Embedding theorems for functional spaces associated with a class of Hermitian forms Текст научной статьи по специальности «Математика»

УДК 517.98

Embedding Theorems for Functional Spaces Associated with a Class of Hermitian Forms

Anastasiya S. Peicheva*

Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 28.05.2016, received in revised form 10.06.2016, accepted 14.11.2016 We prove embedding theorems into the scale of Sobolev-Slobodetskii spaces for functional spaces associated with a class of Hermitian forms. More precisely we consider the Hermitian forms constructed with the use of the first order differential matrix operators with injective principal symbol. The results are valid for both coercive and non-coercive forms.

Keywords: non-coercive Hermitian forms, embedding theorems, matrix elliptic operators. DOI: 10.17516/1997-1397-2017-10-1-83-95.

It is well known that integro-differential Hermitian forms are closely related to the generalized setting of mixed boundary value problems for strongly elliptic equations and systems, as well as the existence and uniqueness theorems for such problems (see, for example, [1-6], and other). We prove embedding theorems into the scale of Sobolev-Slobodetskii spaces for functional spaces associated with one class of Hermitian forms. More precisely, we consider the Hermitian forms constructed with the use of the first order matrix differential operators with the injective principal symbol. The results are valid for both coercive and non-coercive forms.

1. Function spaces

Let D be a bounded domain with Lipschitz boundary in Euclidean space R", n > 2, with coordinates x = (xi,..., xn). For some multi-index a = (ai,..., an) we will write da for the

dH

partial derivative —^-„ a . We consider the complex-valued functions defined over the

dxa1 ■ ■ ■ dxOr

domain D and its closure D. We also fix an open (in the induced topology) connected set S with piecewise smooth boundary dS on the hypersurface dD.

Let Cs(D, S), s G Z+, be the set of s-times continuously differentiable functions in D, which are disappearing in some (one-sided) neighborhood S in D. We will write Lq(D), 1 < q < for standard normed Lebesgue spaces of functions over D. We also write Hs(D), s c N, for the Sobolev space of functions whose weak derivatives up to the order s belong to L2(D). Similarly, Hs(dD), s c N, stand for the Sobolev space on the boundary of domain dD of functions whose weak derivatives up to the order s belong to L2(dD). Let the space Hq(D) stand for the closure of space C^(D) in Hs(D). For positive non-integer s we denote by Hs(D) the Sobolev-Slobodetskii space, see, for instance, [7]. More precisely, in the case of 0 < s < 1 the space Hs(D) consists of

* peichevaas@mail.ru © Siberian Federal University. All rights reserved

functions f G L2(D), D c 1" such that the functional

2 \ 1/2

I (\ 1,112 + [ If (x) - f (V)? d d \

\hs(d) = (j\f \\l2(d) + jd x - y\n+2s dxdV)

is finite, that is defining the norm in this space. Similarly, we define the function space Hs(dD), 0 < s < 1, on the space of functions defined on dD.

For non-integer s > 1 we set s = [s] + A, where [s] is integer part of the s. Then Hs(D) consists of elements of H{s\D) such that daf G HX(D) for all multi-induces |a| = [s]. This is a Banach space with the norm

\\H s(D) = (\\f \\H M{D) + E I9", WHhd))

a\ = [s]

1/2

and even a Hilbert space with the evident scalar product. The closure of Cs(D,S) in the space Hs(D) is denoted by Hs(D,S). In particular, H 1(D,dD) = Hl(D).

As usual, for the function space B we denote by B" the Cartesian product of n copies of this

( " N 1/2

space. If B is a normed space, then we will provide B" with the norm = ( \\uj\\b) ■

Therefore, Bn is a Hilbert space, if B is the one.

2. The embedding theorems for coercive forms

Let A(x, d) be a homogeneous differential the first order matrix operator with injective symbol in a domain X c 1", i.e.

n

A = £ Aj (x)dj, j=i

here Aj (x) are (l x fc)-matrices, whose components are complex-valued CX(X)-functions and

Aj(x)j = k for all x G X,Z G Rn \ {0}.

rang

j=1

Moreover, we require that the following Uniqueness Property in the small on X holds:

if Au = 0 in a domain U C X and u = 0 on an open subset V C U, then u = 0 in U. (1) Let D c X. We consider the following Hermitian form on the space [C 1(D, S)]fc:

(u,v)+ = (Au, Av){L2(D)Y + (a0u,v)[L2 (D)]k + (b0u,v)[l2(dD\s)]k ,

where a0(x) is a Hermitian non-negative (k x k)-matrix with entries aO'^ G L^(D) and (k x k)-matrix b0 is a Hermitian non-negative (k x k)-matrix with measurable bounded components on dD \ S. We denote by H +(D) the completion of the space [C 1(D, S)]fc with respect to the norm || • || + , which is induced by the inner product (■, ■)+ (in those cases where such a form is defined).

Let A*(x) be the adjoint matrix for the matrix Aj(x) and

"

A* = dj (A*(x)■), j=1

be the formal adjoint for A. Then the second order differential operator

n

A*A = - £ di(A*Ajdj) ij = 1

is strongly elliptic in X, i.e. for all x G X, w G D x (Ck \ 0), C G D x (Rn \ 0) det ^ A*Aj (x)CiCj =0; J w* i ^ A*Aj (x)CiCj ) w\ > 0.

i,j=1 I \i,j=1 J J

Then the form (■, ■)+ is related to a mixed problem for the operator A*A.

Lemma 2.1. The Hermitian form (■, ■)+ defines a scalar product on [C 1(D,S)]k if one of the following conditions is fulfilled:

1) a0 ^ c0Ik on U with a constant c0 > 0 and a non-empty open subset U of D;

2) the relatively open set S c dD is not empty;

3) b0 ^ c1Ik on V with a constant c1 > 0 and a non-empty relatively open subset V of dD \ S. Besides, in these cases we see that:

a) the embedding [H 1(D,S)]k ^ H+(D) is continuous, if the entries of the matrix b0 belong L^(3D \ S);

b) the embedding H+(D) ^ [L2(D)]k is continuous; moreover, in this case the elements of H+(D) belong [H1oc(D U S)]k and vanish on S.

Proof. To prove that it is a scalar product, we only need to check that (u, u)+ =0 for a function u G [C 1(D,S)]k implies u = 0 in D.

From 1) and a0u = 0 it follows, that u = 0 on some subset V. Then, the Uniqueness Property (1) for the operator A and the assertion of Au = 0 imply that u = 0.

From conditions 2) or 3) it follows, that any vector u from [C 1(D, S)]k, satisfying (u, u)+ = 0, vanishes in an open non-empty subset r of S c dD. Since u also satisfied Au = 0 in D, then we obtain a Cauchy problem:

Au = 0 in D, \ u = 0 on r.

It follows from the Uniqueness Theorem for the Cauchy problem for systems with an injective symbol, that u = 0 in D (see, for instance, [8, Proposition 4.3.3]).

This proves that the Hermitian form (■, ■)+ defines a scalar product. Further, if the elements of b0 belong to L^(dD\S) then, by the Trace Theorem for the Sobolev spaces (see, for example, [7]), we find that for all u G [C 1(D, S)]k we have following:

\(b0u,u)[L2(dD\S)]k I ^ cllull2i2(dD\S)]k < c\\u\\2{Hi{D)]k ,

where the positive constant c is independent of u. On the other hand, since the components of matrices a0 and Ai (1 < i < n) belong to LX(D), we see that for all u G [C1 (D,S)]k the following is true:

\(a0u,u)[L2(D)]k \ < c0|u|2i2(D)]fc < c0||u||2Hi(D)]fc , \(Ai Aj dj u,diu)[L2(D)]k \ < ci,j\^u\fL2(D)]k < ci,j ||u||2Hi(D)]k ,

where the positive constants c0, cij are independent of u. That is, the space [H 1(D,S)]k is continuously embedded into the H +(D). If

a0 > cl > 0 in D (2)

then for all u e [C 1(D, S)]k we obtain

W^^ > (aou,u)[l2{d)]k > c\\u\\2[L2{D)]k,

i.e. the space H+(D) is continuously embedded into [L2(D)]k.

Since the principal symbol of the operator A is injective then the solutions Au = 0 are infinitely differentiable functions in D. In addition, due to the Uniqueness property (1) the operator A is injective on [Co°(D)]k. Then the Garding inequality for the strongly elliptic operator A* A yields

\u\lHl{D)]k < c\Au\lL2iD)]k for all u e [H0(D)]k, (3)

with positive constant c does not depend of u. We take an domain G c D, such that G C D U S and we fix a function ^ e C 1(D) is vanishes outside the compact set G and also such that 0(x) = 1 for all x e G. Then 4rn e [C1 (D,dD)]k for all u e [C 1(D,S)]k and, according to (3), we have:

\u\[Hi(G)]k < \\M\lHi(D)]k < 4A(0u)IIiL2(D)]i (4)

for all u e [C 1(D, S)]k, with constant c does not depend of u. Finally, as (2) is fulfilled, we obtain that

\A(^u)f{L2{D)]i < 2\^AuflL2{D)]l + 2\(A^)uf{L2(d)] < c\uf+ (5)

for all u e [C 1(D,S)]k, with constant c does not depend of u. In particular, inequalities (4), (5) mean that any sequence {uv} c [C 1(D,S)]k, convergent in the space H+(D), converges in the [H 1(G)]k, too. This proves the statement b). □

As we have seen in the proof of Lemma 2.1, if S = dD then space H +(D) is embedded continuously to [H1 (D,S)]k (cf. [9]). Therefore, we are primarily interested in the case S = dD. Everywhere below we assume that H+(D) is embedded continuously to [L2(D)]k, i.e.

\u\[L2(D)]k < c iNk

for all u e [H1 (D, S)]k, where the constant c does not depend of u. It follows from Lemma 2.1, that this condition is not too restrictive. The Hermitian form (•, •)+ is called coercive if there exists a constant c > 0 such that

Mh^d)]* < c HuH+ for all u e [H 1(D,S)]k

that is, the space H+(D) is continuously embedded into the space [H 1(D,S)]k. Let i be the natural (continuous) embedding:

i: H+(D) ^ [L2(D)]k. (6)

Also, we will need Sobolev spaces with negative smoothness. More precisely, we denote by H-(D) the completion of the space [H 1(D,S)]k by the norm

,, ,, \(v,u)L2(D)\

iM- = suP -1| || •

veH+(D) !M\ + v=0

As is known, (see, for example, [10, Lemma 2.3]) the space H-(D) can be identified with the space dual to H +(D) with respect to pairing

{v,u) = \im(v,uv)[L2(D)]k • (7)

The space [L2(B)\k is continuously embedded into H (D); corresponding embedding we will denote by l ( see, for instance, [10, Lemma 2.2]).

We define [H-s(D)]k and [H-s(D)]k as the dual spaces to [Hs(D)]k and [H§(D)]k respectively, relatively to the pairing (7). By the construction, the following embedding is continuous:

[H-s(D)]k ^ [Hi-s(D)]k, s > 0.

For the coercive case the following two statements on the embedding are, probably, well-known (cf. [5], [6]).

Lemma 2.2. We assume that following estimate

A*Aj(x) CiO) w > C |w|2|C|2 (8)

(j=A;Aj (x)

is fulfilled for all (x,w) G D x (Cn \ {0}), Z G D x (Rn \ {0}), where c is positive constant independent on (x,w) and Z■ Then the embeddings

H+(D) ^ [H 1 (D, S)]k, [H- 1 (D)]k ^ H-(D)

are continuous, if one of the following conditions is fulfilled:

1) there is a positive constant c> 0 such that inequality (2) is true;

2) the set dD \ S has at least one interior point in the relative topology dD and

Wbou\\[L2{dD\s)]k > ci |M|[L2(aD\s)]fc for M u G H 1 (дD, S); (9)

3) the set S contains a not empty, open (in the relative topology of dD) subset■ Proof■ Suppose, that condition 1) is fulfilled. Then

\\u\\ + > (aou,u)[L2(D)]k > cWuflL2(D)]k

for all u G [C 1(D, S)]k and from condition (8) it follows, that

n

l(Au,Au)[L2(D)]k 1 > W^W^D)]* , i=1

here positive constants c, c are independent of u. But this means that for all u G [C 1(D, S)]k

(u,u)+ > ^hWfmiD)]*,

so, the statement of theorem is true, if condition 1).

Let the 2) be fulfilled. We suggest, that for any natural number m there is a such um G [C 1(D,S)]k, that the inequality

(um ,um)+ < \\um \\ [H1 (D)]fc (10)

is true. Consider the bounded sequence < ,.V'm.. >. We can extract a weakly convergent subse-

l \\umW )

quence <vm. = m > to an element v0 G [H 1(D)]k. By the Trace Theorem for Sobolev-l Wum\U j£N

Slobodetskii spaces, the sequence of traces {vmj\qD}jeN converges weakly in [H 1/2(dD)]k to the

trace v0idD £ [H 1/2(dD)]k and {Avm.}jeN converges weakly to the element Av0 in [L2(D)]k. In addition, by Rellich Theorem, the space [H 1/2(dD)]k is compactly embedded in [L2(dD)]k and the [H 1(D)]k is compactly embedded in [L2(D)]k. Hence, the sequence {vmj}jeN converges to v0 in [L2(D)]k and the sequence {vm.\dD}jeN converges to the trace v0\dD in [L2(dD)]k. According to (8), (10) and 2)

\\vmj\\[L2(8D)]k < \\bovmj\\{L2(dD)]k < \\vmj ||+ ^ 0, mj ^

n

J2Wdivmi \\2L2(D)]k < \Avm3\2L2(D)]' ^ 0 mj ^ +X>-

i=1

Since the weak and strong limits are coincided (when both exist), then Av0 = 0 in D and v0 = 0 on S. More other the condition of uniqueness in the small implies that v0 = 0. So it means, that

\\vmj\\[L2(D)]k ^ 0, mj ^ \\vm.\[Hi(D)]k ^ 0, mj ^

which contradicts the equality \\vmj\\[Hi(D)]k = 1.

For conditions 3) the proof is similar. □

In particular, under the hypothesis of Lemma 2.2 the Hermitian form (•, •)+ is coercive, and the embedding (6) is compact.

Lemma 2.3. Under the assumption S = dD, the following embeddings are continuous:

H+(D) ^ [H 1(D, dD)]k, ([H 1(D, dD)]k)' ^ H-(D). In particular, the Hermitian form (•, •)+ is coercive and the embedding (6) is compact. Proof. The statement is well known (see, for example, [4]). □

3. The embedding theorem for non-coercive forms

Now we obtain an embedding theorem for the space H +(D) under weaker assumptions than in the Lemmata 2.2, 2.3.

Theorem 3.1. We assume that the coefficients of the matrix Ai are infinitely smooth in the closure of some neighborhood X of the compact set D and there is a constant c > 0 such that (9) is done. Then

1) the space H +(D) is continuously embedded into [L2(D)]k, if there is a constant c1 > 0 such that (2) is satisfied;

2) the space H+(D) is continuously embedded into [H 1/2-e(D)]k with an arbitrary e > 0, if for all u £ [C^(X)]k, the following is true

(AU,AU)[L2(X)]1 > m \\u\\2L2(X)]k (11)

with a constant m > 0 is independent of the function u.

Moreover, if dD £ C2, then H +(D) is continuously embedded into [H 1/2(D)]k.

Proof. The statement 1) follows immediately from the definition of the norm \\ • \\ + .

Let dX £ C. This assumption does not bear loss of generality, although for most of the set goals it will be enough to dX was a Lipschitz surface. Since the operator A* A is strongly elliptic, then the classical Garding inequality and the condition of uniqueness in the small mean, that there exists the Green function G of Dirichlet Problem (see, for example, [2,11], [12, Th. 3.3] and [13,

Theorem 2.26]). More precisely, as above, we define the space [H 1(X)]k as the dual space to [H 1(X,dX)]k with respect to [L2(X)]k-pairing. Clearly, the space [H-1(X)]k is continuously embedded into [H-1(X)]k. As usual, the space [C 1(X, dX)]k is dense in the space [H 1(X, dX)]k, then the operator A* A extends to a continuous mapping of [H 1(X, dX)]k into [H-1(X)]k with using Hermitian form

n

J2 (Aidiu, Aj dj v)[L2 (x)]' = (Au, Av)[L2(x)y, u,v e [H 1(X,dX )]k. (12)

i,j = 1

In particular, form (12) induces a generalized setting of the Dirichlet Problem for the operator A* A b X. Thus, there is a bounded linear operator

G : [H-1(X)]k — [H 1(X, dX)]k,

satisfying

GA*A = I, A*AG = I (13)

on [H1 (X, dX)]k and [H-1(X)]k respectively (see [12, Th. 3.3] or [13, Th. 2.26]).

Applying the Trace Theorem for Sobolev spaces (see, for example, [1]), we introduce the so-called Poisson operator P : [H 1/2(dX)]k — [H 1(X)]k, which satisfies

P o t1 + GA* A = I,

where t1 denotes the operator of the trace [H 1(X )]k — [H 1/2(dX )]k (cf. [12, Lem. 4.3] or [13, Cor. 2.31]). If the boundary of X is a Lipschitz surface, then the Green and Poisson operators posses adequate regularity properties. More precisely,

G : [Hs-1(X)]k [Hs+1(X)]k, djG : [Hs-1(X)]k ^ [Hs(X)]k, P : [Hs+1/2(dX)]k - [Hs+1(X)]k

for all 0 < s < 1/2 (see, for example, [6, § 12]). In particular, if dX is C2 -smooth, then the mappings

G : [L2(X)]k - [H2(X)]k, P : [H3/2(dX)]k - [H2(X)]k are also continuous.

Let vA be so-called a co-normal derivative to dX with respect to the operator A:

n

va = A*(x)vi (x)A, (14)

i=1

here v(x) = (v1(x),..., vn(x)) is unit outward normal to dX at the point x e dX. If X is a domain with Lipschitz boundary, then the normal v(x) exists almost everywhere on dX.

Lemma 3.1. Let X be a domain with Lipschitz boundary. If dX e C2, then the operator P is continuously mapping [L2(dX)]k into [Hl/2(X)]k.

Proof. The proof of this statement is similar to the proof of the corresponding statement for weighted Sobolev spaces in [10, Lemma 7.7]. □

We continue the proof of the theorem. Let e+ will be an operator of extension by zero of the domain D on X and by r+ we denote an operator of restriction from the set X to domain D. Evidently, e+ is the bounded linear operator from [L2(D)]k into [L2(X)]k and r+ is bounded linear operator from [Hs(X)]k into [Hs(D)]k for all s e R.

If (11) is not valid, then according to the condition of the theorem, it follows, that a0 > c1I in D with some constant c1 > 0, and hence H + (D) is continuously embedded into [L2(B)\k. Therefore, the norm || • || + is not weaker than the norm || • ||p on [H 1(D, S)]k

1/2

||u||p = {^M^DY + M^^D)]*^ ■

Since the coefficients of the matrix Ai(x) are continuous up to the boundary of the domain D, from Green's formula the next follows

,, n ,, n

~y2 A*(x)vi(x)AuV ds = y2 (A*Ajdjudv + di(A*Ajdju)v) dx (15)

hD i=i jDi,j=i

for all u e [H2(D)]k and v e [H 1(D)]k. We denote by

Gd : H-1(D)]k ^ [H 1(D,dD)]k,

Green's operator of the Dirichlet Problem for the operator A* A in D. Properties of the operator Gd are similar to the properties of the operator G discussed above for the domain X. In a similar way we introduce the operator Poisson PD.

Combining the formulas (13), (15) we get the following

n

Ml (A*Aj dj Gd A*Au,diGD A* Au^w + i,j = 1

n

+ E (A*Aj dj PD u diPDu) [L2 (D)]k + P V-fL^idD)]* (16) i,j=1

if u e [H 1(D,S)]k. On the other hand, it follows from the Garding inequality that for all u e [H 1(D,S)]k we have

n

HGd A*Auf[Hi{D)]* < cJ2 (A*Aj dj Gd A*Au,diGD A* AU)[L2{d)]* ■ (17)

i,j=1

Using (13), (16) and (17) we see that any sequence {uc [H 1(D,S)]k, which converges to a function u in the space H +(D), it can be represented as

uM = Gd A* AuM + Pd u^, (18)

where the sequence {GDA*Auconverges in [H 1(D,dD)]k c [H 1(D,S)]k to the element uG. Therefore, the sequence {PDu^} converges to some element uP in H +(D) and

u = ug + up = Gd A* Au + Pd u, (19)

here PDu is the integral of Poisson of the trace of the function u,\qd G [L2(dD)]k for u e H+(D). Hence this theorem depends of the behavior of the element uP = PDu.

Since the coefficients of Ai are smooth in a neighborhood of D, we can assume without loss of generality that X is a region with smooth boundary. In this way, if A*Au e [L2(X)]k and u = 0 on dX, then the element u belongs to the [H2(X)]k. Therefore, from a priori estimates, it follows that G generates a bounded operator

r+ Ge+ : [L2(D)]k ^ [H2(D)]k,

where operators r+ and e+ are defined above.

Let s > 0. It is clear that any element u € [H s(D)]k extends up to the element U € [H-s(X)]k by the equality

(U,v)x = (u,v)D

for all v € [Hs(X)]k. Since the element U vanishes in X \ D, it is natural to denote it by e+u. On this way we define a linear operator e+ : [H-s(D)]k ^ [H-s(X)]k, s > 0.

The support of the distribution e+ u belongs to the D. Therefore, using the continuity of pseudo-differential operators on the compact closed manifolds, we conclude that r+Ge+ extends to a linear bounded operator

r+Ge+ : [H-1/2(D)]k ^ [H3/2(D)]k.

Hence the operators

dj (r+Ge+) : [Ht-1/2(D)]k ^ [H1/2+t(D)]k, vA (r+Ge+) : [Ht-1/2(D)]k ^ [He(3D)]k (20)

are also bounded by the Trace Theorem for Sobolev spaces in Lipschitz domains. Note that when e = 0 the statement becomes invalid, as the elements of space [H 1/2(D)]k may have no traces on dD c X.

From (15) and the continuity property (20) it follows that

(v,u)[l2(D)]k = (A*AG(e+v),u)[l2(D)]k (va(r+Ge+)v,u)[l2(dD)]k +

n p

+ V A* AjdjG(e+v)3ju dx (21) , 1 Jd

i,j=1

for all u € [H 1(D, S)]k and v € [L2(D)]k.

Now we claim that the norm || • ||p is not weaker, than the norm || • Wh[i/2-e(D)]k on [H 1(D, S)]k. Indeed,

\(v,u)[L2iD)]k \ \(v»,u)L2D)]k I (22) MlH 1/2-<(D)]k = SUP IMI--= SUP -\T\\- (22)

ve[H'-1/2(D)]k WvW[H'-1/2(D)]k ve[H'-1/2(D)]k WvW[H'-1/2(D)]k v=0 v=0

for all u € [H 1(D, S)]k, where {vis any sequence of smooth functions in D, which approximates v in the space [He-1/2(D)]k. Using the formula (21) for u and v = vM, the expression on the right side of (22) is equal to

n ,,

\ 2_, A*AjdjG(e+v)diudx + (va(r+Ge+)v,u)^l2(QD)]k ij=1JD

Then

\(vA(r+Ge+)v,u)[L2(dD)]k \ < WvA(r+Ge+)vW[H-(D)]k\u\[H-'{8D)]k <

< c WvW[H'-i/2(D)]k \u\[H-'(8D)]k (23)

for all u € [H 1(D, S)]k and v € [He-1/2(D)]k, where the last inequality follows from (20). Here c denotes a constant, which independents of u and v. It follows from the generalized Cauchy inequality that

n 2 / n w n \

| X A*Aj(x)zizj\ < ]T A*Aj(x)zizM ]T A*Aj(x)Ci j (24)

i,j=1

for all z,Z e Cn. The application (24) leads us to the following:

i n „ / n „ \ 1/2

| ^^ A*AjdjG(e+v)diudx < c I ^^ A*Ajdjudiudx\ HvHh^-i/2(D)]* (25) i,j=1 D i,j=1 D

with a constant c independent on u and v.

Using (22), (23) and (25) we conclude that there are positive constants c and C such that

cmh^^-d)]* < mp < C||u||+

for all u e [H 1(D,S)]k. In particular, this provides a continuous embedding H+(D) ^ [H 1/2-e(D)]k, as required.

Finally, let dD e C2. The sequence {uv} from (18) converges in H+(dD), and the norm in this space is not weaker than the norm || • ||p, then it converges also in the [L2(dD)]k. Lemma 3.1 implies, that the sequence {PDuv} convergence to some element uP in [H 1/2(D)]k. The continuously embedding of space H+(D) into [H 1/2(D)]k follows from (19). □

For the case where the operator A* A is scalar, this theorem is similar to [10, Theorem 7.4]. In the work [14] a special case corresponding to the factorization A* A for Lame system of linear elasticity theory was considered. Actually the present proof follows the same general scheme, taking into the account the matrix nature of A* A.

Example 3.1. Denote by L the Lame type operator in R2:

C0(x, d) = —^(x)IA — (A(x) + /i,(x))Vdiv,

where n and A are real-valued functions in LX(D), with ^ > k, (2n + A) > k for a constant k > 0. If the functions n, A belong to C0,1 (D) then there is a formally non-negative self-adjoint operator CA(x,d) = A*A, that differs from C0(x,d) by lower order terms. Of course, there are several possible choices of A. For instance, one may consider

= f ^ rot \ V + Adiv J ■

If b0 > 0 is a matrix with constant entries, then we have a continuous embedding H +(D) ^ [H1/2 (D)]2 (see Theorem 3.1). Moreover, H+(D) is not embedded continuously into [Hs(D)]2 for any s e (1/2,1] (see, for instance, [14, Example 4.5]).

The operator for this case is responsible not for the stress/viscosity on the boundary but for a more large class of interactions with dD. Interpreting the Lame system as a linearization of the stationary version of the Navier-Stokes' type equations for compressible fluids, we see that the operator (vA + b0) reflects rather the vorticity and the source density on conormal directions to dD \ S. This means that the operator is more fit to study problems related to models with the turbulent flows than some others operators (see [14, Example 4.5]). This is useful when we consider the boundary value problems for the Lame operator with boundary-type operator (vA + b0). Then it is natural that the class of the possible solutions to these problem extends to H +(D) due to the loss of the regularity of solutions near dD \ S (for more details see [14]).

Example 3.2. Let D be the unit circle in R2(= C) and S be the part of its boundary where arrg(z) e [0, 2n] \ [—n/2,n/2]. Let A = d be the Cauchy-Riemann operator in R2, i.e. it is the (2 x 2)-matrix

i d \

0=( t dy ■ (26)

\ dy dx J

The formal adjoint d* of d is the (2 x 2)-matrix is conjugated to (26) with respect to the usual Hermitian structure in the space L2(R2). Then an easy computation shows that d*d amounts to

d 2 c)2

the unit (2 x 2)-matrix I2 multiple of the Laplace operator A = ——^ + in R2. In the space

dx2 dy2

d 1 ( d d \ C the Cauchy-Riemann operator just is — = ^ ( q—+ V— q~ J •

We assume that a0 = 0 and b0 > 0 is a (2 x 2)-matrix with constant elements. Thus, H +(D) is continuously embedded into [H 1/2(D)]2 (see Theorem 3.1). Then the norm of the element u e H+(D) will have the following form:

||u||+ = ^A^iD)]2 + Po-U^LzidD^S)]2 ■

Fix a function $ of the class CX(D), which identically equals to zero in the neighborhood of S and equals to 1 on the part where arg(z) e [—n/4,n/4]. We define a new vector function ue(x,y) = $(x,y)ve(x,y), where

v,

(x,y) =

+l){l+e)/2 v=0 4

. ^ (x + V-iy) 4

£> 0.

V IVTo (4v + 1)(1+e)/2jJ

For brevity, the further argument we will carry out in the space C. In C the element ve(x,y) is

„4v

vF(z

(z) = E

z

In other words

0 (4v +\)(1+e)/2 if arg z e

£ > 0.

—n n

T' 4 0, if arg z e US,

where US is some neighborhood S. As ve independent of z, the Leibniz rule implies, that

dve =0, due = (d$)ve. (27)

We will show that the series ue converges in the space H +(D) and will find its norm. To the end, we recall that

i 0, k = v,

\ n/(v + 1), k = v.

Hence

(zk ,zV )[L2(D)]2

(28)

lkl^

E-

VellL2(D)]2 - ^ (4v +1)(2+e)->

v=0 v 1

and therefore the series of ve converges in L2(D) and in H +(D), because dve = 0. Using (27) we see that ue converges in the space H +(D), too.

On the other hand, by direct calculation with the use of (28) (see, for instance, [15, Lemma 1.4]), we obtain:

IlVell

j

ell[H s(D)]-

const ^^

2s — 1

n(4v + 1)

0 < s < 1.

(29)

u

n

Hence the series ve converges in Hs(D) for all 0 ^ s ^ 1/2. Then the series uE converges in Hs(D) for all 0 < s < 1/2.

If s > ^ we have 2s — 1 > 0. For any s > then it follows from (29) that there is a number 0 < ss < 2s — 1 such that the series vEs and uFs do not converge in Hs(D). Thus, uFs belongs to the space H+(D) but for any s € (1/2,1] the element uEs does not belong Hs(D). Therefore, the space H+(D) is embedded continuously into [H 1/2(D)]2 (see Theorem 3.1), but it is not embedded continuously into [Hs(D)]2, with any s € (1/2,1].

In the case when S = 0 the similar result was proved in [16, Theorem 1].

The work was supported by the Russian President's grant for the support of leading scientific schools NSh.-9149.2016.1.

References

[1] S.L.Sobolev, Some applications of functional analysis in mathematical physics: scientific. ed., Nauka, Moscow, 1988 (in Russian).

[2] O.A.Ladyzhenskaya, N.N.Uraltseva, Linear and quasi-linear elliptic equations, Nauka, Moscow, 1973 (in Russian).

[3] J.L.Lions, Non-Homogeneous Boundary Value Problems and Applications. Vol. 1, Berlin-Heidelberg-New York, Springer-Verlag, 1972.

[4] M.I.Vishik, About strongly elliptic systems of differential equations, Mat. sb., 29(71) (1951), no. 3 (in Russian).

[5] M.S.Agranovich, Mixed problems in Lipschitz domains for strongly elliptic systems of 2nd order, Funks. analiz i ego pril., 45(2011), no. 2, 1-22 (in Russian).

[6] M.S.Agranovich, Spectral Problems in Lipschitz Domains, Sovr. probl. mat. Fundament. naprav., 39(2011), 11-35 (in Russian).

[7] L.N.Slobodeckii, The generalized Sobolev spaces and their application to boundary value problems for differential equations in partial derivatives, Uch. zap. Leningr. gos. ped. inst., 197(1958), 54-112 (in Russian).

[8] N.N.Tarkhanov, The Cauchy problem for solutions of elliptic equations, Berlin, Acad. Verl., Vol. 7, 1995.

[9] K.Yoshida, Functional analysis, Springer-Verlag, 1965.

[10] A.A.Shlapunov, N.N.Tarkhanov, The Sturm-Liouville problems in weighted spaces in domains with smooth edges. I., Siberian Advances in Mathematics, 26(2016), no. 1, 30-76.

[11] M.Schechter, Negative norms and boundary problems, Ann. Math., 72(1960), no. 3, 581-593.

[12] B.-W.Schulze, A.A.Shlapunov, N.N.Tarkhanov, Green integrals on manifolds with cracks, Annals of Global Analysis and Geometry, 24(2003), 131-160.

[13] A.A.Shlapunov, N.N.Tarkhanov, Duality by reproducing kernels, Int. J. of Math. and Math. Sc., 6(2003), 327-395.

[14] A.S.Peycheva, A.A.Shapunov, On the completeness of root functions of Sturm-Liouville problems for the Lame system in weighted spaces, ZAMM (Z. Angew. Math. Mech.), 95(2015), no. 11, 1202-1214.

[15] A.A.Shlapunov, Spectral decomposition of Green's integrals and existence of Ws'2 -solutions of matrix factorizations of the Laplace operator in a ball, Rend. Sem. Mat. Univ. Padova, 96(1996), 237-256.

[16] A.Polkovnikov, A.Shlapunov, On the spectral properties of a non-coercive mixed problem associatedwith 9-operator, J. Siberian Fed. Univ., Math. and Phys., 6(2013), no. 2, 247-261.

Теоремы вложения для функциональных пространств, ассоциированных с одним классом эрмитовых форм

Анастасия С. Пейчева

Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041

Россия

Мы докажем теоремы вложения в шкалу пространств Соболева-Слободецкого для функциональных пространств, ассоциированных с одним классом эрмитовых форм. Более точно мы рассматриваем эрмитовы формы, построенные с помощью матричных дифференциальных операторов первого порядка с инъективным главным символом. Конечные результаты получаются как для коэрцитивных, так и для некоэрцитивных форм.

Ключевые слова: некоэрцитивная эрмитова форма, теоремы вложения, эллиптические матричные операторы.