Abstract
The paper is devoted to the theory of quasielliptic operators. We consider scalar and homogeneous quasielliptic operators \(\mathcal{L}(D_x)\) with lower terms in the whole space \({\mathbb R}^n\). Our aim is to study mapping properties of these operators in weighted Sobolev spaces. We introduce a special scale of weighted Sobolev spaces \(W^l_{p,q,\sigma }({\mathbb R}^n)\). These spaces coincide with known spaces of Sobolev type for some parameters l, q, \(\sigma \). We investigate mapping properties of the operators \(\mathcal{L}(D_x)\) in the spaces \(W^l_{p,q,\sigma }({\mathbb R}^n)\). We indicate conditions for unique solvability of quasielliptic equations and systems in these spaces, obtain estimates for solutions and formulate an isomorphism theorem for quasielliptic operators. To prove our results we construct special regularizers for quasielliptic operators.
G. Demidenko—The work is supported in part by the Program of the Presidium of the Russian Academy of Sciences (project no. 0314-2015-0011).
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1 Introduction
In the paper a class of quasielliptic operators \(\mathcal{L}(D_x)\) is considered in the whole space \({\mathbb R}^n\). This class belongs to the classes of quasielliptic operators introduced by S.M. Nikol’skii [1] and L.R. Volevich [2]. Our aim is to study mapping properties of the operators \(\mathcal{L}(D_x)\) in special weighted Sobolev spaces \(W^l_{p,q,\sigma }({\mathbb R}^n)\) and to establish isomorphism theorems.
The first isomorphism theorems for scalar elliptic operators were proved by L.A. Bagirov and V.A. Kondratiev [3], M. Cantor [4, 5], R.C. McOwen [6, 7]. Isomorphism theorems for matrix homogeneous elliptic operators were proved by Y. Choquet-Bruhat and D. Christodoulou [8], R.B. Lockhart and R.C. McOwen [9].
As a rule, isomorphism theorems for elliptic operators are not trivial. For example, consider the elliptic operator
where \(\triangle \) is the Laplace operator. If \(\varepsilon > 0\) then the mapping is an isomorphism. However, the mapping
is not an isomorphism. Taking into account results of L.D. Kudryavtsev [10], L.A. Bagirov and V.A. Kondratyev [3], L. Nirenberg and H.F. Walker [11], M. Cantor [4, 5], it is necessary to use weighted Sobolev spaces for proving isomorphism theorems for the Laplace operator. The first isomorphism theorems for the Laplace operator were proved by M. Cantor [4] and R.C. McOwen [6]. They used special weighted Sobolev spaces.
It should be noted that isomorphism theorems for matrix elliptic operators can be more complicated. For example, consider the Stokes operator
This operator is elliptic in the Douglis–Nirenberg sense. One can prove an isomorphism theorem for the Stokes operator (see [12]). However, it is necessary to use a product of special weighted Sobolev spaces with different components of smoothness vectors and different weights.
The first isomorphism theorems for matrix homogeneous quasielliptic operators were proved by G.V. Demidenko [13, 14]. The investigations [13, 14] were continued by G.N. Hile [15]. In the present paper we consider a more general class of quasielliptic operators \(\mathcal{L}(D_x)\) in \({\mathbb R}^n\).
2 Quasielliptic Operators
First we consider the following scalar differential operator
where the coefficients \(a_\beta \) are constants. Suppose that its symbol \(L(i\xi )\), \(\xi \in {\mathbb R}^n\), satisfies the following conditions.
Condition 1
The symbol \(L(i\xi )\) is homogeneous with respect to a vector \(\alpha = (\alpha _1,\ldots ,\alpha _n)\), \(1/\alpha _j \in {\mathbb N}\), \(j = 1,\ldots ,n\); i.e.,
Condition 2
The equality
holds if and only if \(\xi = 0\).
Definition 1
The differential operator \(L(D_x)\) is called quasielliptic, if its symbol satisfies Conditions 1, 2.
This class of operators belongs to the class of differential operators introduced by S.M. Nikol’skii [1].
Quasielliptic operators \(L(D_x)\) whose symbols \(L(i\xi )\) are homogeneous with respect to a vector \(\alpha \) are usually called quasielliptic operators without lower terms. Such operators can be written in the form
Examples of such operators are elliptic operators, 2b-parabolic operators without lower terms, etc.
Note that the symbol of the quasielliptic operator (1) satisfies the following estimate
where \(c_1\), \(c_2 > 0\) are constants.
We now consider the differential operators
where \(L(D_x)\) is the quasielliptic operator (1). We will call operators of the form (2) quasielliptic operators with lower terms. Denote the differential operator corresponding to the lower terms by
Condition 3
Suppose that the symbol of the differential operator (2) satisfies the estimate
where \(0 \le q < 1\), \(c_3\), \(c_4 > 0\) are constants.
Example 1
Consider the differential operator
We have
Obviously, Conditions 1–3 are fulfilled for \(q = k/m\).
We now consider the matrix differential operator
where the coefficients \(A_\beta \) are constant \((m\times m)\)-matrices with real or complex entries. Suppose that its symbol \(L(i\xi )\) satisfies the following condition.
Condition 4
The equality
holds if and only if \(\xi = 0\).
Definition 2
The matrix differential operator (5) is called homogeneous quasielliptic operator if its symbol satisfies Condition 4.
This class of operators belongs to the class of differential operators introduced by L.R. Volevich [2]. Examples of such operators are homogeneous elliptic operators, 2b-parabolic operators without lower terms, parabolic operators with ‘opposite times directions’, etc.
We now consider matrix differential operators of the form
where \(L(D_x)\) is the matrix quasielliptic operator of the form (5), the coefficients \(A'_\beta \) are constant \((m\times m)\)-matrices.
We will call operators of the form (6) homogeneous quasielliptic operator with lower terms. Suppose that its symbol \(\mathcal{L}(i\xi )\) satisfies the following condition.
Condition 5
Suppose that the symbol of the differential operator (6) satisfies the estimate
where \(0 \le q < 1\), \(c_5\), \(c_6 > 0\) are constants.
Example 2
Consider the parabolic operator with ‘opposite times directions’
where \(\triangle '\) is the Laplace operator in \({\mathbb R}^{n-1}\) and \(\alpha \beta > 0\). Obviously,
Consequently, Conditions 4, 5 are fulfilled for
Isomorphism theorems for quasielliptic operators of the forms (1), (5) without lower terms were proved in [13,14,15]. In our paper we study mapping properties of quasielliptic operators of the forms (2), (6). Particularly, we formulate isomorphism theorems for these operators.
3 Weighted Sobolev Spaces
We introduce the weighted Sobolev spaces \(W^l_{p,q,\sigma }({\mathbb R}^n)\). Using these spaces, one can solve the problem on isomorphism for quasielliptic operators \(\mathcal{L}(D_x)\) of the form (2) or (6).
Definition 3
Let \(l = (1/\alpha _1, \ldots , 1/\alpha _n)\), \(1/\alpha _j \in {\mathbb N}\), \(j = 1,\ldots ,n\), \(1< p < \infty \), \(0 \le q \le 1\), \(\sigma \ge 0\). Denote by \(W^l_{p,q,\sigma }({\mathbb R}^n)\) the weighted Sobolev space of functions \(u \in L_{loc}({\mathbb R}^n)\) having the weak derivatives \(D^{\nu }_x u\), \(\nu \alpha \le 1\), such that
Here \(\langle x \rangle ^2 = \sum \limits _{j=1}^n x_j^{2/\alpha _j}\).
Introduce the norm
The weighted Sobolev spaces \(W^l_{p,q,\sigma }({\mathbb R}^n)\) coincide with well-known spaces for some parameters l, q, \(\sigma \). We consider several examples.
Example 3
Obviously, the space \(W^l_{p,q,0}({\mathbb R}^n) = W^l_{p,0,\sigma }({\mathbb R}^n)\) is the Sobolev space \(W^l_p({\mathbb R}^n)\).
Example 4
The space \(W^l_{p,1,\sigma }({\mathbb R}^n)\) coincides with the space \(W^l_{p,\sigma }({\mathbb R}^n)\) introduced in [16]. Indeed, by definition [16],
Example 5
In the isotropic case \(1/\alpha _1 = \ldots = 1/\alpha _n = \overline{l}\) the norm (8) for \(q = \sigma = 1\) is equivalent to the norm
Then, from the work [10] of L.D. Kudryavtsev it follows that the space \(W^l_{p,1,1}({\mathbb R}^n)\) for \(p > n\) coincides with the Sobolev space
where
Example 6
Consider the Nirenberg–Walker–Cantor space \(M^p_{\ell ,k}({\mathbb R}^n)\) [4, 11] whose norm is defined as
Clearly, by (9) the space \(W^l_{p,1,1}({\mathbb R}^n)\) coincides with the space \(M^p_{\overline{l},-\overline{l}}({\mathbb R}^n)\) in the isotropic case \(1/\alpha _1 = \ldots = 1/\alpha _n = \overline{l}\) for \(q = \sigma = 1\), \(p > 1\).
Definition 4
Denote by \(\mathop {W}\limits ^{\circ }{\!}^l_{p,q,\sigma }({\mathbb R}^n)\) the completion of \(C^\infty _0({\mathbb R}^n)\) with respect to the norm (8).
From Definitions 3 and 4 it follows that the space \(\mathop {W}\limits ^{\circ }{\!}^l_{p,q,\sigma }({\mathbb R}^n)\) is embedded in the space \(W^l_{p,q,\sigma }({\mathbb R}^n)\). One can show that the strict embedding holds
for sufficiently large \(\sigma > 1\).
In the next theorem we indicate the condition when these spaces coincide. Note that theorems of such type are very important in the theory of differential operators.
Theorem 1
If \(0 \le \sigma \le 1\) then \(\mathop {W}\limits ^{\circ }{\!}^l_{p,q,\sigma }({\mathbb R}^n)\, =\, W^l_{p,q,\sigma }({\mathbb R}^n)\).
Definition 5
Denote by
the space of integrable functions with the norm
Thereafter we will say that a vector-function
belongs to the weighted Sobolev space \(W^l_{p,q,\sigma }({\mathbb R}^n)\), if every its component \(u_j\) belongs to \(W^l_{p,q,\sigma }({\mathbb R}^n)\). By definition,
Analogously, a vector-function
belongs to the weighted space \(L_{p,\gamma }({\mathbb R}^n)\), if every its component \(f_j\) belongs to \(L_{p,\gamma }({\mathbb R}^n)\) and
4 Mapping Properties of the Operators (2), (6)
Consider the quasielliptic operator \(\mathcal{L}(D_x)\) defined by (2) or (6). Introduce the notation \(|\alpha | = \sum \limits ^n_{j=1} \alpha _j\).
The following theorems hold.
Theorem 2
Let \(\beta =(\beta _1,\ldots ,\beta _n)\), \(1 \ge \beta \alpha \ge q\). Then the following estimate is satisfied for every \(U \in C^\infty _0({\mathbb R}^n)\)
where the constant \(c_\beta > 0\) does not depend on U.
Theorem 3
Let \(\beta =(\beta _1,\ldots ,\beta _n)\), \(q > \beta \alpha \ge 0\) and
Then the following estimate is satisfied for every \(U \in C^\infty _0({\mathbb R}^n)\)
where the constant \(c_\beta > 0\) does not depend on U.
Theorem 4
Let
Then for every \(F \in L_{p,(\sigma - 1)q}({\mathbb R}^n)\) there exists a unique \(U \in W^l_{p,q,\sigma }({\mathbb R}^n)\) such that
Moreover, the estimate holds
with a constant \(c > 0\) independent of F.
Theorem 5
Let \(|\alpha |/p > q\). Then the mapping
is an isomorphism.
Remark 1
Theorems 4, 5 are analogs of some theorems in [13,14,15] for quasielliptic operators without lower terms.
We illustrate Theorem 5 by using the differential operator (4):
Taking into account Example 1, we have
Consequently, by Theorem 5 the mapping
is an isomorphism for \(p \in (1,\, \frac{n}{2k})\), \(n > 2k\).
Consider the critical cases in (4): \(k = 0\) and \(k = m\).
In the first case \(k = 0\) we have \(\triangle ^0 = I\), \(q = 0\) and \(W^l_{p,0,1}({\mathbb R}^n) = W^l_{p}({\mathbb R}^n)\). Then (10) is rewritten in the form
Therefore the isomorphism theorem gives the classical result.
In the second case \(k = m\), we have \(q = 1\) and \(W^l_{p,1,1}({\mathbb R}^n) = W^l_{p,1}({\mathbb R}^n)\). Then (10) is rewritten in the form
The isomorphism theorem for \(p \in \left( 1,\, \frac{n}{2m}\right) \), \(n > 2m\) follows from [7].
5 Elements of Used Technique
To prove of the above results we use a technique of integral representations for regularizers of differential operators. Our technique is based on the special representation by S.V. Uspenskii [17] for integrable functions:
where
Applying the integral representation (10), we construct the following integral operators
where \(l^{j,k}(i\xi )\) are entries of the inverse matrix \((\mathcal{L}(i\xi ))^{-1}\). In the case of \(m = 1\) we write \((\mathcal{L}(i\xi ))^{-1} F(y)\) instead of the sum
In the present paper we use the operators \(P_{j,h}\) for \(h \ll 1\) in order to construct regularizers of the quasielliptic operators (2), (6). Using these regularizers, we indicate the conditions for unique solvability of the quasielliptic equations and systems in the weighted Sobolev spaces, obtain the estimates for the solutions and formulate the isomorphism theorem for the quasielliptic operators.
References
Nikol’skii, S.M.: The first boundary problem for a general linear equation. Sov. Math. Dokl. 3, 1388–1390 (1962)
Volevich, L.R.: Local properties of solutions to quasielliptic systems. Mat. Sb. 59, 3–52 (1962). (in Russian)
Bagirov, L.A., Kondratyev, V.A.: On elliptic equations in \({\mathbb{R}}^n\). Differ. Uravn. 11, 498–504 (1975). (in Russian)
Cantor, M.: Spaces of functions with asymptotic conditions on \({\mathbb{R}}^n\). Indiana Univ. Math. J. 24, 897–902 (1975)
Cantor, M.: Elliptic operators and decomposition of tensor fields. Bull. AMS 5, 235–262 (1981)
McOwen, R.C.: The behavior of the laplacian on weighted sobolev spaces. Comm. Pure Appl. Math. 32, 783–795 (1979)
McOwen, R.C.: On elliptic operators in \({\mathbb{R}}^n\). Comm. Partial Diff. Eqn. 5, 913–933 (1980)
Choquet-Bruhat, Y., Christodoulou, D.: Elliptic systems in \(H_{s,\sigma }\) spaces on manifolds which are euclidean at infinity. Acta Math. 146, 129–150 (1981)
Lockhart, R.B., McOwen, R.C.: Elliptic differential operators on noncompact manifolds. Ann. Scuola Norm. Sup. Pisa. Cl. Sci. 12, 409–447 (1985)
Kudryavtsev, L.D.: Direct and reverse embedding theorems. Applications to solution of elliptic equations by the variational method. Trudy Mat. Inst. Steklov. Akad Nauk SSSR 55, 1–182 (1959). (in Russian)
Nirenberg, L., Walker, H.F.: The null spaces of elliptic partial differential operators in \({\mathbb{R}}^n\). J. Math. Anal. Appl. 42, 271–301 (1973)
Demidenko, G.V.: On one class of matrix differential operators. Sib. Math. J. 45, 86–99 (2004)
Demidenko, G.V.: On quasielliptic operators in \({\mathbb{R}}^n\). Sib. Math. J. 39, 884–893 (1998)
Demidenko, G.V.: Isomorphic properties of quasi-elliptic operators. Russ. Acad. Sci. Dokl. Math. 59, 102–106 (1999)
Hile, G.N.: Fundamental solutions and mapping properties of semielliptic operators. Math. Nachr. 279, 1538–1572 (2006)
Demidenko, G.V.: On weighted sobolev spaces and integral operators determined by quasi-elliptic operators. Russ. Acad. Sci. Dokl. Math. 49, 113–118 (1994)
Uspenskii, S.V.: On the representation of functions defined by a class of hypoelliptic operators. Proc. Steklov Inst. Math. 117, 343–352 (1972)
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Demidenko, G. (2017). Mapping Properties of One Class of Quasielliptic Operators. In: Giri, D., Mohapatra, R., Begehr, H., Obaidat, M. (eds) Mathematics and Computing. ICMC 2017. Communications in Computer and Information Science, vol 655. Springer, Singapore. https://doi.org/10.1007/978-981-10-4642-1_29
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