Carleman estimates and unique continuation property for abstract elliptic equations

The unique continuation theorems for elliptic differential-operator equations with variable coefficients in vector-valued L_p-space are investigated. The operator-valued multiplier theorems, maximal regularity properties and the Carleman estimates for the equations are employed to obtain these results. In applications the unique continuation theorems for quasielliptic partial differential equations and finite or infinite systems of elliptic equations are studied.

AMS: 34G10; 35B45; 35B60.

The aim of this article, is to present a unique continuation result for solutions of a differential inequalities of the form:

where

here a_ij are real numbers, A = A (x), A_k = A_k (x) and V (x) are the possible linear operators in a Banach space E.

Jerison and Kenig started the theory of L_p Carleman estimates for Laplace operator with potential and proved unique continuation results for elliptic constant coefficient operators in [1]. This result shows that the condition V ∈ L_n/2,loc is in the best possible nature. The uniform Sobolev inequalities and unique continuation results for second-order elliptic equations with constant coefficients studied in [2]. This was latter generalized to elliptic variable coefficient operators by Sogge in [3]. There were further improvement by Wolff [4] for elliptic operators with less regular coefficients and by Koch and Tataru [5] who considered the problem with gradients terms. A comprehensive introductions and historical references to Carleman estimates and unique continuation properties may be found, e.g., in [5]. Moreover, boundary value problems for differential-operator equations (DOEs) have been studied extensively by many researchers (see [6-18] and the references therein).

In this article, the unique continuation theorems for elliptic equations with variable operator coefficients in E-valued L_p spaces are studied. We will prove that if , , V ∈ L_μ (Rⁿ; L(E)), p, μ∈ (1, ∞) and satisfies (1), then u is identically zero if it vanishes in a nonempty open subset, where is an E-valued Sobolev-Lions type space. We prove the Carleman estimates to obtain unique continuation. Specifically, we shall see that it suffices to show that if , then

In the Hilbert space L₂ (Rⁿ; H), we derive the following Carleman estimate

Any of these inequalities would follow from showing that the adjoint operator L_t (x; D) = e^twL (x; D) e^-tw satisfies the following relevant local Sobolev inequalities

uniformly to t, where L_0t = e^twL₀e^-tw. In application, putting concrete Banach spaces instead of E and concrete operators instead of A, we obtain different results concerning to Carleman estimates and unique continuation.

Let R and C denote the sets of real and complex numbers, respectively. Let

Let E and E₁ be two Banach spaces, and L (E, E₁) denotes the spaces of all bounded linear operators from E to E₁. For E₁ = E we denote L (E, E₁) by L (E). A linear operator A is said to be a φ-positive in a Banach space E with bound M > 0 if D (A) is dense on E and

with λ ∈ S_φ, φ ∈ (0, π], I is identity operator in E. We will sometimes use A + ξ or A_ξ instead of A + ξI for a scalar ξ and (A + ξI)^-1 denotes the inverse of the operator A + ξI or the resolvent of operator A. It is known [19, §1.15.1] that there exist fractional powers A^θ of a positive operator A and

We denote by L_p (Ω; E) the space of all strongly measurable E-valued functions on Ω with the norm

By L_p,q (Ω) and let us denoted, respectively, the (p, q)-integrable function space and Sobolev space with mixed norms, where 1 ≤ p, q < ∞, see [20].

Let E₀ and E be two Banach spaces and E₀ is continuously and densely embedded E.

Let l be a positive integer.

We introduce an E-valued function space (sometimes we called it Sobolev-Lions type space) that consist of all functions u ∈ L_p (Ω; E₀) such that the generalized derivatives are endowed with the

The Banach space E is called an UMD-space if the Hilbert operator dy is bounded in L_p (R, E), p ∈ (1, ∞) (see e.g., [21,22]). UMD spaces include, e.g., L_p, l_p spaces and Lorentz spaces L_pq, p, q ∈ (1, ∞).

Let E₁ and E₂ be two Banach spaces. Let S (Rⁿ; E) denotes a Schwartz class, i.e., the space of all E-valued rapidly decreasing smooth functions on Rⁿ. Let F and F^-1denote Fourier and inverse Fourier transformations, respectively. A function Ψ ∈ C^m (Rⁿ; L (E₁, E₂)) is called a multiplier from L_p (Rⁿ; E₁) to L_q (Rⁿ; E₂) for p, q ∈ (1, ∞) if the map u → Ku = F^-1 Ψ (ξ) Fu, u ∈ S (Rⁿ; E₁) is well defined and extends to a bounded linear operator

We denote the set of all multipliers from L_p (Rⁿ; E₁) to L_q (Rⁿ; E₂) by . For E₁ = E₂ = E and q = p we denote by M_p (E). The L_p-multipliers of the Fourier transformation, and some related references, can be found in [19, § 2.2.1-§ 2.2.4]. On the other hand, Fourier multipliers in vector-valued function spaces, have been studied, e.g., in [23-28].

A set K ⊂ L (E₁, E₂) is called R-bounded [22,23] if there is a constant C such that for all T₁, T₂, . . . , T_m ∈ K and u_1,u₂, . . . , u_m ∈ E₁, m ∈ N

where {r_j} is a sequence of independent symmetric {-1, 1}-valued random variables on [0,1]. The smallest C for which the above estimate holds is called a R-bound of the collection K and denoted by R (K).

Let

For any r = (r₁, r₂, . . . , r_n), r_i ∈ [0, ∞) the function (iξ)^r, ξ ∈ Rⁿ will be defined such that

where

Definition 2.1. The Banach space E is said to be a space satisfying a multiplier condition with respect to p, q ∈ (1, ∞) (with respect to p if q = p) when for Ψ ∈ C⁽ⁿ⁾ (Rⁿ; L (E₁, E₂)) if the set

is R-bounded, then .

Definition 2.2. The φ-positive operator A is said to be a R-positive in a Banach space E if there exists φ ∈ [0, π) such that the set

is R-bounded.

Remark 2.1. By virtue of [29] or [30] UMD spaces satisfy the multiplier condition with respect to p ∈ (1, ∞).

Note that, in Hilbert spaces every norm bounded set is R-bounded. Therefore, in Hilbert spaces all positive operators are R-positive. If A is a generator of a contraction semigroup on L_q, 1 ≤ q ≤ ∞ [31], A has the bounded imaginary powers with , or if A is a generator of a semigroup with Gaussian bound in E ∈ UMD then those operators are R-positive (see e.g., [24]).

It is well known (see e.g., [32]) that any Hilbert space satisfies the multiplier condition with respect to p ∈ (1, ∞). By virtue of [33] Mikhlin conditions are not sufficient for operator-valued multiplier theorem. There are however, Banach spaces which are not Hilbert spaces but satisfy the multiplier condition (see Remark 2.1).

Let , be a collection of multipliers in . We say that H_k is a uniform collection of multipliers if there exists a constant M > 0, independent on h ∈ K, such that

for all h ∈ K and u ∈ S (Rⁿ; E₁).

We set

In view of [17, Theorem A₀], we have

Theorem 2.0. Let E₁ and E₂ be two UMD spaces and let

If

uniformly with respect to h ∈ K then Ψ_h (ξ) is a uniformly collection of multipliers from L_p (Rⁿ; E₁) to L_q (Rⁿ; E₂).

Let

Embedding theorems in Sobolev-Lions type spaces were studied in [13-18,32,34]. In a similar way as [17, Theorem 3] we have

Theorem 2.1. Suppose the following conditions hold:

(1) E is a Banach space satisfying the multiplier condition with respect to p, q ∈ (1, ∞) and A is a R-positive operator on E;

(2) l is a positive and α_k are nonnegative integer numbers such that 0 ≤ μ ≤ 1 - ϰ, t and h are positive parameters.

Then the embedding

is continuous and there exists a positive constant C_µ such that for

the uniform estimate holds

Moreover, for the following uniform estimate holds

Consider at first the equation with constant coefficients

where and A is the possible unbounded operator in a Banach space E.

Let and t is a positive parameter.

Remark 3.1. It is clear to see that

where . Let L_0t (x, ξ) is the principal operator symbol of L_0t (x, D) on the domain B₀, i.e.,

where

Our main aim is to show the following result:

Remark 3.2. Since Q(ξ) ∈ S (φ) for all φ ∈ [0, π) due to positivity of A, the operator function A + |ξ^||², ξ ∈ Rⁿ is uniformly positive in E. So there are fractional powers of A+|ξ^||² and the operator function is positive in E (see e.g., [19, §1. 15.1]).

First, we will prove the following result.

Theorem 3.1. Suppose A is a positive operator in a Hilbert space H. Then the following uniform Sobolev type estimate holds for the solution of Equation (3)

By virtue of Remark 3.1 it suffices to prove the following uniform coercive estimate

for .

To prove the Theorem 3.1, we shall show that L_0t (x, D) has a right parametrix T, with the following properties.

Lemma 3.1. For t > 0 there are functions K = K_t and R = R_t so that

where δ denotes the Dirac distribution. Moreover, if we let T = T_t be the operator with kernel K, i.e.,

and R is the operator with kernel R (x, y), then for large t > 0, the adjoint of these operators satisfy the following estimates

Proof. By Remark 3.2 the operator function is positive in E for all ξ ∈ Rⁿ. Since tw₁ + iξ₁ ∈ S(φ), due to positivity of A, for the factor has a bounded inverse for all ξ ∈ Rⁿ, t > 0 and

Therefore, we call G_t (x, ξ) the regular factor. Consider now the second factor

By virtue of operator calculus and fractional powers of positive operators (see e.g., [19, §1.15.1] or [35]) we get that - [tw₁ + iξ₁] ∉ S (φ) for ξ₁ = 0 and tw₁ = |ξ^||, i.e., the operator B_t (x, ξ) does not has an inverse, in the following set

So we will called B_t the singular factor and the set Δ_t call singular set for the operator function B_t. The operator cannot be bounded in the set Δ_t. Nevertheless, the operator , and hence , can be bounded when (x, ξ) is sufficiently far from Δ_t. For instance, if we define

by properties of positive operators we will get the same estimate of type (10) for the singular factor B_t. Hence, using this fact and the resolvent properties of positive operators we obtain the following estimate

where the constant C is independent of x, ξ, t and ^cΓ_t denotes the complement of Γ_t.

Let such that, β(ξ) = 0 if and β (ξ) = 0 near the origin. We then define

and notice that β₀ (ξ) = 0 on Γ_t. Hence, if we define

and recall (11), then by [31] it follows from standard microlocal arguments that

where R_0t belongs to a bounded subset of S^-1 which is independent of t. Since operator also has the same property, it follows that for all

By reasoning as in [31] we get that tR_0t belongs to a bounded subset of S⁰. So, we have the following estimate

Moreover, the Remark 3.2, positivity properties of A and, (11) and (12) imply that, the operator functions and are uniformly bounded. Then, if we let T₀ be the operator with kernel K₀ (x, y), by using the Minkowski integral inequality and Plancherel's theorem we obtain

For inverting L_0t (x, D) on the set Γ_t we will require the use of Fourier integrals with complex phase. Let β₁ (ξ) = 1 - β₀ (ξ). We will construct a Fourier integral operator T₁ with kernel

so that the analogs of (16) and the estimates (7)-(9) are satisfied. Since is uniformly bounded on Γ_t, we should expect to construct the phase function Φ in (13) using the factor B_t (x, ξ). Specifically, we would like Φ to satisfy the following equation

The Equation (14) leads to complex eikonal equation (i.e., a non-linear partial differential equation with complex coefficients).

Since w₁ (x) = 1 + x₁, w₁ (y) = 1 + y₁, we have

is a solution of (15). To use this we get

Next, if we set

then it follows from L_0t (x, ξ) = G_t (x, ξ)B_t (x, ξ) and (14) that

Consequently, (16)-(18) imply that

By reasoning as in [3] we obtain that the first and second summands in (19) belong to a bounded subset of S⁰. So, we see that the equality (5) must hold. Now we let K (x, y) = K₀ (x, y) + K₁ (x, y) and R (x, y) = R₀ (x, y) + R₁ (x, y), where

Due to regularity of kernels, by using of Minkowski and Hölder inequalities we get the analog estimate as (7) and (9) for the operators T₀ and R₁₀. Thus, in order to finish the proof, it suffices to show that for f ∈L₂ (B₀; E) one has

However, since R_1,1 ≈ tT₁, we need only to show the following

By using the Minkowski inequalities we get

where . The estimates (13) and (16) imply that

where

Consequently, it follows from Plancherel's theorem that

Note that for every N we have

Since A is a positive operator in E, we have

when . Then by using the above estimate it not easy to check that

i.e.,

Moreover, it is clear that

Thus from (24) by using the above relations and Young's inequality we obtain the desired estimate

Moreover, by using the estimate (10) and the resolvent properties of the positive operator A we have

The last two estimates then, imply the estimates (20)-(22).

Proof of Theorem 3.1: The estimates (7)-(9) imply the estimate (5), i.e., we obtain the assertion of the Theorem 3.1.

Consider the following DOE

where and A is the possible unbounded operator in a Banach space E and a_ij are

real-valued smooth functions in B_ε = {x ∈ Rⁿ, |x| < ε}.

Condition 4.1. There is a positive constant γ such that for all ξ ∈ Rⁿ, x ∈

The main result of the section is the following

Theorem 4.1. Let E be a Banach space satisfies the multiplier condition and A be a R-positive operator in E. Suppose the Condition 4.1 holds, n ≥ 3, p = and p' is the conjugate of p, w = and a_ij ∈ C^∞ (B_ε)_. Then for ( B_ε; E(A)) and the following estimates are satisfied:

Proof. As in the proof of Theorem 3.1, it is sufficient to prove the following estimates

where,

Consequently, since w₁ ≃ 1 on B_ε, it follows that, if we let Q_t (εx, D) be the differential operator whose adjoint equals

then it suffices to prove the following

The desired estimates will follow if we could constrict a right operator-valued parametrix T, for Q_t* (εx, D) satisfying L_p estimates. these are contained in the following lemma.

Lemma 4.1. For t > 0 there are functions K = K_t and R = R_t, so that

where δ denotes the Dirac distribution. Moreover, if we let T = T_t be the operator with kernel K (x, y) and R be the operator with kernel R (x, y), then if ε and are sufficiently small, the adjoint of these operators satisfy the following uniform estimates

Proof. The key step in the proof is to find a factorization of the operator-valued symbol that will allow to microlocally invert near the set where vanishes. Note that, after making a suitable choice of coordinates, it is enough to show that if L (x, D) is of the form

therefore, we can expressed as

where

where

The ellipticity of Q(x, D) and the positivity of the operator A, implies that the factor G_t (x, ξ) never vanishes and as in the proof of Theorem 3.1 we get that

i.e., the operator function G_t (εx, ξ) has uniformly bounded inverse for (x, ξ) ∈ B_ε ×Rⁿ. One can only investigate the factor B_t (εx, ξ). In fact, if we let

then the operator function B_t (x, ξ) is not invertible for (x, ξ) ∈ Δ_t. Nonetheless, B_t (εx, ξ) and can be have a bounded inverse when (x, ξ) is sufficiently far away. For instance, if we define

by properties of positive operators we will get the same estimate of type (37) for the singular factor B_t. Hence, we using this fact and the resolvent properties of positive operators we obtain the following estimate

As in § 3, we can use (38) to microlocallity invert away from Γ_t . To do this, we first fix as in § 3. We then define

It is clear that β₀ (ξ) = 0 on Γ_t. Consequently, if we define

and recall (37), then we can conclude that standard microlocal arguments give that

where R₀ belongs to a bounded subset of S^-1 that independent of t. Since the adjoint operator also is abstract pseudodifferential operator with this property, by reasoning as in [31, Theorem 6] it follows that

Moreover, the positivity properties of A and the estimate (38) imply that the operator functions are uniformly bounded. Next, let T₀ be the operator with kernel K₀. Then in a similar way as in [31] we obtain that

which also the first estimate is stronger than the corresponding inequality in Lemma 4.1. Finally, since T₀ ∈ S^-2 and it follows from imbedding theorem in abstract Sobolev spaces [17] that

Thus, we have shown that the microlocal inverse corresponding to ^cΓ_t, satisfies the desired estimates.

Let β₁ (ξ) = 1-β₀ (ξ). To invert for (x, ξ) ∈ Γ_t, we have to construct a Fourier integral operator T₁, with kernel

such that the analogs of (39) and (32)-(35) are satisfied. For this step the factorization (36) of the symbol will be used. Since the factor G_t (εx, ξ) has a bounded inverse for (x, ξ) ∈ Γ_t, the previous discussions show that we should try to construct the phase function in (46) using the factor B_t (εx, ξ). We would like Φ (x, y, ξ) to solve the complex eikonal equation

Since B_t (εx, Φ_x) - B_t (εy, ξ) is a scalar function (it does not depend of operator A ), by reasoning as in [3, Lemma 3.4] we get that

where ϕ is real and defined as

while

and

Then we obtain from the above that

Next, if we set

then it follows from (36) and (48) that

for every N when β₁ (ξ) ≠ 0. Consequently, (49), (50) imply that

By reasoning as in Theorem 3.1 we obtain from (51) that

where

while R₁₀ belongs to a bounded subset of S^-1 and tR₁₀ belongs to a bounded subset of S⁰. In view of this formula, we see that if we let K (x, y) = K₀ (x, y) + K₁ (x, y) and R (x, y) = R₀ (x, y)+R₁ (x, y), where R₁ = R₁₀ +R₁₁, then we obtain (31). Moreover, since R₁₀ satisfies the desired estimates, we see from Minkowski inequality that, in order to finish the proof of Lemma 4.1, it suffices to show that for

where .

To prove the above estimates we need the following prepositions for oscillatory integral in E-valued L_p spaces which generalize the Carleson and Sjolin result [36].

Preposition 4.1. Let E be Banach spaces and . Moreover, suppose Φ ∈ C^∞ satisfies | ∇Φ| ≥ γ > 0 on supp A. Then for all λ > 1 the following holds

where C_N-depends only on γ if Φ and A (x) belong to a bounded subset of C^∞ and C^∞ (Rⁿ, L (E)) and A is supported in a fixed compact set.

Proof. Given x₀ ∈ supp A. There is a direction ν ∈ S^n-1 such that |(ν, ∇Φ)| ≥ on some ball centered at x₀. Thus, by compactness, we can choose a partition of unity consisting of a finite number of terms and corresponding unit vectors ν_j such that on supp A and on supp φ_j. For A_j = φ_jA it suffices to prove that for each j

After possible changing coordinates we may assume that ν_j = (1, 0, . . . , 0) which means that on supp φ_j. If let , then . Consequently, if is a adjoint, then

Since our assumption imply that (L*)^N A_j (x) = O (λ^-N), the result follows.

Preposition 4.2. Suppose Φ ∈ C^∞ is a phase function satisfying the non-degeneracy condition det on the support of

Then for the following estimates hold

Proof. In view of [3, Remark 2.1] we have

where |y - z| is small. By using a smooth partition of unity we can decompose A (x, y) into a finite number of pieces each of which has the property that (57) holds on its support. So, by (57) we can assume

on supp A for same C > 0. To use this we notice that

where

Hence, by virtue of Preposition 4.1 and by (58) we obtain that

Consequently, by Young's inequality, the operator with kernel K_λ acts

By (59) we get that

Moreover, it is clear to see that

Therefore, by applying Riesz interpolation theorem for vector-valued L_p spaces (see e.g., [19, § 1.18]) we get the assertion.

In a similar way as in [3, Preposition 3.6] we have.

Preposition 4.3. The kernel K₁ (x, y) can be written as

where, for every fixed N, the operator functions A_j satisfy

and moreover, the phase functions φ_j are real and the property that when ε is small enough, 0 < δ ≤ ε and y₁ ∈ [-ε, ε] is fixed, the dilated functions

in the some fixed neighborhood of the function in the C^∞ topology. Then, the following estimates holds

Proof. By representation of K₁ (x, y) and Φ (x, y, ξ) we have

Then, by using (36) in view of positivity of operator A, by reasoning as in [3, Preposition 3.6] we obtain the assertion.

Let us now show the end of proof of Lemma 4.1. Let be supported in such that and set . Then we define kernels K_1,ν, ν = 0, 1, 2, . . . , as follows

Let T_1,ν denotes the operators associated to these kernels. Then, by positivity properties of the operator A and by Prepositions 4.2, 4.3 we obtain for the following estimates

By summing a geometric series one sees that these estimates imply (52) and (53) for case of α = 0.

Let us first to show (60). One can check that the estimate (59) implies that the L_r norm of is O (t^n-2t ^-n/r). But, if we let r = n/n - 2, it is follows from Young inequality and the fact that that

as desired. To prove the result for ν > 0, set and let be the kernel of the operator . Then, if we fix x₁ and y₁, it follows that the norm of the operator

equal times the norm of the dilated operator

where δ = 2^ν t^-1. By Preposition 4.3, the kernel in last integral equals the complex conjugate of

and, consequently by using the Proposition 4.2, for 0 < δ ≤ ε and for supp we obtain that

This estimate implies

For we set

Then, the desired estimate (60) follows from the above estimate and Young's inequality. The other inequality (61), follows from a similar argument.

Preposition 4.4. The estimates (32)-(34) imply (30).

Proof. Indeed, (31) implies that

and so Minkowski's inequality, (32) and (34) give that

which implies that the first inequality in (30) for sufficiently large t. Moreover, in a similar way, using (32) and (33) we get (30) for α = 0. To prove (30) for |α| = 1, we use (33), (34) and obtain

Hence, the result follows.

Now we can show the end of the proof of Theorem 4.1. Really, we obtain the estimate (30), which implies the estimates (26) and (27). That is the assertion of Theorem 4.1 is hold.

Theorem 4.2. Assume all conditions of Theorem 4.1 are satisfied, then for if and then u is identically 0 if it vanishes in a nonempty open subset.

Proof. Suppose

in a connected open set G, where and . Then, after the possibly change of variables, one sees that Theorem 4.2 would follow if we could show that if

then 0 ∉supp u. Moreover, by making a proper choice of geodesic coordinate system, we may assume L (x, D) as

Then argue as in [29], first set u_ε (x) = u (εx) where ε is chosen small enough so that (26) and (27) hold for B_ε. Let be equal to one when |x| < and set U_ε = ηu_ε. Then if V_ε (x) = V (εx) and

which implies that

Let

If the condition (63) holds, then we can always choose δ to be small enough that

and so that if C is as in (26), (27) and C₀ is as in (64) then

Next, (26), (27) imply

If we recall that , then we see that (64) and Hölder's inequality imply

Thus, by (63) for sufficiently large t > 0 and we can conclude that

finally, since w' (x) = 1 + x₁> 0 on B_ε , this forces U_ε (x) = 0 for x ∈ S_δ and so 0 ∉ supp u which completes the proof.

Consider the differential operator

where a_ij are real-valued functions numbers, A = A (x), A_k = A_k (x), V (x) are the possible linear operators in a Banach space E.

By using Theorem 4.2 and perturbation theory of linear operators we obtain the following result

Theorem 4.3. Assume:

(1) all conditions of Theorem 4.1 are satisfied;

(2)

Then, for D^αu ∈ L_p,loc (B₀; E) if ||P (x, D) u||_E ≤ ||Vu||_E and , then u is identically 0 if it vanishes in a nonempty open subset.

Proof. By condition (2) and by Theorem 2.1, for all ε > 0 there is a C (ε) such that

Then, by using (29) and the above estimate we obtain the assertion.

Let Ω ⊂ R^l be an open connected set with compact C^2m-boundary ∂Ω. Let us consider the BVP for the following elliptic equation

where u = (x, y), , . Let .

Let Q denotes the operator generated by the problem (64), (65).

Theorem 5.1. Let the following conditions be satisfied;

(1) for each |α| = 2m and for each |α| = k < 2m with r_k ≥ q and

(2) b_jβ ∈ C^2m-mj (∂Ω) for each j, β and m_j < 2m, , for |β| = m_j, y^|∈ ∂G, where σ = (σ₁, σ₂, . . . , σ_n) ∈ R^m is a normal to ∂G ;

(3) for let

(4) for each y₀ ∈ ∂Ω local BVP in local coordinates corresponding to y₀

has a unique solution ϑ ∈C₀ (R₊) for all h = (h₁, h₂, . . . , h_m) ∈ R^m, and for ξ¹ ∈ R^l-1 with

(5) Condition 4.1 holds, a_ij ∈ C^∞ (B_ε), n ≥ 3, and p' is the conjugate of p and

(6) d_k ∈ L_∞ (Rⁿ × Ω).

Then:

(a) for sufficiently large b > 0, t ≥ t₀ and for the Carleman type estimate

holds for .

(b) for and the differential inequality

has a unique continuation property.

Proof. Let E = L_q (Ω). Consider the following operator A which is defined by

For x ∈ Rⁿ also consider operators

The problem (5.1), (5.2) can be rewritten in the form (4.1), where u (x) = u (x, .), f (x) = f (x, .) are functions with values in E = L_q (Ω). Then by virtue of [24, Theorems 3.6 and 8.2] the (1) condition of Theorem 4.1 is satisfied. Moreover, by using the embedding and interpolation properties of Sobolev spaces (see e.g., [19, §4]) we get that there is ε > 0 and a continuous function C (ε) such that

Due to positive of the operator A, then we obtain that

Then it is easy to get from the above estimate that (2) condition of the Theorem 4.3 is satisfied. By virtue of (5) condition, (2) condition of the Theorem 4.1 is fulfilled too. Hence, by virtue of Theorems 4.1 and 4.3 we obtain the assertions.

Consider the following infinity systems of PDE

Let

Let O denotes the operator generated by the problem (66).

Theorem 6.1. Let the following conditions are satisfied:

(1) a_k ∈ C_b (Rⁿ), a_k (x) ≠ 0, x ∈ Rⁿ, k = 1, 2, . . . , n and the Condition 4.1 holds;

(2) there are 0 < ν < such that

a.e. for x ∈ Rⁿ.

Then:

(a) for sufficiently large b > 0, t ≥ t₀ and for the Carleman type estimate

holds for .

(b) for and the differential inequality

has a unique continuation property.

Proof. Let E = l_q and A, A_k (x) be infinite matrices, such that

It is clear to see that this operator A is R-positive in l_q and all other conditions of Theorems 4.1 and 4.3 are hold. Therefore, by virtue of Theorems 4.1 and 4.3 we obtain the assertions.

The author declares that they have no competing interests.

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