Maximal Sobolev regularity for infinite dimensional problem . 78

by assumption D_if ∈ W^1,2(O, µ) and if c < 1/4, by using (1.28), we have Z

O

(D_if (x))²x²_ie^−|x|2/2dx

= Z

{x∈O: cx²_i>log |Dif (x)|}

(Dif (x))²x²_ie^−|x|2/2dx +

Z

{x∈O: cx²_i≤log |Dif (x)|}

(D_if (x))²x²_ie^−|x|2/2dx

≤ Z

O

e^2cx2ix²_ie^−|x|2/2dx + Z

O

1

c|D_if |²log |D_if |e^−|x|2/2dx

≤ C₁+ 1 c

Z

O

|∇D_if |dµ + 1 2

Z

O

(D_if )²dµ log

Z

O

(D_if )²dµ



. Summing over i from 1 to n we have h∇f, xi ∈ L²(O, µ).

3.2 Maximal Sobolev regularity for infinite

where ef ∈ C_b^∞(G). Let γ_G be the induced measure γ ◦ π⁻¹_G in G; if G is identified with R^q through the isomorphism x 7→ (bh₁(x), . . . , bh_q(x)) then γ_G is the standard Gaussian measure in R^q.

We recall that L^Ωn is the Ornstein-Uhlenbeck operator associated to the quadratic form E_Ωn,γ while L^O is Ornstein-Uhlenbeck operator associated to the quadratic form E_O,γG.

Proposition 44. Let v be the weak solution of the finite dimensional problem λv − L^Ov = ef|O in O

Then u(x) := v(π_G(x)) is the weak solution of λu − L^Ωnu = f|Ω_nin Ω_n

Proof. We remark that the space X can be split as X = G × eX where X = (I − πe _G)(X), and γ = γ_G⊗eγ_G whereeγ_G= γ ◦ (I − π_G)⁻¹ is the measure induced on eX by the projection I − π_G. Let ϕ ∈ W^1,2(Ω_n, γ), then

Z

Ωn

(λu(x)ϕ(x) + h∇Hu(x), ∇Hϕ(x)iH) γ(dx)

= Z

Ωn

λv(π_G(x))ϕ(π_G(x) + (I − π_G)(x))

+ h∇Hv(πG(x)), ∇Hϕ(πG(x) + (I − πG)(x))iHγ(dx)

= Z

O× eX

λv(ξ)ϕ(ξ + y)e

+ h∇v(ξ), ∇ϕ(ξ + y)iγe _G(dξ)eγ_G(dy) (where ϕ(· + y) ∈ We ^1,2(O, γ_G))

= Z

O× eX

f (ξ)e ϕ(ξ + y)γe _G(dξ)eγ_G(dy)

= Z

Ωn

f (πe _G(x))ϕ(x)γ(dx)

= Z

Ωn

f (x)ϕ(x)γ(dx), and the statement follows.

Proposition 45. The function u satisfies kuk_W^2,2_(Ω,γ)≤ K where

K := Ckf k_L²_(Ω1,γ) and C is the constant of Theorem 18.

Proof. We recall that u(x) = v(π_G(x)). Then kuk²_W^2,2_(Ω,γ)≤ kuk²_W2,2(Ωn,γ)

= Z

Ωn

|u(x)|² +

∞

X

i=1

|D_iu(x)|²+

∞

X

i,j=1

|D_iju(x)|² γ(dx)

= Z

O

|v(ξ)|²+

q

X

i=1

|∂iv(ξ)|²

+

q

X

i,j=1

|∂_ijv(ξ)|²

!

µ(dξ) (By using Theorem 18)

≤ C²k ef k²_L2(O,µ) = C²kf k²_L2(Ωn,γ)≤ C²kf k²_L2(Ω1,γ)

If we consider the sequence {un}_n∈N of weak solutions of the problems λψ − L^Ωnψ = f|Ωn in Ωn.

By Proposition 45 it follows

ku_nk_W^2,2_(Ω,γ) ≤ K.

Possibly replacing u_nby a subsequence, there exists u ∈ W^2,2(Ω, γ) such that u_n * u in W^2,2(Ω, γ).

Proposition 46. The function u is the weak solution of (1).

Proof. We know that for all ϕ ∈ F C_b¹(X) Z

Ωn

λu_nϕ dγ + Z

Ωn

h∇_Hu_n, ∇_Hϕi_Hdγ = Z

Ωn

f ϕ dγ.

We claim that

n→∞lim Z

Ωn

λu_nϕ dγ = Z

Ω

λuϕ dγ.

Indeed,

Z

Ωn

λu_nϕ dγ = Z

Ω

λu_nϕ dγ + Z

Ω\Ωn

λu_nϕ dγ; (3.10) by the weak convergence

n→∞lim Z

Ω

λu_nϕ dγ = Z

Ω

λuϕ dγ

while     Z

Ω\Ωn

λunϕ dγ    

≤ λ

Z

Ω\Ωn

|un|²dγ

1/2Z

Ω\Ωn

|ϕ|²dγ

1/2

≤ λK

Z

Ω\Ωn

|ϕ|²dγ

1/2

that goes to zero as n → ∞ by the absolute continuity of the integral. Now we claim that

n→∞lim Z

Ωn

h∇_Hu_n, ∇_Hϕi_Hdγ = Z

Ω

h∇_Hu, ∇_Hϕi_Hdγ.

In fact, Z

Ωn

h∇_Hu_n, ∇_Hϕi_Hdγ = Z

Ω

h∇_Hu_n, ∇_Hϕi_Hdγ + Z

Ω\Ωn

h∇_Hu_n, ∇_Hϕi_Hdγ.

By the weak convergence in W^1,2(Ω, γ)

n→∞lim Z

Ω

h∇_Hu_n, ∇_Hϕi_Hdγ = Z

Ω

h∇_Hu, ∇_Hϕi_Hdγ while

    Z

Ω\Ωn

h∇_Hu_n, ∇_Hϕi_Hdγ    

≤ λ

Z

Ω\Ωn

|∇_Hu_n|²_Hdγ

1/2Z

Ω\Ωn

|∇_Hϕ|²_Hdγ

1/2

≤ λK

Z

Ω\Ωn

|∇_Hϕ|²_Hdγ

1/2

that goes to zero as n → ∞.

Moreover,

n→∞lim Z

Ωn

f ϕ dγ = Z

Ω

f ϕ dγ.

Therefore letting n → ∞ in (3.10) we get that u satisfies (1.25).

Finally we give the maximal regularity result.

Theorem 19. If u is the weak solution of λu − L^Ωu = f on Ω then u ∈ W^2,2(Ω, γ) and

kuk_W^2,2_(Ω,γ) ≤ Ckf k_L²_(Ω,γ) Proof. By Proposition 45 it follows

kunk_W^2,2_(Ω,γ) ≤ Ckf k_L²_(Ωn,γ) (3.11) where C = C(λ) is the constant of the Theorem 18.

We remark that

n→∞lim kf k_L²_(Ωn,γ) = kf k_L²_(Ω,γ) since γ(Ω_n\Ω) → 0.

By the weak convergence of u_n to u we have kuk_W^2,2_(Ω,γ) ≤ lim sup

n→∞

ku_nk_W^2,2_(Ω,γ). Letting n → ∞ in (3.11) we get our claim.

3.3 The Neumann boundary condition

In this section we put under Assumption 1 and we prove that the weak solution u of (1) satisfies a Neumann type boundary condition.

First we prove a useful lemma.

Proposition 47. If u ∈ L^p(∂Ω, ρ) and Z

∂Ω

uϕ dρ = 0 ∀ϕ ∈ F C_b¹(X), then u = 0 ρ−a.e. in ∂Ω.

Proof. Since the map

v 7→

Z

∂Ω

uv dρ

is continuous from W^1,q(Ω, γ) to R for all q > p⁰, and F C_b¹(X) is dense in W^1,q(Ω, γ), it follows that

Z

∂Ω

uψ dρ = 0 ∀ψ ∈ W^1,q(Ω, γ).

In particular, since the restrictions to Ω of the Lipschitz continuous and bounded functions ψ : X → R belong to W^1,q(Ω, γ), we have

Z

∂Ω

uψ dρ = 0 ∀ψ ∈ Lip_b(X).

Lemma 13 yields Z

∂Ω

uψ dρ = 0 ∀ψ ∈ L^q(∂Ω, ρ) and this implies that u = 0 ρ−a.e..

Now we are ready to prove that the weak solution of (1) satisfies a bound-ary condition similar to the Neumann boundbound-ary condition.

Proposition 48. If u is the weak solution of λu − Lu = f on Ω then h∇_Hu(x), ∇_Hg(x)

|∇_Hg(x)|_Hi_H = 0 ρ − a.e x ∈ ∂Ω. (3.12) Proof. We fix ϕ ∈ F C_b¹(X). We denote by u_n the solution to

λψ − L^Ωnψ = f_|Ωn in Ωn. (3.13) We recall that u_nis a cylindrical function and, thanks to the result of Section 1.8, we have u_n ∈ W^2,2(Ω_n, γ). We multiply the differential equation (3.13) by ϕ and we integrate on Ω, getting

Z

Ω

(λu_n− L^Ωnu_n)ϕ dγ = Z

Ω

f ϕ dγ.

We recall that L^Ωnu_n is cylindrical, then

L^Ωnu_n(x) =

q

X

i=1

∂_iiu_n(x) − bh_i(x)∂_iu_n(x).

Therefore, by using (1.20), we obtain Z

Ω

λϕu_n dγ + Z

Ω

h∇_Hu_n, ∇_Hϕi_Hdγ = Z

Ω

f ϕ dγ + Z

∂Ω

h∇_Hu_n, ∇Hg

|∇_Hg|_Hi_Hϕ dρ, where

h∇_Hu_n, ∇_Hϕi_H =

q

X

i=1

∂_iu_n∂_iϕ, and

h∇Hun, ∇HgiH =

q

X

i=1

∂iun∂ig.

As in the previous section we have

n→∞lim Z

Ω

λϕu_n dγ = Z

Ω

λϕu dγ, and

n→∞lim Z

Ω

h∇_Hu_n, ∇_Hϕi_Hdγ = Z

Ω

h∇_Hu, ∇_Hϕi_Hdγ,

We claim that the map v 7→

Z

∂Ω

h∇_Hv, ∇_Hg

|∇_Hg|_Hi_Hϕ dρ

from W^2,2(Ω, γ) to R belongs to (W^2,2(Ω, γ))⁰. Indeed, the function x 7→ h∇_Hv(x), ∇_Hg(x)

|∇_Hg(x)|_Hi_Hϕ(x) =: F (x)

belongs to W^1,q(Ω, γ) for all q ∈ (1, 2). Moreover kF k_W^1,q_(Ω,γ) ≤ eCkvk_W^2,2_(Ω,γ), and the trace operator is linear and continuous from W^1,q(Ω, γ) to L¹(∂Ω, ρ).

Therefore, since u_n* u in W^2,2(Ω, γ),

n→∞lim Z

∂Ω

h∇_Hu_n, ∇_Hg

|∇Hg|H

i_Hϕ dρ = Z

∂Ω

h∇_Hu, ∇_Hg

|∇Hg|H

i_Hϕ dρ.

Then we have Z

Ω

λuϕ dγ + Z

Ω

h∇Hu, ∇HϕiHdγ = Z

Ω

f ϕ dγ + Z

∂Ω

h∇Hu, ∇Hg

|∇_Hg|_HiHϕ dρ and since u is a weak solution of (1) we get

Z

∂Ω

h∇Hu, ∇_Hg

|∇_Hg|_HiHϕ dρ = 0

for all ϕ ∈ F C_b¹(X). By using Proposition 47 we obtain (3.12).

Therefore, if u ∈ D(L) then u ∈ W^2,2(Ω, γ) and u satisfies the Neumann boundary condition (3.12).

Appendix A

Density properties

In this appendix we show some density results for which we thank Simone Ferrari. Let (Y, d) be a complete metric space and let ρ be a finite Radon measure defined on the Borel sets of Y . Let BU C(Y ) be the set of real value uniformly bounded continuous functions and let Lip_b(Y ) be the set of Lipschitz bounded functions.

Lemma 12. Let f : Y → R be a bounded ρ−measurable function. Then for all ε > 0 there exists g ∈ BU C(Y ) such that

ρ({x ∈ Y : f (x) 6= f_ε(x)}) < ε and

sup

x∈Y

|g(x)| ≤ 2 sup

x∈Y

|f (x)|.

Proof. We fix ε > 0. Since ρ is a Radon measure then there exists K₀, compact subset of Y , such that ρ(Y \ K₀) < ε. By the Lusin theorem there exists a function f0 ∈ C(K0) = BU C(K0) such that:

ρ({x ∈ K₀ : f₀(x) 6= f|K₀(x)}) < ε and

sup

x∈K0

|f₀(x)| ≤ sup

x∈K0

|f (x)| ≤ sup

x∈Y

|f (x)|.

We define the following function, studied in [22],

g(x) =







f (x) if x ∈ K₀

inf

y∈K0

f₀(y) d(x, y)

d(x, K0) if x 6∈ K₀

then g is a BU C extension of f₀ to the whole Y . We remark that for x 6∈ K₀ there exists y_ε ∈ K₀ such that

d(x, K₀) = inf

y∈K0

d(x, y) ≥ d(x, y_ε) − ε, therefore for x 6∈ K₀ we have

|g(x)| =    

y∈Kinf0

f₀(y) d(x, y) d(x, K₀)

   

≤ sup

x∈Y

|f (x)| d(x, y_ε)

d(x, K₀) ≤ sup

x∈Y

|f (x)|d(x, K₀) + ε d(x, K₀) for all ε. Then for all x 6∈ K₀ we have

g(x) ≤ sup

y∈Y

|f (y)|.

Finally sup

x∈Y

|g(x)| = sup

x∈Y

|g_|K0(x) + g_{|Y \K0}(x)| ≤ sup

x∈K0

|g(x)| + sup

x∈Y \K0

|g(x)|

= sup

x∈K0

|f₀(x)| + sup

x∈Y \K0

|g(x)| ≤ 2 sup

x∈Y

|f (x)|.

Moreover

ρ({x ∈ Y : g(x) 6= f (x)}) ≤ ρ({x ∈ K₀ : g(x) 6= f (x)}) + ρ({x ∈ Y \ K₀ : g(x) 6= f (x)})

≤ ρ({x ∈ K0 : f0(x) 6= f (x)}) + ρ(Y \ K0) < 2ε.

Lemma 13. The subspace Lip_b(Y ) is dense in L^p(Y, ρ) with respect the norm k · k_L^p_(Y,ρ).

Proof. Let f ∈ L^p(Y, ρ). For k ∈ N we put

f_k(x) =







k if f (x) > k f (x) if f (x) ∈ [−k, k]

− k if f (x) < −k

so that f_k(x) is bounded and measurable. Then by Lemma 12 there exists fe_k ∈ BU C(Y ) such that

ρ({x ∈ Y : efk(x) 6= fk(x)}) ≤ 1 2^k

Then by [25] there exists g_k ∈ Lip_b(Y ) such that kg_k− ef_kk_L^∞_{(Y )} ≤ 1

2^k. Now we estimate

kg_k− f k_L^p_(Y,ρ)≤ kg_k− ef_kk_L^p_(Y,ρ)+ k ef_k− f_kk_L^p_(Y,ρ)+ kf_k− f k_L^p_(Y,ρ), where

kg_k− ef_kk_L^p_(Y,ρ)=

Z

Y

|g_k(x) − ef_k(x)|^pρ(dx)

1/p

≤ kg_k− ef_kk_L^∞_{(Y )}ρ(Y )^1/p ≤ ρ(Y )^1/p 2^k . Concerning the second term we recall that

sup

x∈Y

| efk(x)| ≤ 2 sup

x∈Y

|fk(x)| = 2k then

k ef_k− f_kk_L^p_(Y,ρ) =

Z

Y

| ef_k(x) − f_k(x)|^pρ(dx)

1/p

=

Z

{x∈Y : efk(x)6=fk(x)}

| ef_k(x) − f_k(x)|^pρ(dx)

1/p

≤ 3k ρ({x ∈ Y : ef_k(x) 6= f_k(x)})^1/p ≤ 3k 2^k/p.

Finally we remark that since f_k → f ρ-a.e. for k → ∞, and |f_k(x)| ≤

|f (x)| ∈ L^p(Y, ρ), then the Lebesgue theorem yields kfk− f k_L^p_(Y,ρ) → 0, k → ∞.

Appendix B

Hermite polynomials and Wiener chaos decomposition

In this chapter we introduce the Hermite polynomials and their link with the Ornstein-Uhlenbeck operator in the whole space X. We refer to the books [3] for a detailed treatment.

First we define the Hermite polynomials in finite dimension.

Definition 32. For k ∈ N, we define the Hermite polynomials of degree k by

H_k(x) := (−1)^k

√k! exp x² 2

 d^k dx^k

 exp



−x² 2



. (B.1)

We set H₀(t) = 1.

The first Hermite polynomials are H₁(x) = x, H₂(x) = 1

√2(x²− 1), H₃(x) = 1

√6(x³− 3x), H₄(x) = 1 2√

6(x⁴− 6x² + 3).

Now give some properties of Hermite polynomials.

Proposition 49. The Hermite polynomials on R satisfy the following prop-erties:

1. {H_k}_k∈N∪{0} is an orthonormal basis of L²(R, γ1);

2. H_k⁰(x) = √

kH_k−1(x), for all k ≥ 1;

3. H_k⁰(x) = xHk(x) −√

k + 1Hk+1(x), for all k ≥ 1;

4. H_k⁰⁰(x) = −H_k⁰(x) − kH_k(x), for all k ≥ 1.

Definition 33. Let α = (k₁, . . . , k_n) be a multi-index. We define the Hermite polynomials on Rⁿ as

H_α(x) = H_k1(x₁) · · · H_kn(x_n), where H_ki is the Hermite polynomial on R of degree ki .

We set

X_k:= span{H_α : |α| = k} in L²(Rⁿ, γ_n).

Proposition 50. For all h, k ∈ N ∪ {0} with h 6= k, we have Xh⊥X_k. Proof. It is sufficient to prove that for |α| = k and |β| = h, with h 6= k, we

have Z

Rⁿ

H_α(x)H_β(x)γ_n(dx) = 0.

Recalling Definition 33, we have Z

Rⁿ

H_α(x)H_β(x)γ_n(dx) =

n

Y

i=1

Z

R

H_αi(x_i)H_βi(x_i)γ₁(dx_i) = 0,

since h 6= k implies that there exists j ∈ {1, . . . , n} such that α_j 6= β_j.

Proposition 51. {H_α}_{α∈(N∪{0})}ⁿ is an orthonormal basis of L²(Rⁿ, γ_n). More-over

L²(Rⁿ, γn) =

∞

M

k=0

Xk.

B.1 Hermite polynomials in infinite dimension

Let X be a separable Banach space endowed with a non-degenerate centered Gaussian measure γ. Let {h_n}_n∈N be an orthonormal basis of H such that {bh_n}_n∈N⊂ X^∗ is a basis of X_γ^∗.

Definition 34. The Hermite polynomials are function of the type x 7→ H_k1(bh₁(x)) · · · H_km(bh_m(x)), x ∈ X,

with m ∈ N and k1, . . . , k_m ∈ N ∪ {0}.

Proposition 52. The space generated by the Hermite polynomials is dense in L²(X, γ).

Proposition 53. The Hermite polynomials are orthonormal in L²(X, γ).

Proof. We fix n, m ∈ N, n ≥ m. Let α, β be two multi-indices, with |α| = m and |β| = n. Then

Z

X

Hα1(bh1(x)) · · · Hαm(bhm(x)) · Hβ1(bh1(x)) · · · Hβn(bhn(x)) γ(dx)

= Z

Rⁿ

H_α1(ξ₁) · · · H_αm(ξ_m) · H_β1(ξ₁) · · · H_βn(ξ_n) γ_n(dξ)

= hH_{(α1,...,αm,0,...,0)}, H_βi_L2(Rⁿ,γn) =

(1 if (α, 0) = β 0 otherwise .

Now we consider α ∈ (N ∪ {0})^N, α = (α₁, . . . , α_n, . . .), and

|α| =

∞

X

n=1

α_n.

Then |α| < ∞ if and only if α_n 6= 0 for a finite number if n. In this case we put

H_α(x) =

∞

Y

n=1

H_αn(bh_n(x)).

For k ∈ N ∪ {0} we set

Xk:= span{Hα: |α| = k} in L²(X, γ).

Then

L²(X, γ) = M

k∈N∪{0}

X_k. (B.2)

The identity (B.2) is called Wiener chaos decomposition. For f ∈ L²(X, γ), the orthogonal projection of f on X_k is given by

I_k(f ) = X

|α|=k

hf, H_αi_L2(X,γ)· H_α.

We define a family of operators (T (t))_t≥0⊂ L(L²(X, γ)) as T (t)f (x) =

Z

X

f

e^−tx +√

1 − e^−2ty γ(dy).

One can prove that (T (t))_t≥0 is a C₀−semigroup named Ornstein-Uhlenbeck semigroup (see [3]).

Let L be the infinitesimal generator of (T (t))_t≥0 in L²(X, γ), with D(L) :=



f ∈ L²(X, γ) : ∃ lim

t→0⁺

T (t)f − f

t in L²(X, γ)

 . We call L the Ornstein-Uhlenbeck operator.

Proposition 54. If L is the infinitesimal generator of (T (t))t≥0, then D(L) =u ∈ W^1,2(X, γ) : ∃g ∈ L²(X, γ) :

Z

X

h∇_Hu, ∇_Hϕi_Hdγ = Z

X

g ϕ dγ ∀ϕ ∈ W^1,2(X, γ)

 , moreover

D(L) = W^2,2(X, γ), with equivalence of the norms.

The following property links the Ornstein-Uhlenbeck semigroup and the Hermite polynomials.

Proposition 55. For all t ≥ 0 and f ∈ L²(X, γ), we have T (t)f =

∞

X

k=0

e^−ktI_k(f ).

As a consequence of Proposition 55, we have D(L) =

(

f ∈ L²(X, γ) :

∞

X

k=0

k²kI_k(f )k²_L2(X,γ) < ∞ )

, and

Lf =

∞

X

k=0

−kI_k(f ), ∀f ∈ D(L).

Therefore −k is eigenvalue of L, and the corresponding eigenspace is X_k∩ D(L).

Lemma 14. For all k ∈ N, we have X^k ⊂ D(L).

Proof. If f ∈ Xk, then Ik(f ) = f while In(f ) = 0 for all n 6= k. Moreover the function T (t)f = e^−ktf is differentiable at t = 0.

Proposition 56. The following identity W^1,2(X, γ) =

(

f ∈ L²(X, γ) :

∞

X

k=0

kkI_k(f )k²_L2(X,γ) < ∞ )

, holds.

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