We formulate the elliptic differential operator with infinite number of variables and investigate that it is well defined on infinite tensor product of spaces of square integrable functions. Under suitable conditions, we prove Garding's inequality for this operator.

1. Introduction

In order to solve the Dirichlet problem for a differential operator by using Hilbert space methods (sometimes called the direct methods in the calculus of variations), Garding's inequality represents an essential tool [1, 2]. For strongly elliptic differential operators, Garding's inequality was proved by Gärding [3] and its converse by Agmon [4]. One can find a proof for Garding's inequality and its converse in the work of Stummel [5] for strongly semielliptic operators. Two examples for strongly elliptic and semielliptic operators are studied in [6]. More recent results on this subject can be found in [7, 8] for a class of differential operators containing some non-hypoelliptic operators which were first introduced by Dynkin [9] and for differential operators in generalized divergence form (see also [10, 11]).

The aim of this work is to study the existence of the weak solution of the Dirichlet problem for a second-order elliptic differential operator with infinite number of variables.

2. Some Function Spaces

In this paper, we will consider spaces of functions of infinitely many variables, see [12, 13]. For this purpose we introduce the product measure dρ(x)=(p1(x1)dx1)×(p2(x2)dx2)×⋯=(dρ1(x1))×(dρ2(x2))×⋯,(pk(xk)dxk=dρk(xk),k=1,2,…)
defined on the space R∞=R1×R1×⋯ of points x=(xk)k=1∞,xk∈R1, where (pk)1∞ is a fixed sequence of weights, such thatC2(R1)∋pk(t)>0,∫R1pk(t)dt=1.
For k =1,2,…, we putR∞=R1×R1×⋯×R1︸k-1×R1×⋯
We can write x∈R∞, by x=(xk,x̃), where x̃=(x1,…,xk-1,xk+1,…)
and dρ(x)=dρ(xk)×dρ(x̃).

With respect to dρ we construct on R∞ the Hilbert space of functions of infinitely many variables
L2(R∞)=L2(R∞,dρ(x)),
which can be understood as the infinite tensor product
⨂k=1,e∞L2(R1,dρk(xk))
with the identity stabilization e=(e(k))1∞, e(k)∈L2(R1,dρk(xk)), e(k)=1. To say that the function f∈L2(R∞,dρ(x)) is cylindrical, it means that there exist an m=1,2,…, and an fc∈L2(Rm,dρ(m)(x(m))),(x(m)=(x1,…,xm)),(dρ(m)(x(m))=⨂1mdρk(xk)), such that f(x)=fc(x(m)), x∈R∞.

On the collection of functions which are l=1,2,… times continuously differentiable up to the boundary Γ of Rm for sufficiently large m, we introduce the scalar product (u,v)l=∑|α|≤l(Dαu,Dαv)L2(R∞,dρ(x)),
whereDα=∂|α|(∂x1)α1(∂x2)α2⋯,|α|=∑i=1∞αi.
The differentiation is taken in the sense of generalized functions, and after the completion we obtain the Sobolev spaces W2l(R∞), l=1,2,….

Sobolev space of order l on R∞ is defined byW2l(R∞)={u∣Dαu∈L2(R∞,dρ(x))∀α,|α|≤l},

W2l(R∞) endowed with the scalar product (2.7) forming a dense subspace of L2(R∞,dρ(x)), with‖u‖L2(R∞,dρ(x))≤‖u‖W2l(R∞)
for u∈W2l(R∞).

We use the technique of [13] to construct chains of spaces W2l(R∞)⊆L2(R∞,dρ(x))=W20(R∞)⊆W2-l(R∞),l=0,1,…,where W2-l(R∞) are the duals of W2l(R∞).

3. Elliptic Differential Operator with Infinite Number of Variables

Consider (ak)k=1∞ to be a sequence of nonnegative locally bounded functions in R∞ (i.e., they are bounded on each compact subset) with derivatives (∂/∂xk)ak∈Lp,loc for any p≥1 and k=1,2,…, and for a suitable x0∈R∞ it satisfies the following conditions:

there exists a constant c1>0 such that
∑k=1∞ak(x0)≥c1,

let c1 be the constant in condition (1), and there is n0 belonging to ℕ such that
maxk∈Nsupx∈R∞|ak(x)-ak(x0)|≤c12n0.

Now, we define on L2(R∞,dρ(x)) an elliptic differential operator with infinitely many variables(Lu)(x)=-∑k=1∞1pk(xk)∂∂xk(ak(x)∂∂xk(pk(xk)u(x)))=-∑k=1∞Dk(akDku)(x),u∈W21(R∞),
where(Dku)(x)=1pk(xk)∂∂xk(pk(xk)u(x)).

Theorem 3.1.

Assume that (pk)k=1∞ satisfy the condition that
∑k=1∞(Dk21)(x)
converges in L2(R∞,dρ(x)). Then the operator L in (3.3) is well defined and admits a closure in L2(R∞,dρ(x)).

Proof.

The mapping
L2(R1,dxk)∋U(xk)⟼u(xk)=pk-1/2(xk)U(xk)∈L2(R1,dρk(xk))
is an isometry between the two spaces of square integrable functions. It carries (∂U/∂xk)(xk) into the sandwiched (by means of pk) derivative
(Dku)(xk)=pk-1/2(xk)∂∂xk(pk1/2(xk)u(xk))=(∂u∂xk)(xk)+(DkI)(xk)u(xk),
and it carries
∂∂xk(ak(x)∂U∂xk(xk))
into the corresponding Dk derivative:
Dk(ak(x)Dku)(xk)=pk-1/2(xk)∂∂xk(pk1/2(xk)ak(x)(Dku)(xk))=∂∂xk[ak(x)pk-1/2(xk)∂∂xk(pk1/2(xk)u(xk))]+(Dk1)(xk)ak(x)(Dku)(xk)=ak(x)∂2u∂xk2(xk)+∂ak∂xk(x)∂u∂xk(xk)+∂∂xk[ak(x)(Dk1)(xk)u(xk)]+(Dk1)(xk)ak(x)(Dku)(xk)=∂∂xk(ak(x)∂u∂xk(xk))+∂∂xk[ak(x)(Dk1)(xk)u(xk)]+(Dk1)(xk)ak(x)∂u∂xk(xk)+ak(x)(Dk1)2(xk)u(xk)=∂∂xk(ak(x)∂u∂xk(xk))+ak(x)u(xk)∂∂xk(Dk1)(xk)+(Dk1)(xk)[ak(x)∂u∂xk(xk)+∂ak∂xk(x)u(xk)]+(Dk1)(xk)ak(x)∂u∂xk(xk)+ak(x)(Dk1)2(xk)u(xk)=∂∂xk(ak(x)∂u∂xk(xk))+(Dk1)(xk)[2ak(x)∂u∂xk(xk)+u(xk)∂ak∂xk(x)]+(Dk21)(xk)ak(x)u(xk).
Denote by Cc,0∞(R∞) the linear span of the set of all cylindrical infinitely differentiable finite functions dense in W2l(R∞), that is, all the functions u∈W2l(R∞) of the form
R∞∋x⟼u(x)=uc(x1,…,xn),
where n depends on u and uc∈C0∞(Rn),n=1,2,…. Condition (3.5) implies that Dk1,Dk21∈L2(R1,dρk(xk)), (see [13, Lemma (3.2)]). We note that the action of L on the function u(x)=uc(x(n)) has the form
(Lu)(x)=-∑k=1nDk(akDkuc)(x)-uc(x)[∑k=n+1∞(akDk21)(x)+∑k=n+1∞(Dk1)(x)∂ak∂xk(x)],
then in view of condition (3.5), the operator Cc,0∞(R∞)∋u(x)↦(Lu)(x)=-∑k=1∞Dk(akDku)(x)∈L2(R∞,dρ(x)) is well defined in L2(R∞,dρ(x)) and admits a closure which is again denoted by L.

4. A Garding Inequality

In our consideration, we have an operator of the form(Lu)(x)=-∑k=1∞Dk(akDku)(x)
with u∈W21(R∞).

Lemma 4.1.

The operator L is Hermitian.

Proof.

It is sufficient to verify the Hermitianness on functions of the form u(x)=uc(x(n)),v(x)=vc(x(m)), whereuc∈C0∞(Rn),vc∈C0∞(Rm); for example, we take it that m≤n.

Using (3.11), we obtain
(Lu,v)L2(R∞,dρ(x))=-∫Rn[∑k=1n(DkakDkuc)(x)]vc(x)̅p1(x1)⋯pn(xn)dx1⋯dxn-∫R∞(∑k=n+1∞[ak(x)(Dk21)(x)+(Dk1)(x)∂ak∂xk(x)]uc(x))vc(x)̅dρ(x)=-∑k=1n∫Rn(DkakDkuc)(x)vc(x)̅p1(x1)⋯pn(xn)dx1⋯dxn-∑k=n+1∞∫R∞([ak(x)(Dk21)(x)+(Dk1)(x)∂ak∂xk(x)]uc(x))vc(x)̅dρ(x)=-∑k=1nAk-∑k=n+1∞Bk,
where
Ak=∫Rn(DkakDkuc)(x)vc(x)̅p1(x1)⋯pn(xn)dx1⋯dxn=∫Rn-1(∫R1(DkakDkuc)(x)vc(x)̅pk(xk)dxk)p1(x1)⋯pk-1(xk-1)×pk+1(xk+1)⋯pn(xn)dx1⋯dxk-1dxk+1⋯dxn=∫Rn-1∫R11pk(xk)∂∂xk(ak(x)∂∂xk(pk(xk)uc(x)))×vc(x)̅pk(xk)dxkp1(x1)⋯pk-1(xk-1)×pk+1(xk+1)⋯pn(xn)dx1⋯dxk-1dxk+1⋯dxn=∫Rn-1∫R1(∂∂xk(pk(xk)uc(x)))(-ak(x)∂∂xk(pk(xk)vc(x)̅))×dxkp1(x1)⋯pk-1(xk-1)pk+1(xk+1)⋯pn(xn)dx1⋯dxk-1dxk+1⋯dxn=∫Rn-1∫R1uc(x)1pk(xk)∂∂xk(ak(x)∂∂xk(pk(xk)vc(x)̅))×pk(xk)dxkp1(x1)⋯pk-1(xk-1)×pk+1(xk+1)⋯pn(xn)dx1⋯dxk-1dxk+1⋯dxn=∫Rnuc(x)(DkakDkvc)(x)̅p1(x1)⋯pn(xn)dx1⋯dxn,Bk=∫R∞uc(x)[ak(x)(Dk21)(x)+(Dk1)(x)∂ak∂xk(x)]vc(x)̅dρ(x).
Hence, we have
(Lu,v)L2(R∞,dρ(x))=-∑k=1n∫Rnuc(x)(DkakDkvc)(x)̅p1(x1)⋯pn(xn)dx1⋯dxn-∑k=n+1∞∫R∞uc(x)([ak(x)(Dk21)(x)+(Dk1)(x)∂ak∂xk(x)]vc(x)̅)dρ(x)=(u,Lv)L2(R∞,dρ(x)).

Now, we can define on W21(R∞) the bilinear formB(u,v)=(Lu,v)L2(R∞,dρ(x)),
where L∈ℒ(W21(R∞),W2-1(R∞))B(u,v)=-∑k=1∞∫R∞1pk(xk)∂∂xk(ak(x)∂∂xk(pk(xk)u(x)))v(x)̅dρ(x)=-∑k=1∞∫R∞1pk(xk)∂∂xk(ak(x)∂∂xk(pk(xk)u(x)))v(x)̅pk(xk)dxkρ(x̃)=-∑k=1∞∫R∞∂∂xk(ak(x)∂∂xk(pk(xk)u(x)))v(x)̅pk(xk)dxkdρ(x̃)=∑k=1∞∫R∞ak(x)∂∂xk(pk(xk)u(x))∂∂xk(v(x)̅pk(xk))dxkdρ(x̃)=∑k=1∞∫R∞ak(x)pk(xk)1pk(xk)∂∂xk(pk(xk)u(x))1pk(xk)×∂∂xk(pk(xk)v(x)̅)dxkdρ(x̃),
thenB(u,v)=∑k=1∞∫R∞ak(x)Dku(x)Dkv(x)̅dρ(x).

Lemma 4.2.

The bilinear form (4.7) is continuous on W21(R∞).

Proof.

For u,v∈Cc,0∞(R∞),
|B(u,v)|≤∑k=1∞∫R∞ak(x)|Dku(x)||Dkv(x)|dρ(x)≤maxk∈Nsupx∈R∞ak(x)∑k=1∞∫R∞|Dku(x)||Dkv(x)|dρ(x)≤maxk∈Nsupx∈R∞ak(x)∑k=1∞(∫R∞|Dku(x)|2dρ(x))1/2(∫R∞|Dkv(x)|2dρ(x))1/2≤c∑k=1∞‖Dku‖L2(R∞,dρ(x))‖Dkv‖L2(R∞,dρ(x))≤c‖u‖W21(R∞)‖v‖W21(R∞).
Thus B has a continuous extension onto W21(R∞) which is again denoted by B.

Theorem 4.3.

Suppose that L is given as in (4.1). In particular assume that (3.5) holds. Then there exist positive constants c0>0 and c1≥0 such that
B(u,u)≥c0‖u‖W21(R∞)2-c1‖u‖L2(R∞,dρ(x))2
holds for all u∈W21(R∞).

Proof.

For u∈Cc,0∞(R∞),
B(u,u)=∑k=1∞∫R∞ak(x)|Dku(x)|2dρ(x)=∑k=1∞ak(x0)∫R∞|Dku(x)|2dρ(x)-∑k=1∞∫R∞(ak(x0)-ak(x))|Dku(x)|2dρ(x)≥∑k=1∞ak(x0)∫R∞|Dku(x)|2dρ(x)-∑k=1∞∫R∞|ak(x0)-ak(x)||Dku(x)|2dρ(x),
and using conditions (1) and (2),
B(u,u)≥c1∑k=1∞∫R∞|Dku(x)|2dρ(x)-maxk∈Nsupx∈R∞|ak(x0)-ak(x)|∑k=1∞∫R∞|Dku(x)|2dρ(x)≥c1[‖u‖W21(R∞)2-‖u‖L2(R∞,dρ(x))2]-c12n0∑K=1∞∫R∞|Dku(x)|2dρ(x)≥c1[‖u‖W21(R∞)2-‖u‖L2(R∞,dρ(x))2]-c12n0‖u‖W21(R∞)2=c1(1-12n0)‖u‖W21(R∞)2-c1‖u‖L2(R∞,dρ(x))2,
and with c0=c1(1-1/2n0), we finally obtain (4.9).

5. Conclusions

In view of our recent achievement, we recommend to extend this approach to include the linear partial differential operators in generalized divergence form ∑α,β∈ΓDα(aαβ(·)Dβ), where Γ is finite, and nonempty collection of α=(α1,…,αn), αi=1,2,…, and aαβ(α,β∈Γ×Γ) are real locally bounded functions on R∞.

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