Abstract
The unique continuation theorems for elliptic differentialoperator equations with variable coefficients in vectorvalued L_{p}space are investigated. The operatorvalued 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.
Keywords:
Carleman estimates; unique continuation; embedding theorems; Banachvalued function spaces; differential operator equations; maximal L_{p}regularity; operatorvalued Fourier multipliers; interpolation of Banach spaces1 Introduction
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 secondorder 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 A comprehensive introductions and historical references to Carleman estimates and unique continuation properties may be found, e.g., in [5].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 differentialoperator equations (DOEs) have been studied extensively by many researchers (see [618] and the references therein).
In this article, the unique continuation theorems for elliptic equations with variable operator coefficients in Evalued L_{p }spaces are studied. We will prove that if , , V ∈ L_{μ }(R^{n}; L(E)), p, μ∈ (1, ∞) and satisfies (1), then u is identically zero if it vanishes in a nonempty open subset, where is an Evalued SobolevLions 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 L_{2 }(R^{n}; H), we derive the following Carleman estimate
Any of these inequalities would follow from showing that the adjoint operator L_{t }(x; D) = e^{tw}L (x; D) e^{tw }satisfies the following relevant local Sobolev inequalities
uniformly to t, where L_{0t } = e^{tw}L_{0}e^{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.
2 Notations, definitions, and background
Let R and C denote the sets of real and complex numbers, respectively. Let
Let E and E_{1 }be two Banach spaces, and L (E, E_{1}) denotes the spaces of all bounded linear operators from E to E_{1}. For E_{1 }= E we denote L (E, E_{1}) by L (E). A linear operator A is said to be a φpositive 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 Evalued 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 E_{0 }and E be two Banach spaces and E_{0 }is continuously and densely embedded E.
Let l be a positive integer.
We introduce an Evalued function space (sometimes we called it SobolevLions type space) that consist of all functions u ∈ L_{p }(Ω; E_{0}) such that the generalized derivatives are endowed with the
The Banach space E is called an UMDspace 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 E_{1 }and E_{2 }be two Banach spaces. Let S (R^{n}; E) denotes a Schwartz class, i.e., the space of all Evalued rapidly decreasing smooth functions on R^{n}. Let F and F^{1}denote Fourier and inverse Fourier transformations, respectively. A function Ψ ∈ C^{m }(R^{n}; L (E_{1}, E_{2})) is called a multiplier from L_{p }(R^{n}; E_{1}) to L_{q }(R^{n}; E_{2}) for p, q ∈ (1, ∞) if the map u → Ku = F^{1 }Ψ (ξ) Fu, u ∈ S (R^{n}; E_{1}) is well defined and extends to a bounded linear operator
We denote the set of all multipliers from L_{p }(R^{n}; E_{1}) to L_{q }(R^{n}; E_{2}) by . For E_{1 }= E_{2 }= 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 vectorvalued function spaces, have been studied, e.g., in [2328].
A set K ⊂ L (E_{1}, E_{2}) is called Rbounded [22,23] if there is a constant C such that for all T_{1}, T_{2}, . . . , T_{m }∈ K and u_{1,}u_{2}, . . . , u_{m }∈ E_{1}, 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 Rbound of the collection K and denoted by R (K).
Let
For any r = (r_{1}, r_{2}, . . . , r_{n}), r_{i }∈ [0, ∞) the function (iξ)^{r}, ξ ∈ R^{n }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) }(R^{n}; L (E_{1}, E_{2})) if the set
Definition 2.2. The φpositive operator A is said to be a Rpositive in a Banach space E if there exists φ ∈ [0, π) such that the set
is Rbounded.
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 Rbounded. Therefore, in Hilbert spaces all positive operators are Rpositive. 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 Rpositive (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 operatorvalued 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 (R^{n}; E_{1}).
We set
In view of [17, Theorem A_{0}], we have
Theorem 2.0. Let E_{1 }and E_{2 }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 }(R^{n}; E_{1}) to L_{q }(R^{n}; E_{2}).
Let
Embedding theorems in SobolevLions type spaces were studied in [1318,32,34]. 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 Rpositive 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
3 Carleman estimates for DOE
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 B_{0}, 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 ∈ R^{n }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
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 ∈ R^{n}. Since tw_{1 }+ iξ_{1 }∈ S(φ), due to positivity of A, for the factor has a bounded inverse for all ξ ∈ R^{n}, 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  [tw_{1 }+ iξ_{1}] ∉ S (φ) for ξ_{1 }= 0 and tw_{1 }= 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 ^{c}Γ_{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 β_{0 }(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 S^{0}. 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 T_{0 }be the operator with kernel K_{0 }(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 β_{1 }(ξ) = 1  β_{0 }(ξ). We will construct a Fourier integral operator T_{1 }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 nonlinear partial differential equation with complex coefficients).
Since w_{1 }(x) = 1 + x_{1}, w_{1 }(y) = 1 + y_{1}, 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 S^{0}. So, we see that the equality (5) must hold. Now we let K (x, y) = K_{0 }(x, y) + K_{1 }(x, y) and R (x, y) = R_{0 }(x, y) + R_{1 }(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 T_{0 }and R_{10}. Thus, in order to finish the proof, it suffices to show that for f ∈L_{2 }(B_{0}; E) one has
However, since R_{1,1 }≈ tT_{1}, 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.
4 L_{p}Carleman estimates and unique continuation for equation with variable coefficients
Consider the following DOE
where and A is the possible unbounded operator in a Banach space E and a_{ij }are
realvalued smooth functions in B_{ε }= {x ∈ R^{n}, x < ε}.
Condition 4.1. There is a positive constant γ such that for all ξ ∈ R^{n}, 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 Rpositive 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 w_{1 }≃ 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 operatorvalued 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 operatorvalued 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 }×R^{n}. 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 β_{0 }(ξ) = 0 on Γ_{t}. Consequently, if we define
and recall (37), then we can conclude that standard microlocal arguments give that
where R_{0 }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 T_{0 }be the operator with kernel K_{0}. 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 T_{0 }∈ S^{2 }and it follows from imbedding theorem in abstract Sobolev spaces [17] that
Thus, we have shown that the microlocal inverse corresponding to ^{c}Γ_{t}, satisfies the desired estimates.
Let β_{1 }(ξ) = 1β_{0 }(ξ). To invert for (x, ξ) ∈ Γ_{t}, we have to construct a Fourier integral operator T_{1}, 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 β_{1 }(ξ) ≠ 0. Consequently, (49), (50) imply that
By reasoning as in Theorem 3.1 we obtain from (51) that
where
while R_{10 }belongs to a bounded subset of S^{1 }and tR_{10 }belongs to a bounded subset of S^{0}. In view of this formula, we see that if we let K (x, y) = K_{0 }(x, y) + K_{1 }(x, y) and R (x, y) = R_{0 }(x, y)+R_{1 }(x, y), where R_{1 }= R_{10 }+R_{11}, then we obtain (31). Moreover, since R_{10 }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
To prove the above estimates we need the following prepositions for oscillatory integral in Evalued 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^{∞ }(R^{n}, L (E)) and A is supported in a fixed compact set.
Proof. Given x_{0 }∈ supp A. There is a direction ν ∈ S^{n1 }such that (ν, ∇Φ) ≥ on some ball centered at x_{0}. 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_{j}A 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 nondegeneracy 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 vectorvalued 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 K_{1 }(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 y_{1 }∈ [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 K_{1 }(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^{n2}t ^{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 x_{1 }and y_{1}, 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
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 C_{0 }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 + x_{1}> 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 realvalued 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;
Then, for D^{α}u ∈ L_{p,loc }(B_{0}; 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.
5 Carleman estimates and unique continuation property for quasielliptic PDE
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
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^{2mmj }(∂Ω) for each j, β and m_{j }< 2m, , for β = m_{j}, y^{}∈ ∂G, where σ = (σ_{1}, σ_{2}, . . . , σ_{n}) ∈ R^{m }is a normal to ∂G ;
(4) for each y_{0 }∈ ∂Ω local BVP in local coordinates corresponding to y_{0}
has a unique solution ϑ ∈C_{0 }(R_{+}) for all h = (h_{1}, h_{2}, . . . , h_{m}) ∈ R^{m}, and for ξ^{1 }∈ R^{l1 }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_{∞ }(R^{n }× Ω).
Then:
(a) for sufficiently large b > 0, t ≥ t_{0 }and for the Carleman type estimate
(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 ∈ R^{n }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.
6 Carleman estimates and unique continuation property for infinite systems of elliptic equations
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 }(R^{n}), a_{k }(x) ≠ 0, x ∈ R^{n}, k = 1, 2, . . . , n and the Condition 4.1 holds;
(2) there are 0 < ν < such that
a.e. for x ∈ R^{n}.
Then:
(a) for sufficiently large b > 0, t ≥ t_{0 }and for the Carleman type estimate
(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 Rpositive 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.
Competing interests
The author declares that they have no competing interests.
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