Garding's Inequality for Elliptic Differential Operator with Infinite Number of Variables

Abstract

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)=𝑝₁𝑥₁𝑑𝑥₁×𝑝₂𝑥₂𝑑𝑥₂=×⋯𝑑𝜌₁𝑥₁×𝑑𝜌₂𝑥₂𝑝×⋯,_𝑘𝑥_𝑘𝑑𝑥_𝑘=𝑑𝜌_𝑘𝑥_𝑘,𝑘=1,2,… (2.1) defined on the space 𝑅^∞=𝑅¹×𝑅¹×⋯ of points 𝑥=(𝑥_𝑘)^∞_𝑘=1,𝑥_𝑘∈𝑅¹, where (𝑝_𝑘)^∞₁ is a fixed sequence of weights, such that𝐶²𝑅¹∋𝑝_𝑘(𝑡)>0,_𝑅1𝑝_𝑘(𝑡)𝑑𝑡=1. (2.2) For k =1,2,…, we put𝑅^∞=𝑅¹×𝑅¹×⋯×𝑅¹_𝑘−1×𝑅¹×⋯ (2.3) We can write 𝑥∈𝑅^∞, by 𝑥=(𝑥_𝑘,𝑥), where 𝑥=𝑥₁,…,𝑥_𝑘−1,𝑥_𝑘+1,… (2.4) and 𝑑𝜌(𝑥)=𝑑𝜌(𝑥_𝑘)×𝑑𝜌(𝑥).

With respect to 𝑑𝜌 we construct on 𝑅^∞ the Hilbert space of functions of infinitely many variables 𝐿₂𝑅^∞=𝐿₂𝑅^∞,𝑑𝜌(𝑥), (2.5) which can be understood as the infinite tensor product ^∞_𝑘=1,𝑒𝐿₂𝑅¹,𝑑𝜌_𝑘𝑥_𝑘 (2.6) with the identity stabilization 𝑒=(𝑒^(𝑘))^∞₁, 𝑒^(𝑘)∈𝐿₂(𝑅¹,𝑑𝜌_𝑘(𝑥_𝑘)), 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 𝑓_𝑐∈𝐿₂(𝑅^𝑚,𝑑𝜌_(𝑚)(𝑥^(𝑚))),(𝑥^(𝑚)=(𝑥₁,…,𝑥_𝑚)),(𝑑𝜌_(𝑚)(𝑥^(𝑚))=⨂^𝑚₁𝑑𝜌_𝑘(𝑥_𝑘)), such that 𝑓(𝑥)=𝑓_𝑐(𝑥^(𝑚)), 𝑥∈𝑅^∞.

On the collection of functions which are 𝑙=1,2,… times continuously differentiable up to the boundary Γ of 𝑅^𝑚 for sufficiently large 𝑚, we introduce the scalar product (𝑢,𝑣)_𝑙=_{|𝛼|≤𝑙}𝐷^𝛼𝑢,𝐷^𝛼𝑣_{𝐿2(𝑅∞,𝑑𝜌(𝑥))}, (2.7) where𝐷^𝛼=𝜕^|𝛼|𝜕𝑥₁^𝛼1𝜕𝑥₂^𝛼2⋯,|𝛼|=^∞_𝑖=1𝛼_𝑖. (2.8) The differentiation is taken in the sense of generalized functions, and after the completion we obtain the Sobolev spaces 𝑊^𝑙₂(𝑅^∞), 𝑙=1,2,….

Sobolev space of order 𝑙 on 𝑅^∞ is defined by𝑊^𝑙₂𝑅^∞=𝑢∣𝐷^𝛼𝑢∈𝐿₂𝑅^∞,𝑑𝜌(𝑥)∀𝛼,|𝛼|≤𝑙, (2.9)

𝑊^𝑙₂(𝑅^∞) endowed with the scalar product (2.7) forming a dense subspace of 𝐿₂(𝑅^∞,𝑑𝜌(𝑥)), with‖𝑢‖_𝐿2(𝑅^∞_{,𝑑𝜌(𝑥))}≤‖𝑢‖_𝑊^𝑙_2(𝑅^∞₎ (2.10) for 𝑢∈𝑊^𝑙₂(𝑅^∞).

We use the technique of [13] to construct chains of spaces 𝑊^𝑙₂𝑅^∞⊆𝐿₂𝑅^∞,𝑑𝜌(𝑥)=𝑊⁰₂𝑅^∞⊆𝑊₂^−𝑙𝑅^∞,𝑙=0,1,…, (2.11)where 𝑊_𝟐^−𝐥(𝑅^∞) are the duals of 𝑊^𝐥_𝟐(𝑅^∞).

3. Elliptic Differential Operator with Infinite Number of Variables

Consider (𝑎_𝑘)^∞_𝑘=1 to be a sequence of nonnegative locally bounded functions in 𝑅^∞ (i.e., they are bounded on each compact subset) with derivatives (𝜕/𝜕𝑥_𝑘)𝑎_𝑘∈𝐿_𝑝,loc for any 𝑝≥1 and 𝑘=1,2,…, and for a suitable 𝑥₀∈𝑅^∞ it satisfies the following conditions:(1)there exists a constant 𝑐₁>0 such that ^∞_𝑘=1𝑎_𝑘𝑥₀≥𝑐₁, (3.1)(2)let 𝑐₁ be the constant in condition (1), and there is 𝑛₀ belonging to ℕ such that max_𝑘∈ℕsup_{𝑥∈𝑅∞}𝑎_𝑘(𝑥)−𝑎_𝑘𝑥₀≤𝑐₁2𝑛₀. (3.2)

Now, we define on 𝐿₂(𝑅^∞,𝑑𝜌(𝑥)) an elliptic differential operator with infinitely many variables(𝐿𝑢)(𝑥)=−^∞_𝑘=11𝑝_𝑘𝑥_𝑘𝜕𝜕𝑥_𝑘𝑎_𝑘(𝑥)𝜕𝜕𝑥_𝑘𝑝_𝑘𝑥_𝑘𝑢(𝑥)=−^∞_𝑘=1𝐷_𝑘𝑎_𝑘𝐷_𝑘𝑢(𝑥),𝑢∈𝑊¹₂𝑅^∞, (3.3) where𝐷_𝑘𝑢𝑥)=1(𝑝_𝑘𝑥_𝑘𝜕𝜕𝑥_𝑘𝑝_𝑘𝑥_𝑘𝑢(𝑥). (3.4)

Theorem 3.1. Assume that (𝑝_𝑘)^∞_𝑘=1 satisfy the condition that ^∞_𝑘=1𝐷²_𝑘1(𝑥) (3.5) 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 𝐿₂𝑅¹,𝑑𝑥_𝑘∋𝑈𝑥_𝑘⟼𝑢𝑥_𝑘=𝑝_𝑘^−1/2𝑥_𝑘𝑈𝑥_𝑘∈𝐿₂𝑅¹,𝑑𝜌_𝑘𝑥_𝑘 (3.6) is an isometry between the two spaces of square integrable functions. It carries (𝜕𝑈/𝜕𝑥_𝑘)(𝑥_𝑘) into the sandwiched (by means of 𝑝_𝑘) derivative 𝐷_𝑘𝑢𝑥_𝑘=𝑝_𝑘^−1/2𝑥_𝑘𝜕𝜕𝑥_𝑘𝑝_𝑘^1/2𝑥_𝑘𝑢𝑥_𝑘=𝜕𝑢𝜕𝑥_𝑘𝑥_𝑘+𝐷_𝑘𝐼𝑥_𝑘𝑢𝑥_𝑘, (3.7) and it carries 𝜕𝜕𝑥_𝑘𝑎_𝑘(𝑥)𝜕𝑈𝜕𝑥_𝑘𝑥_𝑘 (3.8) into the corresponding 𝐷_𝑘 derivative: 𝐷_𝑘𝑎_𝑘(𝑥)𝐷_𝑘𝑢𝑥_𝑘=𝑝_𝑘^−1/2𝑥_𝑘𝜕𝜕𝑥_𝑘𝑝_𝑘^1/2𝑥_𝑘𝑎_𝑘(𝑥)𝐷_𝑘𝑢𝑥_𝑘=𝜕𝜕𝑥_𝑘𝑎_𝑘(𝑥)𝑝_𝑘^−1/2𝑥_𝑘𝜕𝜕𝑥_𝑘𝑝_𝑘^1/2𝑥_𝑘𝑢𝑥_𝑘+𝐷_𝑘1𝑥_𝑘𝑎_𝑘(𝑥)𝐷_𝑘𝑢𝑥_𝑘=𝑎_𝑘(𝑥)𝜕²𝑢𝜕𝑥²_𝑘𝑥_𝑘+𝜕𝑎_𝑘𝜕𝑥_𝑘(𝑥)𝜕𝑢𝜕𝑥_𝑘𝑥_𝑘+𝜕𝜕𝑥_𝑘𝑎_𝑘(𝑥)𝐷_𝑘1𝑥_𝑘𝑢𝑥_𝑘+𝐷_𝑘1𝑥_𝑘𝑎_𝑘(𝑥)𝐷_𝑘𝑢𝑥_𝑘=𝜕𝜕𝑥_𝑘𝑎_𝑘(𝑥)𝜕𝑢𝜕𝑥_𝑘𝑥_𝑘+𝜕𝜕𝑥_𝑘𝑎_𝑘(𝑥)𝐷_𝑘1𝑥_𝑘𝑢𝑥_𝑘+𝐷_𝑘1𝑥_𝑘𝑎_𝑘(𝑥)𝜕𝑢𝜕𝑥_𝑘𝑥_𝑘+𝑎_𝑘(𝑥)𝐷_𝑘1²𝑥_𝑘𝑢𝑥_𝑘=𝜕𝜕𝑥_𝑘𝑎_𝑘(𝑥)𝜕𝑢𝜕𝑥_𝑘𝑥_𝑘+𝑎_𝑘(𝑥)𝑢𝑥_𝑘𝜕𝜕𝑥_𝑘𝐷_𝑘1𝑥_𝑘+𝐷_𝑘1𝑥_𝑘𝑎_𝑘(𝑥)𝜕𝑢𝜕𝑥_𝑘𝑥_𝑘+𝜕𝑎_𝑘𝜕𝑥_𝑘(𝑥)𝑢𝑥_𝑘+𝐷_𝑘1𝑥_𝑘𝑎_𝑘(𝑥)𝜕𝑢𝜕𝑥_𝑘𝑥_𝑘+𝑎_𝑘(𝑥)𝐷_𝑘1²𝑥_𝑘𝑢𝑥_𝑘=𝜕𝜕𝑥_𝑘𝑎_𝑘(𝑥)𝜕𝑢𝜕𝑥_𝑘𝑥_𝑘+𝐷_𝑘1𝑥_𝑘2𝑎_𝑘(𝑥)𝜕𝑢𝜕𝑥_𝑘𝑥_𝑘+𝑢𝑥_𝑘𝜕𝑎_𝑘𝜕𝑥_𝑘(𝑥+𝐷_2𝑘)1𝑥_𝑘𝑎_𝑘(𝑥)𝑢𝑥_𝑘. (3.9) Denote by 𝐶^∞_𝑐,0(𝑅^∞) the linear span of the set of all cylindrical infinitely differentiable finite functions dense in 𝑊^𝑙₂(𝑅^∞), that is, all the functions 𝑢∈𝑊^𝑙₂(𝑅^∞) of the form 𝑅^∞∋𝑥⟼𝑢(𝑥)=𝑢_𝑐𝑥₁,…,𝑥_𝑛, (3.10) where 𝑛 depends on 𝑢 and 𝑢_𝑐∈𝐶^∞₀(𝑅^𝑛),𝑛=1,2,…. Condition (3.5) implies that 𝐷_𝑘1,𝐷²_𝑘1∈𝐿₂(𝑅¹,𝑑𝜌_𝑘(𝑥_𝑘)), (see [13, Lemma (3.2)]). We note that the action of 𝐿 on the function 𝑢(𝑥)=𝑢_𝑐(𝑥^(𝑛)) has the form (𝐿𝑢)(𝑥)=−^𝑛_𝑘=1𝐷_𝑘𝑎_𝑘𝐷_𝑘𝑢_𝑐(𝑥)−𝑢_𝑐(𝑥)^∞_𝑘=𝑛+1𝑎_𝑘𝐷²_𝑘1(𝑥)+^∞_𝑘=𝑛+1𝐷_𝑘1(𝑥)𝜕𝑎_𝑘𝜕𝑥_𝑘(𝑥), (3.11) then in view of condition (3.5), the operator 𝐶^∞_𝑐,0(𝑅^∞)∋𝑢(𝑥)↦(𝐿𝑢)(𝑥)=−∑^∞_𝑘=1𝐷_𝑘(𝑎_𝑘𝐷_𝑘𝑢)(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=1𝐷_𝑘𝑎_𝑘𝐷_𝑘𝑢(𝑥) (4.1) with 𝑢∈𝑊¹₂(𝑅^∞).

Lemma 4.1. The operator 𝐿 is Hermitian.

Proof. It is sufficient to verify the Hermitianness on functions of the form 𝑢(𝑥)=𝑢_𝑐(𝑥^(𝑛)),𝑣(𝑥)=𝑣_𝑐(𝑥^(𝑚)), where𝑢_𝑐∈𝐶^∞₀(𝑅^𝑛),𝑣_𝑐∈𝐶^∞₀(𝑅^𝑚); for example, we take it that 𝑚≤𝑛.
Using (3.11), we obtain (𝐿𝑢,𝑣)_𝐿2(𝑅^∞_{,𝑑𝜌(𝑥))}=−_𝑅𝑛^𝑛_𝑘=1𝐷_𝑘𝑎_𝑘𝐷_𝑘𝑢_𝑐(𝑥)𝑣_𝑐(𝑥)𝑝₁𝑥₁⋯𝑝_𝑛𝑥_𝑛𝑑𝑥₁⋯𝑑𝑥_𝑛−_𝑅∞^∞_𝑘=𝑛+1𝑎_𝑘(𝑥)𝐷²_𝑘1(𝑥)+𝐷_𝑘1(𝑥)𝜕𝑎_𝑘𝜕𝑥_𝑘(𝑥)𝑢_𝑐(𝑥)𝑣_𝑐(𝑥)𝑑𝜌(𝑥)=−^𝑛_{𝑘=1𝑅𝑛}𝐷_𝑘𝑎_𝑘𝐷_𝑘𝑢_𝑐(𝑥)𝑣_𝑐(𝑥)𝑝₁𝑥₁⋯𝑝_𝑛𝑥_𝑛𝑑𝑥₁⋯𝑑𝑥_𝑛−^∞_{𝑘=𝑛+1𝑅∞}𝑎_𝑘(𝑥)𝐷²_𝑘1(𝑥)+𝐷_𝑘1(𝑥)𝜕𝑎_𝑘𝜕𝑥_𝑘(𝑥)𝑢_𝑐(𝑥)𝑣_𝑐(𝑥)𝑑𝜌(𝑥)=−^𝑛_𝑘=1𝐴_𝑘−^∞_𝑘=𝑛+1𝐵_𝑘, (4.2) where 𝐴_𝑘=_𝑅𝑛𝐷_𝑘𝑎_𝑘𝐷_𝑘𝑢_𝑐(𝑥)𝑣_𝑐(𝑥)𝑝₁𝑥₁⋯𝑝_𝑛𝑥_𝑛𝑑𝑥₁⋯𝑑𝑥_𝑛=_{𝑅𝑛−1𝑅1}𝐷_𝑘𝑎_𝑘𝐷_𝑘𝑢_𝑐(𝑥)𝑣_𝑐(𝑥)𝑝_𝑘𝑥_𝑘𝑑𝑥_𝑘𝑝₁𝑥₁⋯𝑝_𝑘−1𝑥_𝑘−1×𝑝_𝑘+1𝑥_𝑘+1⋯𝑝_𝑛𝑥_𝑛𝑑𝑥₁⋯𝑑𝑥_𝑘−1𝑑𝑥_𝑘+1⋯𝑑𝑥_𝑛=_{𝑅𝑛−1𝑅1}1𝑝_𝑘𝑥_𝑘𝜕𝜕𝑥_𝑘𝑎_𝑘(𝑥)𝜕𝜕𝑥_𝑘𝑝_𝑘𝑥_𝑘𝑢_𝑐(𝑥)×𝑣_𝑐(𝑥)𝑝_𝑘𝑥_𝑘𝑑𝑥_𝑘𝑝₁𝑥₁⋯𝑝_𝑘−1𝑥_𝑘−1×𝑝_𝑘+1𝑥_𝑘+1⋯𝑝_𝑛𝑥_𝑛𝑑𝑥₁⋯𝑑𝑥_𝑘−1𝑑𝑥_𝑘+1⋯𝑑𝑥_𝑛=_{𝑅𝑛−1𝑅1}𝜕𝜕𝑥_𝑘𝑝_𝑘𝑥_𝑘𝑢_𝑐(𝑥)−𝑎_𝑘(𝑥)𝜕𝜕𝑥_𝑘𝑝_𝑘𝑥_𝑘𝑣_𝑐(𝑥)×𝑑𝑥_𝑘𝑝₁𝑥₁⋯𝑝_𝑘−1𝑥_𝑘−1𝑝_𝑘+1𝑥_𝑘+1⋯𝑝_𝑛𝑥_𝑛𝑑𝑥₁⋯𝑑𝑥_𝑘−1𝑑𝑥_𝑘+1⋯𝑑𝑥_𝑛=_{𝑅𝑛−1𝑅1}𝑢_𝑐(𝑥)1𝑝_𝑘𝑥_𝑘𝜕𝜕𝑥_𝑘𝑎_𝑘(𝑥)𝜕𝜕𝑥_𝑘𝑝_𝑘𝑥_𝑘𝑣_𝑐(𝑥)×𝑝_𝑘𝑥_𝑘𝑑𝑥_𝑘𝑝₁𝑥₁⋯𝑝_𝑘−1𝑥_𝑘−1×𝑝_𝑘+1𝑥_𝑘+1⋯𝑝_𝑛𝑥_𝑛𝑑𝑥₁⋯𝑑𝑥_𝑘−1𝑑𝑥_𝑘+1⋯𝑑𝑥_𝑛=_𝑅𝑛𝑢_𝑐(𝑥)𝐷_𝑘𝑎_𝑘𝐷_𝑘𝑣_𝑐(𝑥)𝑝₁𝑥₁⋯𝑝_𝑛𝑥_𝑛𝑑𝑥₁⋯𝑑𝑥_𝑛,𝐵_𝑘=_𝑅∞𝑢_𝑐(𝑥)𝑎_𝑘(𝑥)𝐷²_𝑘1(𝑥)+𝐷_𝑘1(𝑥)𝜕𝑎_𝑘𝜕𝑥_𝑘(𝑥)𝑣_𝑐(𝑥)𝑑𝜌(𝑥). (4.3) Hence, we have (𝐿𝑢,𝑣)_𝐿2(𝑅^∞_{,𝑑𝜌(𝑥))}=−^𝑛_{𝑘=1𝑅𝑛}𝑢_𝑐(𝑥)𝐷_𝑘𝑎_𝑘𝐷_𝑘𝑣_𝑐(𝑥)𝑝₁𝑥₁⋯𝑝_𝑛𝑥_𝑛𝑑𝑥₁⋯𝑑𝑥_𝑛−^∞_{𝑘=𝑛+1𝑅∞}𝑢_𝑐(𝑥)𝑎_𝑘(𝑥)𝐷²_𝑘1(𝑥)+𝐷_𝑘1(𝑥)𝜕𝑎_𝑘𝜕𝑥_𝑘(𝑥)𝑣_𝑐(𝑥)(𝑥)=()𝑑𝜌𝑢,𝐿𝑣_𝐿2(𝑅^∞_{,𝑑𝜌(𝑥))}. (4.4)

Now, we can define on 𝑊¹₂(𝑅^∞) the bilinear form𝐵(𝑢,𝑣)=(𝐿𝑢,𝑣)_𝐿2(𝑅^∞_{,𝑑𝜌(𝑥))}, (4.5) where 𝐿∈ℒ(𝑊¹₂(𝑅^∞),𝑊₂⁻¹(𝑅^∞))𝐵(𝑢,𝑣)=−^∞_{𝑘=1𝑅∞}1𝑝_𝑘𝑥_𝑘𝜕𝜕𝑥_𝑘𝑎_𝑘(𝑥)𝜕𝜕𝑥_𝑘𝑝_𝑘𝑥_𝑘𝑢(𝑥)𝑣(𝑥)𝑑𝜌(𝑥)=−^∞_{𝑘=1𝑅∞}1𝑝_𝑘𝑥_𝑘𝜕𝜕𝑥_𝑘𝑎_𝑘(𝑥)𝜕𝜕𝑥_𝑘𝑝_𝑘𝑥_𝑘𝑢(𝑥)𝑣(𝑥)𝑝_𝑘𝑥_𝑘𝑑𝑥_𝑘𝜌(𝑥)=−^∞_{𝑘=1𝑅∞}𝜕𝜕𝑥_𝑘𝑎_𝑘(𝑥)𝜕𝜕𝑥_𝑘𝑝_𝑘𝑥_𝑘𝑢(𝑥)𝑣(𝑥)𝑝_𝑘𝑥_𝑘𝑑𝑥_𝑘(𝑥)=𝑑𝜌_{∞𝑘=1𝑅∞}𝑎_𝑘(𝑥)𝜕𝜕𝑥_𝑘𝑝_𝑘𝑥_𝑘𝑢(𝑥)𝜕𝜕𝑥_𝑘𝑣(𝑥)𝑝_𝑘𝑥_𝑘𝑑𝑥_𝑘(𝑥)=𝑑𝜌_{∞𝑘=1𝑅∞}𝑎_𝑘(𝑥)𝑝_𝑘𝑥_𝑘1𝑝_𝑘𝑥_𝑘𝜕𝜕𝑥_𝑘𝑝_𝑘𝑥_𝑘𝑢(𝑥)1𝑝_𝑘𝑥_𝑘×𝜕𝜕𝑥_𝑘𝑝_𝑘𝑥_𝑘𝑣(𝑥)𝑑𝑥_𝑘𝑑𝜌(𝑥), (4.6) then𝐵(𝑢,𝑣)=^∞_{𝑘=1𝑅∞}𝑎_𝑘(𝑥)𝐷_𝑘𝑢(𝑥)𝐷_𝑘𝑣(𝑥)𝑑𝜌(𝑥). (4.7)

Lemma 4.2. The bilinear form (4.7) is continuous on 𝑊¹₂(𝑅^∞).

Proof. For 𝑢,𝑣∈𝐶^∞_𝑐,0(𝑅^∞), |𝐵(𝑢,𝑣)|≤^∞_{𝑘=1𝑅∞}𝑎_𝑘(𝑥)𝐷_𝑘𝑢(𝑥)𝐷_𝑘𝑣(𝑥)𝑑𝜌(𝑥)≤max_𝑘∈ℕsup_{𝑥∈𝑅∞}𝑎_𝑘(𝑥)^∞_{𝑘=1𝑅∞}𝐷_𝑘𝑢(𝑥)𝐷_𝑘𝑣(𝑥)𝑑𝜌(𝑥)≤max_𝑘∈ℕsup_{𝑥∈𝑅∞}𝑎_𝑘(𝑥)^∞_{𝑘=1𝑅∞}𝐷_𝑘𝑢(𝑥)²𝑑𝜌(𝑥)^1/2_𝑅∞𝐷_𝑘𝑣(𝑥)²𝑑𝜌(𝑥)^1/2≤𝑐^∞_𝑘=1𝐷_𝑘𝑢_{𝐿2(𝑅∞,𝑑𝜌(𝑥))}𝐷_𝑘𝑣_{𝐿2(𝑅∞,𝑑𝜌(𝑥))}≤𝑐‖𝑢‖_𝑊¹_2(𝑅^∞₎‖𝑣‖_𝑊¹_2(𝑅^∞₎. (4.8) Thus 𝐵 has a continuous extension onto 𝑊¹₂(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‖²_{𝑊12(𝑅∞)}−𝑐₁‖𝑢‖²_𝐿2𝑅^∞_{(𝑥),𝑑𝜌} (4.9) holds for all 𝑢∈𝑊¹₂(𝑅^∞).

Proof. For 𝑢∈𝐶^∞_𝑐,0(𝑅^∞), 𝐵(𝑢,𝑢)=^∞_{𝑘=1𝑅∞}𝑎_𝑘(𝑥)𝐷_𝑘𝑢(𝑥)²(𝑥)=𝑑𝜌_∞𝑘=1𝑎_𝑘𝑥_0𝑅∞𝐷_𝑘𝑢(𝑥)²𝑑𝜌(𝑥)−^∞_{𝑘=1𝑅∞}𝑎_𝑘𝑥₀−𝑎_𝑘(𝑥)𝐷_𝑘𝑢(𝑥)²(𝑥)≥𝑑𝜌_∞𝑘=1𝑎_𝑘𝑥_0𝑅∞𝐷_𝑘𝑢(𝑥)²𝑑𝜌(𝑥)−^∞_{𝑘=1𝑅∞}𝑎_𝑘𝑥₀−𝑎_𝑘(𝑥)𝐷_𝑘𝑢(𝑥)²𝑑𝜌(𝑥), (4.10) and using conditions (1) and (2), 𝐵(𝑢,𝑢)≥𝑐₁^∞_{𝑘=1𝑅∞}𝐷_𝑘𝑢(𝑥)²𝑑𝜌(𝑥)−max_𝑘∈ℕsup_{𝑥∈𝑅∞}𝑎_𝑘𝑥₀−𝑎_𝑘(𝑥)^∞_{𝑘=1𝑅∞}𝐷_𝑘𝑢(𝑥)²𝑑𝜌(𝑥)≥𝑐₁‖𝑢‖²_{𝑊12(𝑅∞)}−‖𝑢‖²_𝐿2𝑅^∞_{(𝑥),𝑑𝜌}−𝑐₁2𝑛₀^∞_𝐾=1𝑅^∞𝐷_𝑘𝑢(𝑥)²𝑑𝜌(𝑥)≥𝑐₁‖𝑢‖²_{𝑊12(𝑅∞)}−‖𝑢‖²_𝐿2𝑅^∞_{(𝑥,𝑑𝜌)}−𝑐₁2𝑛₀‖𝑢‖²_{𝑊12(𝑅∞)}=𝑐₁11−2𝑛₀‖𝑢‖²_{𝑊12(𝑅∞)}−𝑐₁‖𝑢‖²_𝐿2𝑅^∞_{(𝑥,𝑑𝜌)}, (4.11) 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∈Γ}𝐷^𝛼(𝑎_𝛼𝛽(⋅)𝐷^𝛽), where Γ is finite, and nonempty collection of 𝛼=(𝛼₁,…,𝛼_𝑛), 𝛼_𝑖=1,2,…, and 𝑎_𝛼𝛽(𝛼,𝛽∈Γ×Γ) are real locally bounded functions on 𝑅^∞.