claim thatϕε∈W^1,2

A^α−1(H, νε)and that

nlim→∞A^(α−1)/2Dϕ_ε,n→A^(α−1)/2Dϕ_ε inL²(H, ν_ε;H ),

which will imply (3.28).

Letm, n∈N; then by (3.24) it follows that

H

A^(α−1)/2Dϕ_ε,n−A^(α−1)/2Dϕ_ε,m ²dνε≤ 2 λ

H

|f_ε,n−f_ε,m|²dνε.

Therefore the sequence{ϕ_ε,n}n∈Nis Cauchy inW^1,2

A^α−1(H, ν_ε)and the conclusion follows. The estimate (3.29) follows

similarly by (3.25).

We conclude this subsection with an integration by parts formula needed later. We set V :=

ψ∈C_b¹(K): |Q^1/2n(y)|⁻¹ψ∈C_b(K)

. (3.30)

Lemma 3.4. Letϕ∈D(N_ε)andψ∈V.Then the following identity holds.

K

Nεϕψdν= −1 2

K

A^(α−1)/2Dϕ, A^(α−1)/2Dψ dν + 1

μ(K)

Σ

A^α−1γ (Dϕ),n(y)

ψdμΣ. (3.31)

Proof. We first notice that the last integral in (3.31) is meaningful since

Σ

A^α−1γ (Dϕ),n(y) ψdμΣ

²

≤A^α−1

Σ

A^(α−1)/2γ (Dϕ) ² Q^1/2n(y) ²dμΣ

Σ

ψ² Q^1/2n(y) ⁻²dμΣ<∞ by (2.5).

Now, taking in account that E(H )is a core for N_ε, it is sufficient to prove (3.31) forϕ ∈E(H ). By the basic integration by parts formula (2.2) we deduce, for anyi∈Nandψ∈V that

K

D_iϕD_iψdν= −

K

D_i²ϕψdν+ 1 μ(K)

Σ

γ (D_iϕ) n(y)

iψdμΣ

+ 1 λ_i

K

xiDiϕψdν.

It follows that a^α_i⁻¹

K

D_iϕD_iψdν= −a_i^α−1

K

D_i²ϕψdν + 1

μ(K)a_i^α−1

Σ

γ (Diϕ) n(y)

iψdμΣ+1 2a_i^α

K

xiDiϕψdν.

Now, summing up oniyields

K

A^(α−1)/2Dϕ, A^(α−1)/2Dψ dν= −

K

Tr

A^α−1D²ϕ ψdν + 1

μ(K)

Σ

A^α−1γ (Dϕ),n(y)

dμΣ+2

K

x, A^αDϕ ψdν,

which is precisely Eq. (3.31).

3.2. Convergence of{ϕ_ε}asε→0

LetN:D(N )⊂L²(K, ν)→L²(K, ν)be the operator defined by N ϕ, ψL²(K,ν)= −¹₂a(ϕ, ψ ) ∀ψ∈W^1,2

A^(α−1)/2(K, ν), ϕ∈D(N ), D(N )=

ϕ∈W^1,2

A^(α−1)/2(K, ν): a(ϕ, ψ ) ≤C|ϕ|L²(K,ν)|ψ|L²(K,ν), ∀ψ∈W^1,2

A^(α−1)/2(K, ν)

. (3.32)

The operatorLis self-adjoint inL²(K, ν)and the Neumann problem (1.1) (or equivalently (1.9)) reduces to

λϕ−N ϕ=f. (3.33)

We are going to show that for each f ∈L²(K, ν)andε→0, ϕ_ε=(λI−N_ε)⁻¹f is convergent inL²(K, ν) to ϕ=(λI−N )⁻¹f and derive so, via the estimate proven in Proposition3.3, high order regularity properties for the solutionϕto (3.33).

We first note that forf ∈C_b(H )we have ϕ_ε(x)=E

_∞

0

e^−λtf

X_ε(t, x)

dt ∀x∈H. (3.34)

Now, by a standard argument it follows that from (3.34) iff ∈C_b¹(H )we have sup

x∈H

Dϕε(x) ≤ 1

λDfC_b(H ) ∀ε, λ >0. (3.35)

Theorem3.5below is the main result of this section.

Theorem 3.5. Letλ >0,f ∈L²(K, ν)and letϕεbe the solution of Eq. (3.26).Then{ϕε}is strongly convergent in L²(K, ν)toϕ=(λI−N )⁻¹f whereNis defined by(3.32).

Moreover,the following statements hold.

(i) limε→0A^(α−1)/2Dϕ_ε=A^(α−1)/2DϕinL²(K, ν;H ), (ii) ϕ∈W^2,2

A^α−1(K, ν)and|A^α/2Dϕ| ∈L²(K, ν), (iii) ϕfulfills the Neumann condition

A^α−1γ Dϕ(x)

,n(x)

=0, μ_Σ a.e.onΣ, (3.36)

whereγ (Dϕ(x))is defined by Proposition2.3.

In particular, sinceNis dissipative Theorem3.5amounts to say that for eachf∈L²(K, ν)the equationλϕ−N ϕ= f has a unique solutionϕsatisfying (ii), (iii).

Proof of Theorem3.5. Without danger of confusion we shall denote again byf the restrictionf|K off toK. In fact eachf ∈L²(K, ν)can be extended by 0 outsideKto a function inL²(H, ν). By this convention, everywhere in the sequel(λI−N )⁻¹f forf ∈L²(H, ν)means(λI−N )⁻¹f|K.

Step1. We have

εlim→0ϕ_ε=(λI−N )⁻¹f inL²(K, ν). (3.37)

In fact by (3.28), (3.29) it follows that there exist a sequence{ε_k} →0 andϕ∈W^1,2

A^α−1(K, ν)such that ϕ_εk→ϕ, weakly inL²(K, ν),

A^(α−1)/2Dϕε_k →A^(α−1)/2Dϕ, weakly inL²(K, ν;H ).

Letψ∈C_b¹(H )and notice that by (3.5) and by (3.26) we have the identity 1

2

H

A^(α−1)/2Dϕε, A^(α−1)/2Dψ dνε=

H

(f−λϕε)ψdνε. Equivalently

1 2

K

A^(α−1)/2Dϕ_ε, A^(α−1)/2Dψ dν+1

2

K^c

A^(α−1)/2Dϕ_ε, A^(α−1)/2Dψ dνε

=

H

(f−λϕε)ψdνε. (3.38)

Since by (3.28) we have

K^c

A^(α−1)/2Dϕε, A^(α−1)/2Dψ dνε

²≤

H

A^(α−1)/2Dϕε ²dνε

K^c

A^(α−1)/2Dψ ²dνε

≤ 2 λ

H

f²dνε

K^c

A^(α−1)/2Dψ ²dνε→0, asε→0, it follows by (3.38) that

1 2

K

A^(α−1)/2Dϕ, A^(α−1)/2Dψ dν=

K

(f−λϕ)ψdν ∀ψ∈C_b¹(K).

Obviously, this identity extends to allψ∈W^1,2

A^α−1(K, ν), which implies thatϕ_ε→(λI−N )⁻¹f weakly inL²(K, ν) asε→0 .

Step2. We have

limε→0ϕ_ε=ϕ inL²(K, ν), limε→0A^(α−1)/2Dϕ_ε=A^(α−1)/2Dϕ inL²(K, ν;K).

We first assume thatf∈C_b¹(H ). Let us start from the identity

H

Nεϕεϕεdνε= −1 2

H

A^(α−1)/2Dϕε ²dνε ∀ϕ∈D(Nε), (3.39)

which follows from (3.5). By (3.26) and (3.39) we see that 1

2

H

A^(α−1)/2Dϕ_ε ²dνε= −

H

(λϕ_ε−f )ϕ_εdνε, (3.40)

which implies in virtue of (3.32), (3.33)

εlim→0

K

1

2 A^(α−1)/2Dϕ_ε ²+λϕ²_ε

dνε=

K

f ϕdν

= −N ϕ, ϕ +λ

K

ϕ²dν

=

K

1

2 A^(α−1)/2Dϕ ²+λϕ²

dν. (3.41)

Here we have used the fact that

εlim→0

K^c

A^(α−1)/2Dϕε ²dνε=0