A NOTE ON WEIGHTED SOBOLEV SPACES RELATED TO WEAKLY AND STRONGLY DEGENERATE DIFFERENTIAL OPERATORS

In this paper we discuss some issues related to Poincaré’s inequality for a special class of weighted Sobolev spaces. A common feature of these spaces is that they can be naturally associated with differential operators with variable diffusion coefficients that are not uniformly elliptic. We give a classification of these spaces in the 1-D case bases on a measure of degeneracy of the corresponding weight coefficient and study their key properties.


Introduction
In this paper we discuss some issues related to Poincaré's inequality for a special class of weighted Sobolev spaces. A common feature of these spaces is that they can be naturally associated with differential operators with variable diffusion coefficients that are not uniformly elliptic, even though they are in general uniformly elliptic in compact subsets of the domain, even though they are in general uniformly elliptic in compact subsets of the space domain, provided that these subsets are at a positive distance from the the so-called zone of degeneracy. This degeneracy zone may occur either on a part of the boundary or on a sub-manifold of the space domain.
Some aspects of this problem with a degeneration at the boundary point x 0 = 0 of 1-D domain Ω = (0, 1) have been recently considered by Alabau-Boussouira, Cannarsa, and Leugering in [1]. In particular, if a ∈ C([0, 1]) ∩ C 1 ((0, 1]) is a given weight coefficient with properties a(x) > 0, ∀ x ∈ (0, 1] and a(0) = 0, (1.1) then the authors in [1] propose to measure the degree of degeneracy of the function a(·) at x = 0 by the parameter µ a which is defined as Moreover, they propose to use the parameter µ a as a main feature for the classification of the weight functions a : Ω → R. They say that a function a(·) has a weak degeneration at x 0 = 0 if µ a ∈ [0, 1), and this function is strongly degenerate if µ a > 1. It can be shown that any weakly degenerate functions a(·) (i.e., 0 ≤ µ a < 1) belongs to the class of Muckenhoupt weights A 2 (Ω), that is, w : Ω → R + belongs to A 2 (Ω) if and this case of degeneration has received a lot of attention in the literature (see, for instance, [4,6,[9][10][11][12]15,16]). At the same time, if µ a > 1, then a(·) ∈ A 2 and in this case we can expect to have many new effects related to the solvability issues of the corresponding boundary value problems and their properties. It is worth noting that such issues as controllability and observability of the corresponding degenerate systems are also closely related to the parameter µ a . In particular, it has been shown in [1] for degenerate wave equations of the form u tt − (a(x)u x ) x = 0 in (0, ∞) × (0, 1) that their observability and boundary controllability no longer hold true if µ a ≥ 2. The same conclusion can be done for the parabolic case (see, for instance, [2,3,7,13]). So, the authors in [1] provide analysis of the above mentioned properties assuming that µ a < 2 and for that they make use of the following weighted Sobolev spaces H 1 a (Ω) = u ∈ L 2 (Ω) u is locally absolutely continuous in (0, 1], √ au x ∈ L 2 (Ω) with norm and H 1 a,0 (Ω) := u ∈ H 1 a (Ω) : u(1) = 0 . It is well known that the loss of uniform ellipticity for operators like Au = − (a(x)u x ) x raises new questions related to the well-posedness of the corresponding evolution equations in suitable functional spaces as well as new estimates for the underlying elliptic equations. With that in mind, the authors in [1] have shown that the Poincaré's inequality for elements of weighted Sobolev space H 1 a,0 (Ω) can be established not only in the case of weakly degenerate weight function a(·), but also if µ a < 2. In particular, if µ a < 2 it has been shown in [1] that At the same time, in problems involving cloaking which, obviously, is incompatible with observability, the degeneracy of the coefficients is quadratic [8], and, hence, in this case we have µ a = 2. Therefore, the purpose of this paper is to study the issues related to the weighted Sobolev space H 1 a,0 (Ω) provided the degeneracy zone of the weight function a(·) is an interior point x 0 of the domain Ω and the measure of degeneracy at this point can be equal to 2 or larger than 2. As a sub-product of our analysis, we show that the classification of degeneracy measure of function a(·) essentially depends on the properties of its derivative a(x) x . In particular, for the weight functions a ∈ C([0, 1]) ∩ C 1 ((0, 1]) with properties (1.1), the following assertions hold true The paper is organized as follows. In Section 2, we introduce our notations, define the new degeneracy parameters A a,i and µ a,i , i = 1, 2, and derive some auxiliary inequalities for the weight function a : Ω → R. In Section 3, we prove the Poincaré's type inequalities for functions in weighted Sobolev space H 1 a,0 (Ω) in the case of weakly degenerate weight functions a(·). In particular, we show that in this case Poincaŕe's inequality (1.2) can be extended to the following one where H 1 a,0 (Ω) = u ∈ L 2 (Ω) : , 2 , , 2 + C Sob 1 + a 2 (1) , and parameters A i,a and µ i,a , i = 1, 2, are given by relations (2.2), (2.3), and (2.4), and satisfy conditions (3.4). We also show in this section that if ( √ a) −1 x ∈ L ∞ (Ω) then max{µ i,a , 2A i,a } < 2 and, hence, Poincaré's inequality (1.3) remains valid.
In Section 4, we proceed in the study of some key results for functions in weighted Sobolev space H 1 a,0 (Ω) provided the weight function a(·) has a large measure of degeneracy at some interior point x 0 ∈ (0, 1). In other words, our key assumption in this section is ( √ a) x ∈ L ∞ (Ω). In this case, we show that the weighted space H 1 a,0 (Ω) is isomorphic to the following one As a result, we derive another type of Poincaré's inequality for elements of H 1 a,0 (Ω). Namely, we establish the following relations (see Theorem 4.2) and Moreover, in this case we can not guarantee that elements of the space H 1 a,0 (Ω) are continuous functions in Ω or even integrable over this domain. Instead we can assert that the following implication holds true: If u ∈ H 1 a,0 (Ω) and (

Assumptions and Preliminaries
Let x 0 ∈ [0, 1] be a given point. We set We denote by C ∞ 0 (R) the locally convex space of all infinitely differentiable functions with compact support. Following the standard way, we define the Banach space W 1,2 0 (Ω; 0) as the closure of C ∞ 0 (R; 0) = {ϕ ∈ C ∞ 0 (R) : ϕ(0) = 0} with respect to the norm We also set C k,α (Ω) for the Hölder space of those functions on Ω having continuous derivatives up to order k and such that the kth derivative is Hölder continuous with exponent α ∈ (0, 1]. It is well known that C k,α (Ω) is a Banach space with respect to the norm and the embedding C k,α (Ω) → C k,0 (Ω) is compact. Let a : Ω → R be a given function with properties (i) a(x 0 ) = 0 and a(x) > 0 for all x ∈ Ω \ {x 0 }; (ii) a ∈ C(Ω) ∩ C 1 loc (Ω 0 ). In what follows, we associate with the function a : Ω → R the following degenerate elliptic operator (2.1) Let G i : Ω → [0, ∞), i = 1, 2, be non-decreasing continuous functions such that G i (0) = 0 and A 1,a := sup By analogy with [1-3], we also set µ 1,a := sup Example 2.1. As an example of function a : Ω → R + with the above indicated properties (i)-(ii), we can consider the following one (see [5,14]).
Here, x 0 = 1 2 . It is easy to check that, in this case, properties (i)-(ii) hold true. Moreover, setting G 1 (x) = k 1 x and G 2 (x) = k 2 x, where k 1 , k 2 are some positive constants, we see that In addition, we have the following properties √ a x ∈ L ∞ (Ω) if p 1 and p 2 are greater than 1, Another example of a weight function a : [0, 1] → R + with x 0 = 1 2 can be described as follows: After some calculus, we arrive at Before proceeding further, we list below some simple properties of function a : Ω → R related to the given characteristics A i,a and µ i,a .
are unknown a priori, we begin with the case when G i (s) ≥ s for all s ∈ Ω. Then the following relations are obvious. Therefore, making use of representation (2.4), we get Integrating this inequality over Arguing in a similar manner, we have It remains to consider the second case: From this, after integration over [0, x], we deduce Arguing in a similar manner, we have

Poincaré Inequality for a Weighted Sobolev Space
We now introduce some weighted Sobolev spaces that are naturally associated with functions a : Ω → R satisfying properties (i)-(ii) and with degenerate elliptic operators like (2.1) (see, for instance, [1, 13]). We denote by H 1 a (Ω) the following space of all functions u ∈ L 2 (Ω) such that It is easy to see that H 1 a (Ω) is a Hilbert space with the scalar product and associated norm Moreover, the Sobolev embedding theorem implies that H 1 We note that this subspace is correctly defined because the compactness of the embedding H 1 a,0 (Ω), then u(·) is a continuous function at x = 0, and, therefore, the condition u(0) = 0 is consistent.
Let us show that, because of the degeneration of the weight function a : Ω → R satisfying properties (i)-(ii), H 1 a,0 (Ω) is a Hilbert space with respect to the scalar product To do so, it is enough to establish some version of Poincaré inequality. We begin with the following observations. Assume that 0 ≤ max{2A 1,a , µ 1,a }, max{2A 2,a , µ 2,a } < 1. Proof. Let u be an arbitrary element of H 1 a (Ω). As follows from Lemma 2.1, if assumption (3.3) holds true, then 1/a(x) ∈ L 1 (Ω). Hence, in view of the representation and the Cauchy-Bunjakovski inequality, we see that the function u is summable over Ω. Hence, u(·) is absolutely continuous in Ω. Assume that the following conditions max{2A 1,a , µ 1,a } < 2 and max{2A 2,a , µ 2,a } < 2 (3.4) hold true. Then Proof. Let u be an arbitrary element of H 1 a,0 (Ω). Then, using direct arguments, for any x ∈ [0, x 0 ), we have .
From this and estimate (2.7), by Fubini's theorem, we obtain Arguing in a similar manner, for any x ∈ (x 0 , 1], we have Then, estimate (2.7) and Fubini's theorem yield the following bound .
Proof. We adapt a reasoning here that can be used to prove Hardy's inequality.
With that in mind, we observe that, for all x ∈ [0, x 0 ], the following transformation is validˆx Hence, From this, we deduce that Taking the limit as x ↑ x 0 in the last relation, we arrive at the estimate By analogy with the previous case, we make use of the following transformation which is valid for each As a result, passing to the limit in the last relation as x ↓ x 0 , we arrive at the following inequality Thus, the announced estimate (3.9) is a direct consequence of (3.11) and (3.13).
Before proceeding further, we notice that if u is an arbitrary element of the standard Sobolev space W 1,2 (d, 1) with d ∈ (x 0 , 1), then u(·) is an absolutely continuous function on [d, 1]. Moreover, by Sobolev embedding theorem, the injection W 1,2 (d, 1) → C 0,1 ([d, 1]) is continuous and there exists a constant C Sob > 0 such that max x∈ [d,1] |u(x)| ≤ C Sob u W 1,2 (d,1) , ∀ u ∈ W 1,2 (d, 1). (3.14) Taking this fact into account and fixing an arbitrary element u ∈ H 1 a,0 (Ω) and d ∈ (x 0 , 1), we note that u ∈ W 1,2 (d, 1) and Utilizing this estimate together with (3.14), we get by (3.14) ≤ C Sob u W 1,2 (d,1) Since d is an arbitrary point of the interval (x 0 , 1), we can pass to the limit in (3.15) as d ↑ 1. As a result, we obtain We are now in a position to prove the main result of this section. Namely, we establish some variant of Poincaré's (or Friedrich's) inequality for weighted Sobolev space H 1 a,0 (Ω) and derive the conditions when this inequality is consistent.
As an obvious consequence of this theorem, we can give the following conclusion.  ⊂ Ω 2 such that a(·) is monotonically decreasing on (x * 1 , x 0 ) and it is a monotonically increasing function on (x 0 , x * 2 ); Proof. Let a : Ω → R be a given function with properties (i)-(iv). Setting and a(x) := ka(x) for all x ∈ Ω, we see that the function a : Ω → R possesses all properties (i)-(iv) and the direct calculations show that µ i,a = µ i, a and A i,a = A i, a , i = 1, 2.
Moreover, in this case, we have Hence, without loss of generality, we can suppose that the function a : As a consequence of this condition, we have So, we can suppose that Since a(·) is a monotonically decreasing function on (x * 1 , x 0 ) and a(·) is a monotonically increasing function on (x 0 , x * 2 ), it follows from (3.23) that . Then, after integration, we obtain Taking into account that a(x 0 ) = 0, we deduce from (3.24)-(3.25) that , and, as a consequence, we have Utilizing the monotonicity property of a(·) around the point x 0 , we deduce from (3.26) that there exists a positive value γ ∈ (0, 2) such that that is, a(x) ∼ |x − x 0 | 2−γ near the degeneration point x 0 . Therefore, in view of representation (2.4), we have µ 1,a := sup Since for the functions a : Ω → R with property (3.27), we can set G i (x) = x, i = 1, 2, it follows from (2.2)-(2.3) and (3.28)-(3.29) that It remains to notice that initial assumption ( √ a) x = C * = const leads to the relation As a result, we have: µ i,a = 2 for i = 1, 2. Combining this fact with (3.30), we arrive at the inequalities (3.4).
Example 3.1. Let the degenerate weight a : Ω → R + be defined by the rule (2.5). Then Therefore, conditions (iii)-(iv) are satisfied with 0 < p 1 , p 2 < 1. Hence, and this statement can be approved by the direct calculations. At the same time, if the weight function a : [0, 1] → R + is defined as in (2.6), then ( √ a) −1 x ∈ L ∞ (Ω) for any p > 0. In this case, as it is indicated in Example 2.1, we have µ i,a = +∞ for i = 1, 2.
The fact that the element y(·) has zero trace at x = 0 is a direct consequence of (4.2) and definition of the space V a,0 (Ω) (4.1).
It is worth to emphasize that, in general, conditions (j)-(jv), that we postulate for the function a : Ω → R + , do not guarantee fulfillment of the inclusion V a,0 (Ω) ⊂ C(Ω) (see Proposition 3.1 for comparison). Indeed, in spite of the fact that the inclusion y ∈ V a,0 (Ω) implies the property ϕ := √ ay ∈ C 0,1/2 (Ω), we see that the function y = 1 √ a ϕ is absolutely continuous in Ω \ {x 0 } and, therefore, y(·) can have a gap at x = x 0 . So, if y ∈ V a,0 (Ω), then y ∈ C(Ω) provided Our next result concerns the comparison of the weighted space V a,0 (Ω) and the space that H 1 a (Ω) that has been considered in the previous section. It is clear that, in general, the spaces H 1 a (Ω) and V a,0 (Ω) differ from each other. However, due to (j)-(jv) properties of the weight function a(·), we can establish the following result. Proof. Let y be an element of H 1 a (Ω). Then taking into account the relation we see that Using the fact that ( √ a) x ∈ L ∞ (Ω), we arrive the estimate On the other hand, representation (4.5) implies that Hence, and announced equivalence of the norms follows.
The next observations are crucial for our further analysis. is the norm on W a (Ω).
Proof. As it follows from (4.8), the function · Wa(Ω) is subadditive and absolutely scalable, that is, · Wa(Ω) is a seminorm on W a (Ω). Let us show that this function is point-separating, i.e., the condition y Wa(Ω) = 0 guarantees the following equalities y(x) = 0 and y x (x) = 0 a.e. in Ω.
Since a(·) vanishes at a single point x = x 0 , we finally have y x (x) = 0 a.e. in Ω.
As for the Friedrich's inequality (4.9), we notice that in view of the following estimateˆΩ we see thatˆΩ Hence, the constant C F in (4.9) is equal to 4.
It is clear now that  Utilizing this representation, Theorem 4.1, and the Friedrich's inequality (4.9) with C F = 4 (see (4.14)), we can give the following conclusion. for all elements u ∈ H 1 a,0 (Ω).
It remains to take into account the representation (4.7).
Returning to representation (4.15), we make use of the following observations. If a : Ω → R is a function satisfying the properties (j)-(jv), then the equality holds true for all u ∈ H 1 a,0 (Ω) and x ∈ Ω \ {x 0 }. Therefore, (1 + C H )) u H 1 a (Ω) .
However, in view of the properties (j)-(jv), this estimate becomes consistent if only ( a(x)) x = const in Ω.
This motivates us to the following conclusion.
Proposition 4.2. Let a : Ω → R be a weight function which is defined as follows where C * > 0 is a given constant. Then and is an equivalent norm to the standard one in H 1 a,0 (Ω).
In conclusion, we would like to emphasize the following fact: If the weight function a(·) satisfies properties (j)-(jv), then the elements of the space H 1 a,0 (Ω) are not necessary continuous functions ( see Proposition 3.1 for comparison).
Example 4.1. Let x 0 = 0.5. Setting a(x) = |x − x 0 | 4 , we see that properties (j)-(jv) hold true. We define the following functions Then, in spite of the fact that the function y : Ω → R has a discontinuity of the second kind at x 0 = 1 2 , the direct calculations show that ( √ a) x ∈ L ∞ (Ω), u ∈ W a (Ω), and y ∈ V a,0 (Ω). Hence, y ∈ H 1 a,0 (Ω) by Theorem 4.1. At the same time, u(x) = a(x) y(x) is the absolutely continuous function in Ω.