Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains

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1 Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains Sergey E. Mikhailov Brunel University West London, Department of Mathematics, Uxbridge, UB8 3PH, UK J. Math. Analysis Appl. (accepted, to appear) Mikhailov-JMAA tex Abstract For functions from the Sobolev space H s 1 (), 2 < s < 3 2, definitions of non-unique generalized and unique canonical co-normal derivative are considered, which are related to possible extensions of a partial differential operator and its right hand side from the domain, where they are prescribed, to the domain boundary, where they are not. Revision of the boundary value problem settings, which makes them insensitive to the generalized co-normal derivative inherent non-uniqueness are given. It is shown, that the canonical co-normal derivatives, although defined on a more narrow function class than the generalized ones, are continuous extensions of the classical co-normal derivatives. Some new results about trace operator estimates and Sobolev spaces characterizations, are also presented. Keywords. Partial differential equation systems, Sobolev spaces, Classical, generalized and canonical co-normal derivatives, Weak BVP settings. 1 Introduction While considering a second order partial differential equation for a function from the Sobolev space H s (), 1 2 < s < 3 2, with a right-hand side from Hs 2 (), the strong co-normal derivative of u defined on the boundary in the trace sense, does not generally exist. Instead, a generalized co-normal derivative operator can be defined by the first Green identity. However this definition is related to an extension of the PDE operator and its right hand side from the domain, where they are prescribed, to the domain boundary, where they are not. Since the extensions are non-unique, the generalized co-normal derivative operator Corresponding author: sergey.mikhailov@brunel.ac.uk, Phone: , Fax:

2 appears to be non-unique and non-linear unless a linear relation between the PDE solution and the extension of its right hand side is enforced. This leads to the need of a revision of the boundary value problem settings, which makes them insensitive to the co-normal derivative inherent non-uniqueness. For functions u from a subspace of H s (), 1 2 < s < 3 2, which can be mapped by the PDE operator into the space H t (), t 1 2, one can still define a canonical co-normal derivative, which is unique, linear in u and coincides with the co-normal derivative in the trace sense if the latter does exist. These notions were developed, to some extent, in [15, 16] for a PDE with an infinitely smooth coefficient on a domain with an infinitely smooth boundary, and a right hand side from H s 2 (), 1 s < 3 2, or extendable to H t (), t 1/2. In [17] the analysis was generalized to the co-normal derivative operators for some scalar PDE with a Hölder coefficient and right hand side from H s 2 (), 1 2 < s < 3 2, on a bounded Lipschitz domain. In this paper updating [18], we extend the previous results on the co-normal derivatives to strongly elliptic second order PDE systems on bounded or unbounded Lipschitz domains with infinitely smooth coefficients, with complete proofs. We also give the week BVP settings invariant to the generalized conormal derivatives non-uniqueness. To obtain these results, some new facts about trace operator estimates and Sobolev spaces characterizations are also proved in the paper. The paper is arranged as follows. Section 2 provides a number of auxiliary facts on Sobolev spaces, traces and extensions, some of which might be new for Lipschitz domains. Particularly, we proved Lemma 2.4 on two-side estimates of the trace operator, Lemma 2.6 on boundedness of extension operators from boundary to the domain for a wider range of spaces, Theorem 2.9 on characterization of the Sobolev space H0 s() = H s () on the (larger than usual) interval 1 2 < s < 3 2, Theorem 2.10 on characterization of the space H t, t > 3 2, Theorem 2.12 on equivalence of Hs 0 () and Hs () for s 1 2, Theorem 2.13 on nonexistence of the trace operator, Lemma 2.15 and Theorem 2.16 on extension of H s () to H s () for all s < 1 2, s 1 2 k. The results of Section 2 are applied in Section 3 to introduce and analyze the generalized and canonical co-normal derivative operators on bounded and unbounded Lipschitz domains, associated with strongly elliptic systems of second order PDEs with infinitely smooth coefficients and right hand side from H s 2 (), 1 2 < s < 3 2. The weak settings of Dirichlet, Neumann and mixed problems (revised versions for the latter two) are considered and it is shown that they are well posed in spite of the inherent non-uniqueness of the generalized co-normal derivatives. It is proved that the canonical co-normal derivative coincides with the classical (strong) one for the cases when they both do exist. The results of Section 3 are generalized to Hölder-Lipschitz coefficients in [14], see also [18]. 2

3 2 Sobolev spaces, trace operators and extensions 2.1 Notations Suppose = + is a bounded or unbounded open domain of R n, which boundary is a simply connected, closed, Lipschitz (n 1) dimensional set. Let denote the closure of and = R n \ its complement. In what follows D() = C comp() denotes the space of Schwartz test functions, and D () denotes the space of Schwartz distributions; H s (R n ) = H2 s(rn ), H s () = H2 s () are the Sobolev (Bessel potential) spaces, where s R is an arbitrary real number (see, e.g., [12]). We denote by H s () the closure of D() in H s (R n ), which can be characterized as H s () = {g : g H s (R n ), supp g }, see e.g. [13, Theorem 3.29]. The space H s () consists of restrictions on of distributions from H s (R n ), H s () := {g : g H s (R n )}, and H s 0 () is closure of D() in Hs (). We recall that H s () coincide with the Sobolev Slobodetski spaces W2 s () for any non-negative s. We denote H s loc () := {g : φg Hs () φ D()}. For infinite (unbounded) domains we will use also the notation H s loc () := {g : φg Hs () φ D()} (for bounded domains H s loc () = Hs ()). Note that distributions from H s () and H s 0 () are defined only in, while distributions from H s () are defined in R n and particularly on the boundary. For s 0, we can identify H s () with the subset of functions from H s (), whose extensions by zero outside belong to H s (R n ), cf. [13, Theorem 3.33], i.e., identify functions u H s () with their restrictions, u H s (). However generally we will distinguish distributions u H s () and u H s (), especially for s 1 2. We denote by H s the subspace of Hs (R n ) (and of H s ()), which elements are supported on, i.e., H s := {g : g Hs (R n ), supp g }. To simplify notations for vector-valued functions, u : C m, we will often write u H s () instead of u H s () m = H s (; C m ), etc. As usual (see e.g. [12, 13]), for two elements from dual complex Sobolev spaces the bilinear dual product, associated with the sesquilinear inner product (, ) := (, ) L2 () in L 2 () is defined as u, v R n := [F 1 u](ξ)[fv](ξ)dξ =: (Fū, Fv) R n =: (ū, v) R n, R n u H s (R n ), v H s (R n ), (2.1) u, v := u, V R n =: (ū, v) if u H s (R n ), v H s (), v = V with V H s (R n ), u, v := U, v R n =: (ū, v) if u H s (R n ), v H s (), u = U with U H s (R n ) (2.2) for s R, where ḡ is the complex conjugate of g, while F and F 1 are the distributional Fourier transform operator and its inverse, respectively, that for integrable functions take form ĝ(ξ) = [Fg](ξ) := e 2πix ξ g(x)dx, R n g(x) = [F 1 ĝ](x) := e 2πix ξ ĝ(ξ)dξ. R n 3

4 For vector-valued elements u H s (R n ) m, v H s (R n ) m, s R, definition (2.1) should be understood as u, v R n := û(ξ) ˆv(ξ)dξ = R n û(ξ) ˆv(ξ)dξ =: (û, ˆv) R n =: (ū, v) R n, R n where û ˆv = û ˆv = m k=1 ûkˆv k is the scalar product of two vectors. Let J s be the Bessel potential operator defined as [J s g](x) = F 1 ξ x {(1 + ξ 2 ) s/2 ĝ(ξ)}. The inner product in H s (), s R, is defined as follows, (u, v) H s (R n ) := (J s u, J s v) R n = (1 + ξ 2 ) s û(ξ)ˆv(ξ)dξ = u, J 2s v, R n R n u, v H s (R n ), (2.3) (u, v) H s () := ((I P )U, (I P )V ) H s (R n ), u = U, v = V, U, V H s (R n ). Here P : H s (R n ) H s (R n \ ) is the orthogonal projector, see e.g. [13, p. 77]. For a general Lipschitz domain, let {ω j } J Rn be a finite open cover of and {φ j (x) D(ω j )} J be a partition of unity subordinate to it, J φ j(x) = 1 for any x. For any j there exists a half-space domain j such that ω j j = ω j and j can be linearly transformed by a rigid translation κ j to a Lipschitz hypograph j = {x R n 1 : x n > ζ j (x )}, where ζ j are some uniformly Lipschitz functions. Let also κ j : R n R n be the Lipschitz-smooth invertible functions (evidently related to ζ j and κ j ) such that R n + x κ j (x) j, while D j (x ) are the Jacobians of the corresponding boundary mappings R n 1 x κ j (x ) j and D j L (R n 1 ). Similar to [19, page 85] we introduce the following definition. DEFINITION 2.1. Let k, be Lipschitz domains. We say that k as k if k are represented using the same system of covering charts ω j as for all sufficiently large k, and lim ζ jk ζ j C k 0,1 ( ω j ) = 0, (2.4) where ζ jk and ζ j are the corresponding Lipschitz functions for the boundary representation. 2.2 Sobolev spaces characterization, traces and extensions To introduce generalized co-normal derivatives in Section 3, we will need several facts about traces and extensions in Sobolev spaces on Lipschitz domain. First we give the following usual definition of the trace operator. DEFINITION 2.2. An operator γ + : H s ( + ) H σ () is a trace operator if for each u H s () and for any sequence ϕ k D() converging to u in H s (), the sequence of the boundary values ϕ k converges to γ + u in H σ (). The trace operator γ : H s ( ) H σ () is defined similarly. If γ + u = γ u we denote them as γu. 4

5 We have the following well-known trace theorem [4, Lemma 3.6]. THEOREM 2.3. If 1 2 < s < 3 2, then the trace operators γ : H s (R n ) H s 1 2 () and γ ± : H s ( ± ) H s 1 2 (), (2.5) are continuous for any Lipschitz domain. Let γ : H 1 2 s () H s (R n ) denote the operator adjoined to the trace operator, γ v, w = v, γw w H s (R n ), v H 1 2 s (). Now we can prove two-side estimates for the trace operator and its adjoined, which particularly imply a statement about the trace operator unboundedness (cf. [12, Chapter 1, Theorem 9.5] for the unboundedness statements in domains with infinitely smooth boundary). LEMMA 2.4. Let be a Lipschitz domain and 1 2 < s 1. Then C C s v H 1 2 s () γ v H s (R n ) C C s v H 1 2 s () v H 1 2 s () (2.6) and thus C C s γ H s (R n ) H s 1 2 () = γ H 1 2 s (R n 1 ) H s (R n ) C C s, (2.7) where C s := (1 + η 2 ) s dη, C and C are positive constants independent of s and v. The norm of the trace operator γ : H s (R n ) H s 1 2 () tends to infinity as s 1 2 since C s, while the operator γ : H 1 2 (R n ) L 2 (), if it does exist, is unbounded. Proof. Let first consider the lemma for the half-space, = R n + = {x R n : x n > 0}, where x = {x, x n }, x R n 1. For v H 1 2 s (R n 1 ), taking into account the uniqueness of the trace operator for s > 1 2, the distributional Fourier transform gives F x ξ {γ v} = F x ξ {v(x )} =: ˆv(ξ ). Then we have, γ v 2 H s (R n ) R = (1 + ξ 2 ) s ˆv(ξ ) 2 dξ n [ ] = (1 + ξ 2 + ξ n 2 ) s dξ n ˆv(ξ ) 2 dξ = C s v 2, (2.8) R n 1 R H 1 2 s (R n 1 ) 5

6 where the substitution ξ n = (1 + ξ 2 ) 1 2 η was used, cf. [3, Chap. 2, Proposition 4.6]. Thus γ H s (R n ) H s 1 = 2 (R n 1 ) γ 1 = H 2 C s s as s 1 (R n 1 ) H s (R n ) 2. On the other hand, by (2.8) the norm γ v H 1 is not finite for any non-zero v. This means the 2 (R n ) operator γ : H 0 (R n 1 ) H 1 2 (R n ) and thus the operator γ : H 1 2 (R n ) H 0 (R n 1 ) is not bounded, which completes the lemma for = R n + with C = C = 1. Let now be a general Lipschitz domain. For v L 2 (), w D(R n ), using the boundary cover and corresponding partition of unity as in Section 2.1 we have, γ v, w R n = v, γw = v(x)w(x)dσ(x) = J J φ j (x)v(x)w(x)dσ(x) = [(φ j v) κ j ](x )[w κ j ](x )D j (x )dx = R n 1 J J D j (φ j v) κ j, γ 0 [w κ j ] R n 1 = γ0[d j (φ j v) κ j ], w κ j R n, where γ 0, γ 0 are the trace operator on Rn + and its adjoined, respectively. Taking into account density of D(R n ) in H s (R n ) and of L 2 () in H 1 2 s (), we have, γ γ v, w R n v H s (R n ) = sup w H s (R n ) w H s (R n ) = sup w H s (R n ) J γ0[d w κ j j (φ j v) κ j ], w H s (R n ) R n (2.9) for any v H 1 2 s (). It is well known (see e.g. [13, Theorem 3.23 and p. 98]) that v 2 H 1 2 s () = J D j (φ j v) κ j 2 H 1 2 s (R n 1 ), 1 2 < s 3 2, (2.10) C w H s (R n ) w κ j H s (R n ) C w H s (R n ), j = 1,..., J, 0 s 1, (2.11) where C, C are some positive constants independent of s. By (2.8) and (2.10), γ 0[D j (φ j v) κ j ] H s (R n ) = C s D j (φ j v) κ j H 1 2 s (R n 1 ) C s v H 1 2 s (). Then (2.9) and (2.11) imply γ v H s (R n ) C J C s v H 1 2 s () v H 1 2 s (), which is the right inequality in (2.6). On the other hand, we have for v L 2 (), w D(R n ), 6

7 φ j γ v, w R n = v, γ(φ j w) = v(x)φ j (x)w(x)dσ(x) = v(x)φ j (x)w(x)dσ(x) = ω j [(φ j v j ) κ j ](x )[w κ j ](x )D j (x )dx = R n 1 D j [(φ j v j ) κ j ], γ 0 [w κ j ] R n 1 = γ0{d j [(φ j v j ) κ j ]}, w κ j R n. By (2.11) this implies, φ j γ v H s (R n ) = sup γ w κ j w H s (R n ) 0{D j [(φ j v) κ j ]}, w H s (R n ) = R n sup γ w κ j w κ j w H s (R n ) 0{D j [(φ j v) κ j ]}, Hs(Rn) w κ j H s (R n ) w R n H s (R n ) C sup γ w κ j 0{D j [(φ j v) κ j ]}, w κ j H s (R n ) = C γ0{d j [(φ j v) κ j ]} H s (R n ), (2.12) w H s (R n ) that is by (2.8) and (2.10), J J φ j γ v 2 2 H s (R n ) C Since R n γ 0{D j [(φ j v) κ j ]} 2 H s (R n ) = C 2 C s J D j [(φ j v) κ j ] 2 = C 2 C H 1 2 s s v 2. (2.13) (R n 1 ) H 1 2 s () for φ j D(R n ), (2.13) gives the left inequality in (2.6). Obviously, (2.6) implies (2.7) for γ and thus for γ. C j γ v H s (R n ) φ j γ v H s (R n ) (2.14) As was shown in the first paragraph of the proof, the functional γ 0 {D j[(φ j v) κ j ]} is not bounded on H 1 2 (R n ) for any non-zero v, then (2.12), (2.14) imply that the operator γ : H 0 () H 1 2 (R n ) and thus the operator γ : H 1 2 (R n ) H 0 () is not bounded. For s > 3/2 the trace operators (2.5) are not continuous on Lipschitz domains, however the following weaker statement holds, which was mentioned in [5] without a proof but can be indeed proved by appropriate estimates of an integral on p. 598 of [5] for this case. LEMMA 2.5. If is a Lipschitz domain and s > 3/2, then the trace operators γ : H s (R n ) H 1 () and γ ± : H s ( ± ) H 1 () are continuous. 7

8 LEMMA 2.6. For a Lipschitz domain there exists a linear bounded extension operator γ 1 : H s 1 2 () H s (R n ), 1 2 s 3 2, which is right inverse to the trace operator γ, i.e., γγ 1g = g for any g H s 1 2 (). (For s = 1 2 the trace operator γ is understood not as in Definition 2.2 but in the non-tangential sense, see, e.g. [8].) Moreover, γ 1 H s 2 1 C, where C is independent of s. () H s (R n ) Proof. For Lipschitz domains and 1 2 < s 1, the boundedness of the extension operator is well known, see e.g. [13, Theorem 3.37]. To prove it for the whole range 1 2 s 3 2, let us consider the Green operator G that solves the Dirichlet Problem for the Laplace equation in and continuously maps H s 1 2 () to H s () if is a bounded domain and to H s loc () if is an unbounded domain. Particularly one can take G = V V 1, where the single layer potential V φ with a density φ = V 1 g Hs 3 2 (), solves the Laplace equation in with the Dirichlet boundary data g and V is the direct value of the operator V on the boundary. The operators V 1 : Hs 1 2 () H s 3 2 () and V : H s 3 2 () H s loc (Rn ) are continuous for 1 2 s 3 2 as stated in [9, 8, 10, 21, 4]. Thus it suffice to take γ 1 = χg, where χ D(R n ) is a cut-off function such that χ = 1 in a sufficiently large open ball such that it includes the boundary. The estimate γ 1 H s 2 1 C, where C is independent of s, then follows. () H s (R n ) Note that continuity of the operator γ was not needed in the proof. Let us denote by E 0 the operator of extension of a function defined in by zero outside to a function defined in R n. THEOREM 2.7. Let be a Lipschitz domain and s 0 while s k for any integer k 0. Then H s () = H s 0() in the sense that u H0 s() for any u H s (), and E 0 v H s () for any v H0 s (). Moreover u H s () u Hs (), E 0v Hs () C v H s (), (2.15) where C depends only on s and on the maximum of the Lipschitz constants of the representation functions ζ j for the boundary, see Section 2.1. Proof. The first claim is proved in [13, Theorem 3.33]. The first estimate in (2.15) is evident, while the second follows from the proofs of the same Theorem 3.33 and Lemma 3.32 in [13]. To characterize the space H0 s() = H s () for 1 2 < s < 3 2, we will need the following statement. LEMMA 2.8. If is a Lipschitz domain and u H s (), 0 < s < 1 2, then dist(x, ) 2s u(x) 2 dx C u 2 H s () (2.16) 8

9 and for a given boundary cover the constant C depends only on s and on the maximum of the Lipschitz constants of the boundary representation functions ζ j, see Section 2.1. Proof. Note first that the lemma claim for u D() follows from the proof of [13, Lemma 3.32]. To prove it for u H s (), let first the domain be such that dist(x, ) < C 0 < (2.17) for all x, which holds true particularly for bounded domains. Let {ϕ k } D() be a sequence converging to u in H s (). If we denote w(x) = dist(x, ) 2s, then w(x) > C 2s 0 > 0. Since (2.16) holds for functions from D(), the sequence {ϕ k } D() is fundamental in the weighted space L 2 (, w), which is complete, implying that ϕ k D() converges in this space to a function u L 2 (, w). Since both L 2 (, w) and H s () are continuously imbedded in the non-weighted space L 2 (), the sequence {ϕ k } converges in L 2 () implying the limiting functions u and u belong to this space and thus coincide. Then from (2.16) for ϕ k we immediately obtain it for arbitrary u H s (). For the unbounded domains for which condition (2.17) is not satisfied, let χ(x) D(R n ) be a cut-off function such that 0 χ(x) 1 for all x, χ(x) = 1 near, while w(x) < 1 for x supp (1 χ). Then (2.17) is satisfied in = supp χ(x) and w(x) u(x) 2 dx = (1 χ(x))w(x) u(x) 2 dx + χ(x)w(x) u(x) 2 dx u 2 L 2 () + w(x) χ(x)u(x) 2 dx u 2 H s () + C χ(x)u 2 H s ( ) C 1 u 2 H s () due to the previous paragraph. Lemma 2.8 allows now extending the following statement known for (ii)], to a wider range of s. < s 1, see [13, Theorem THEOREM 2.9. If is a Lipschitz domain and 1 2 < s < 3 2, then H s 0() = {u H s () : γ + u = 0}. (2.18) Proof. Equality (2.18) for 1 2 < s 1 is stated in [13, Theorem 3.40(ii)]. Let 1 < s < 3 2. If u Hs 0 () then evidently γ+ u = 0 since D is dense in H0 s () and the trace operator γ + is bounded in H s (). To prove that any u H s () with γ + u = 0 belongs to H0 s (), it remains, due to Theorem 2.7, to prove that E 0 u H s (R n ). We remark first of all that E 0 u H 1 (R n ) due to the previous paragraph and Theorem 2.7, and then make estimates similar to those in the proof of [13, Theorem 3.33], E 0 u 2 H s (R n ) E 0u 2 W 1 2 (Rn ) + R n Rn E 0 u(x) E 0 u(y) 2 x y 2(s 1)+n dx dy 9

10 where + R n \ = u 2 W 1 2 () + u(x) 2 dx dy + x y 2(s 1)+n w s 1 (x) := R n \ u(x) u(y) 2 x y 2(s 1)+n dx dy R n \ dy, x, x y 2(s 1)+n u(y) 2 dx dy x y 2(s 1)+n = u 2 W2 s() + 2 w s 1 (x) u(x) 2 dx, and W2 s () is the Sobolev-Slobodetski space. Introducing spherical coordinates with x as an origin, we obtain, w s 1 (x) α n 2(s 1) dist(x, ) 2(s 1) for x, where α n is the area of the unit sphere in R n. Then, taking into account that u H s 1 () and u H s 1 () u H s (), we have by Lemma 2.8, E 0 u 2 H s (R n ) u 2 W s 2 () + 2C u 2 H s () C s u 2 H s (). Theorem 2.7 completes the proof. Let us now give a characterization of the space H t. THEOREM Let be a Lipschitz domain in R n. (i) If t 1 2, then Ht = {0}. (ii) If 3 2 < t < 1 2, then g Ht if and only if g = γ v, i.e., g, W R n = v, γw W H t (R n ), (2.19) with v = γ 1 g Ht+ 1 2 (), i.e., v, w = g, γ 1 w R n w H t 1 2 (), (2.20) where v is independent of the choice of the non-unique operators γ 1, γ 1, and the estimate v H t+ 1 2 () C g H t (R n ) holds with C independent of t. Proof. We will follow an idea in the proof of Lemma 3.39 in [13] (see also [3, Proposition 4.8]), extending it from a half-space to a Lipschitz domain. Let + = and = R n \. For any ϕ D(R n ), let us define ϕ(x) if x ±, ϕ ± (x) = 0 if x ±. Let t > 1 2. Then ϕ± H t ( ± ) (see e.g. [13, Theorem 3.40] and Theorem 2.7 for 1 2 < t 0, for greater t it then follows by embedding), ϕ ϕ + ϕ H t (R n ) = 0, and there exist sequences {ϕ ± k } D(± ) 10

11 converging to ϕ ± in H t ( ± ) as k. Hence g, ϕ R n = lim k g, ϕ + k + ϕ k R n = 0 for any ϕ D(Rn ) proving (i) for t > 1 2 since D(Rn ) is dense in H t (R n ) = [H t (R n )]. Let us prove (ii). For g H t, 3 2 < t < 1 2, let v Ht+ 1 2 () be defined by (2.20), where existence and continuity of γ 1 : H t 1 2 () H t () is proved in Lemma 2.6. Observe that v, w g H t (R n ) w H t 1 2 () γ 1 H t 1 2 () H t (R n ), so v H t+ 1 2 () γ 1 H t 1 2 () H t (R n ) g H t (R n ) C g H t (R n ), where C is independent of t due to Lemma 2.6 if γ 1 is chosen as in that lemma. We also have that g, W R n v, γw = g, ρ R n W H t (R n ), where ρ = W γ 1 γw H t (R n ). Then we have γρ = 0, which due to Theorems 2.7, 2.9 implies ρ ± H t ( ± ), where ρ ± are extensions of ρ ± by zero outside ±, and ρ = ρ + + ρ. Thus there exist sequences {ρ ± k } D(± ) converging to ρ ± in H t ( ± ), implying g, ρ R n = 0 since g H t, and thus ansatz (2.19). To prove that v is uniquely determined by g, i.e., independent of γ 1, let us consider v and v corresponding to different operators γ 1 and γ 1. Then by (2.19), v v, w = γ 1g γ 1g, w = g, γ 1w γ 1w R n = v, γ(γ 1w γ 1w) = 0 w H t 1 2 (). It remains to deal with the case t = 1 2 in (i). Let g 1 H 2. Since 1 H 2 Ht for 3 2 < t < 1 2, then g = γ v for some v H t+ 1 2 () t ( 3 2, 1 2 ), and g H t = γ v H t C C t v 1 H 2 owing to +t () Lemma 2.4. Since C t as t 1 2, this means v H 1 0 as t 1 2 +t () 2 implying v = 0. Combining (2.19) and (2.20) we have the following useful statement. COROLLARY Let be a Lipschitz domain in R n. If g H t with 3 2 < t < 1 2, then g = γ γ 1 g for any choice of γ 1. THEOREM Let be a Lipschitz domain in R n and s 1 2. Then D() is dense in Hs (), i.e., H s () = H s 0 (). Proof. The proof for 0 s 1 2 is available in [13, Theorem 3.40(i)]. To prove the statement for any s 1 2 we remark that if w Hs () = H s () satisfies w, ϕ = 0 for all ϕ D(), then w H s and Theorem 2.10(i) implies w = 0. Hence, D() is dense in H s (), i.e., H s () = H s 0 (). 11

12 Theorem 2.12 implies that for any u D() and s 1 2 there exists a sequence {ϕ k} D() converging to u in H s (). Evidently ϕ k converges to 0 in H σ () for any σ since ϕ k = 0. On the other hand, u D() is the limit in H s () of the sequence {ϕ k } = u, meaning that ϕ k converges in H σ () to u, which is generally non-zero. This leads to the following conclusion of non-existence. COROLLARY For s 1 2 the trace operators γ± : H s ( ± ) H σ (), understood as in Definition 2.2, do not exist for any σ. REMARK (i) Evidently, Corollary 2.13 holds also if the space H σ () is replaced with any Banach space of distributions on. (ii) The trace operator γ ± : B( ± ) H σ () can, of course, still exist on some Banach subspaces on ±, B( ± ) H s ( ± ), s 1 2, with the norms stronger than the norm in Hs ( ± ), particularly on H t ( ± ), t > 1 2. The following two statements give conditions when distributions from H s () can be extended to distributions from H s () and when the extension can be written in terms of a linear bounded operator. The first of them can be considered as a counterpart of Theorem 2.7 for negative s. LEMMA Let be a Lipschitz domain, s < 1 2, s 1 2 k for any integer k > 0. Then for any g H s () there exists g H s () such that g = g and g Hs () C g H s (), where C > 0 does not depend on g. Proof. Any distribution g H s () is a bounded linear functional on H s (). On the other hand, for any v H s 0 () H s () its zero extension ṽ = E 0 v belongs to H s () with ṽ H s () C v H s () (2.21) for s 0, s 1 2 k, by Theorem 2.7. This holds true also for 0 < s < 1 2 since then H s () = [H s ()] = [H s 0 ()] = [ H s ()] = H s () by Theorems 2.12 and 2.7, while the extension ṽ H s () is defined as ṽ, w := v, E 0 w w H s (), 0 < s < 1 2, and by Theorems 2.12 and 2.7, ṽ H s () = sup w H s ()\{0} ṽ, w w H s () v, E 0 w = sup w H s ()\{0} w H s () C sup w H s ()\{0} v, E 0 w E 0 w Hs () C v H s (). giving estimate (2.21). 12

13 Thus the functional g H s () continuous on H s () and thus on H0 s () can be extended by the Hahn-Banach theorem to a functional g H s () continuous on H s () such that g Hs () = g [H s ()] = g [H s. Then by estimate (2.21) for s < 1 0 ()] 2, s 1 2 k, we have, g, v g [H s = sup C sup 0 ()] v H s 0 ()\{0} v H s 0 () which completes the proof. g, ṽ ṽ H ṽ s ()\{0} H s () C g [ H s ()] = C g H s (), THEOREM Let be a Lipschitz domain and 3 2 < s < 1 2, s 1 2. There exists a bounded linear extension operator Ẽs : H s () H s (), such that Ẽs g = g, g H s (). For 1 2 < s < 1 2 the extension operator is unique, (Ẽs ) = Ẽ s and Ẽs g Hs () C g H s (), (2.22) where C depends only on s and on the maximum of the Lipschitz constants of the representation functions ζ j for the boundary, see Section 2.1. Proof. If 0 s < 1 2, then H s () = {E 0 u, u H s ()}, which implies that one can take Ẽs = E 0, where the operator E 0 : H s () H s () of extension by zero is continuous by the Theorems 2.7 and 2.12 with the estimate (2.22) following from estimate (2.15). If 1 2 < s < 0, we define Ẽs as Ẽs g, v := g, E 0 v, g H s (), v H s (), i.e., Ẽs = E 0 = (Ẽ s ), which is continuous with the estimate (2.22) following from the previous paragraph. Theorem 2.10 implies that the extension operator Ẽs : H s () H s () is unique for 1 2 < s < 1 2. Let now 3 2 < s < 1 2. For s in this range, the trace operator γ+ : H s () H s 1 2 () is bounded due to [4, Lemma 3.6] (see also [13, Theorem 3.38]), and there exists a bounded right inverse to the trace operator γ 1 : H s 1 2 () H s (), see Lemma 2.6. Then (I γ 1 γ + ) is a bounded projector from H s () to H s 0 () = H s () due to Theorem 2.7. Thus any functional v H s () can be continuously mapped into the functional ṽ H s () such that ṽ, u = v, E 0 (I γ 1 γ + )u for any u H s (). Since ṽu = vu for any u H s (), we have, Ẽ s := [E 0 (I γ 1 γ + )] : H s () H s () is a bounded extension operator. Since the extension operator Ẽs : H s () H s () is unique for 1 2 < s < 1 2, we will call it canonical extension operator (as opposite to other possible extensions from H s () to H σ (), σ < 1 2 ). For 3 2 < 13

14 s < 1 2, on the other hand, the operator γ 1 : H s 1 2 () H s () in the proof of Theorem 2.16 is not unique, implying non-uniqueness of Ẽs : H s () H s (). We will later need the following two results. LEMMA Let and be open sets, and s 0. If u H s (), then u H s ( ) 0 as the Lebesgue measure of tends to zero. Proof. Let ϕ D(). Then u H s ( ) u ϕ H s ( ) + ϕ H s ( ) u ϕ H s () + ϕ L2 ( ). For any ϵ > 0 we can chose ϕ such that u ϕ H s () < ϵ/2 due to the density of D() in H s () and then chose with sufficiently small measure so that ϕ L2 ( ) < ϵ/2. LEMMA Let k be a sequence of Lipschitz domains converging to a Lipschitz domain and 1 2 < s < 1/2. If u Hs () and ũ k = Ẽs u k, then there exists a constant C independent of u and k such that ũ k Hs ( k ) C u H s () for all sufficiently large k. Proof. By Theorem 2.16, ũ k Hs ( k ) C k u k H s ( k ) C k u H s (), where C k depend only on s and on the maximum of the Lipschitz constants of the representation functions ζ jk for the boundaries k. By (2.4), the Lipschitz constants are bounded and henceforth so are C k. 3 Partial differential operator extensions and co-normal derivatives for infinitely smooth coefficients Let us consider in a system of m complex linear differential equations of the second order with respect to m unknown functions {u i } m i=1 = u : Cm, which for sufficiently smooth u has the following strong form, Au(x) := where f : C m, j i [a ij (x) j u(x)] + i, b j (x) j u(x) + c(x)u(x) = f(x), x, (3.1) := / x j (j = 1, 2,..., n), a(x) = {a ij (x)} n i, = {{akl ij (x)}m k,l=1 }n i,, b(x) = {{b kl i (x)}m k,l=1 }n i=1 and c(x) = {ckl (x)} m k,l=1, i.e., a ij, b i, c : C m m for fixed indices i, j. If m = 1, then (3.1) is a scalar equation. In this paper we assume that a, b, c C (); the case of non-smooth coefficients is addressed in [14], see also [18]. 14

15 The operator A is (uniformly) strongly elliptic in an open domain if there exists a bounded m m matrix-valued function θ(x) such that Re{ ζ θ(x) a ij (x)ξ i ξ j ζ} C ξ 2 ζ 2 i, for all x, ξ R n and ζ C m, where C is a positive constant, see e.g. [7, Definition 3.6.1] and references therein. We say that the operator A is uniformly strongly elliptic in a closed domain if its is uniformly strongly elliptic in an open domain. We will need the strong ellipticity in relation with the solution regularity, starting from Theorem Partial differential operator extensions and generalized co-normal derivative For u H s (), f H s 2 (), s R, equation system (3.1) is understood in the distribution sense as Au, v = f, v v D(), where v : C m and Au, v := E(u, v) v D(), (3.2) E(u, v) = E (u, v) := a ij j u, i v + b j j u, v + cu, v. (3.3) i, Bilinear form (3.3) is well defined for any v D() and moreover, the bilinear functional E : {H s (), H 2 s ()} C is bounded for any s R. Since the set D() is dense in H 2 s (), expression (3.2) defines then a bounded linear operator A : H s () H s 2 () = [ H 2 s ()], s R, Au, v := E(u, v) v H 2 s (). (3.4) Let now 1 2 < s < 3 2. In addition to the operator A defined by (3.4), let us consider also the aggregate partial differential operator Ǎ, defined as, Ǎu, v := Ě(u, v) v H2 s (), (3.5) where Ẽs 1 Ẽs 1 Ě(u, v) = Ě(u, v) := (a ij j u), i v + (b j j u), v Ẽs 1 + (cu), v i, (3.6) and Ẽs 1 : H s 1 () H s 1 () is a bounded extension operator, which is unique by Theorem Note that by (2.2) one can rewrite (3.5) also as (Ǎu, v) := Φ(u, v) v H 2 s (), 15

16 where Φ(u, v) = Ě(u, v) is the sesquilinear form. If s = 1, i.e. u, v H 1 (), evidently Ě(u, v) = E(u, v) = (a ij j u) i v + i, (b j j u) v + cu v dx. The aggregate operator Ǎ : Hs () H s 2 () = [H 2 s ()] is bounded since i v H 1 s (), v H 2 s () H 1 s (). For any u H s (), the functional Ǎu belongs to H s 2 () and is an extension of the functional Au H s 2 () from the domain of definition H 2 s () H 2 s () to the domain of definition H 2 s (). The distribution Ǎu is not the only possible extension of the functional Au, and any functional of the form Ǎu + g, g H s 2 (3.7) gives another extension. On the other hand, any extension of the domain of definition of the functional Au from H 2 s () to H 2 s () has evidently form (3.7). The existence of such extensions is provided by Lemma For u H s (), s > 3 2, the strong (classical) co normal derivative operator T + c u(x) := a ij (x) γ + [ j u(x)]ν i (x) (3.8) i, is well defined on in the sense of traces. Here γ + [ j u] H s 3 2 () L 2 () if 3 2 < s < 5 2, while the outward (to ) unit normal vector ν(x) at the point x belongs to L () for the Lipschitz boundary, implying T + c u L 2 (). Note that for Lipschitz domains one can not generally expect that T + c u belongs to H s (), s > 0, even for infinitely smooth u. We can extend the definition of the generalized co normal derivative, given in [13, Lemma 4.3] for s = 1 (cf. also [11, Lemma 2.2] for the generalized co normal derivative on a manifold boundary), to a range of Sobolev spaces as follows. DEFINITION 3.1. Let be a Lipschitz domain, 1 2 < s < 3 2, u Hs (), and Au = f in for some f H s 2 (). Let us define the generalized co normal derivative T + ( f, u) H s 3 2 () as T + ( f, u), w := Ě(u, γ 1w) f, γ 1 w = Ǎu f, γ 1 w w H 3 2 s (), (3.9) where γ 1 : H 3 2 s () H 2 s () is a bounded right inverse to the trace operator. The notation T + ( f, u) corresponds to the notation T + ( f, u) in [17]. 16

17 THEOREM 3.2. Under the hypotheses of Definition 3.1, the generalized co normal derivative T + ( f, u) is independent of the operator γ 1, the estimate T + ( f, u) H s 3 2 () C 1 u H s () + C 2 f Hs 2 () (3.10) takes place, and the first Green identity holds in the following form, T + ( f, u), γ + v = Ě(u, v) f, v = Ǎu f, v v H 2 s (). (3.11) Proof. For s = 1 the theorem proof is available in [13, Lemma 4.3], which idea is extended here to the whole range 1 2 < s < 3 2. By Lemma 2.6, a bounded operator γ 1 : H 3 2 s () H 2 s () does exist. Then estimate (3.10) follows from (3.9). To prove independence of the co-normal derivative T + ( f, u) of γ 1, let us consider two co-normal derivatives generated by two different operators γ 1 and γ 1. Then their difference is T + ( f, u) T + ( f, u), w = Ǎu f, γ 1w γ 1w w H 3 2 s (). By definition, Ǎu f H s 2, which by Corollary 2.11 implies that Ǎu f, γ 1w γ 1w = Ǎu f, γ 1w γ 1w R n = γ γ 1(Ǎu f), γ 1w γ 1w R n = γ 1(Ǎu f), γγ 1w γγ 1w = γ 1(Ǎu f), w w = 0 w H 3 2 s (). To prove (3.11), let V H 2 s (R n ) be such that v = V implying γ + v = γv. Taking again into account that Ǎu f H s 2, we have by Corollary 2.11, T + ( f, u), γ + v = Ǎu f, γ 1 γ + v = Ǎu f, γ 1 γv R n = γ γ 1(Ǎu f), V R n = Ǎu f, V R n = Ǎu f, v as required. Because of the involvement of f, the generalized co-normal derivative T + ( f, u) is generally non-linear in u. It becomes linear if a linear relation is imposed between u and f (including behavior of the latter on the boundary ), thus fixing an extension of f into H s 2 (). For example, f can be extended as ˇf := Ǎu, which generally does not coincide with f. Then obviously, T + ( ˇf, u) = T + (Ǎu, u) = 0, meaning that the co-normal derivatives associated with any other possible extension f appears to be aggregated in ˇf as ˇf, v = f, v + T + ( f, u), γ + v 17 (3.12)

18 due to (3.11). This justifies the term aggregate for the extension ˇf, and thus for the operator Ǎu. As follows from Definition 3.1, the generalized co-normal derivative is still linear with respect to the couple ( f, u), i.e., T + (α 1 f1, α 1 u 1 ) + T + (α 2 f2, α 2 u 2 ) = T + (α 1 f1 + α 2 f2, α 1 u 1 + α 2 u 2 ) for any complex numbers α 1, α 2. In fact, for a given function u H s (), 1 2 < s < 3 2, any distribution τ Hs 3 2 () may be nominated as a co-normal derivative of u, by an appropriate extension f of the distribution Au H s 2 () into H s 2 (). This extension is again given by the second Green formula (3.11) re-written as follows (cf. [2, Section 2.2, item 4] for s = 1), f, v := Ě(u, v) τ, γ + v = Ǎu γ+ τ, v v H 2 s (). (3.13) Here the operator γ + : H s 3 2 () H s 2 () is adjoined to the trace operator, γ + τ, v := τ, γ + v for all τ H s 3 2 () and v H 2 s (). Evidently, the distribution f defined by (3.13) belongs to H s 2 () and is an extension of the distribution Au into H s 2 () since γ + v = 0 for v H 2 s (). For u C 1 () H 1 (), one can take τ equal to the strong co-normal derivative, T c + u L (), and relation (3.13) can be considered as the classical extension of f = Au H 1 () to f c H 1 (), which is evidently linear. 3.2 Boundary value problems Consider the BVP weak settings for PDE system (3.1) on Lipschitz domain for 1 2 < s < 3 2. The Dirichlet problem: for f H s 2 () and φ 0 H s 1 2 (), find u H s () such that Au, v = f, v v H 2 s (), (3.14) γ + u = φ 0 on. (3.15) The Neumann problem: for ˇf H s 2 (), find u H s () such that Here Au and Ǎu are defined by (3.4) and (3.5), respectively. Ǎu, v = ˇf, v v H 2 s (). (3.16) To set the mixed problem, let D and N = \ D be nonempty, open sub manifolds of, and H s 0 (, D) = {w H s () : γ + w = 0 on D }. We introduce the mixed aggregate operator Ǎ D : H s () [H 2 s 0 (, D )], defined as Ǎ D u, v := Ǎu, v = Ě(u, v) v H2 s 0 (, D ). 18

19 The mixed operator Ǎ D is bounded by the same argument as the aggregate operator Ǎ. For any u H s (), the distribution Ǎ D u belongs to [H 2 s 0 (, D )] and is an extension of the functional Au H s 2 () from the domain of definition H 2 s () = H 2 s 0 () H 2 s 0 (, D ) to the domain of definition H 2 s 0 (, D ), and a restriction of the functional Ǎu H s 2 () from the domain of definition H 2 s () H 2 s 0 (, D ) to the domain of definition H 2 s 0 (, D ). For v H 2 s 0 (, D ), the trace γ + v belongs to H s 1 2 ( N ). If Au = f in for some f H s 2 (), then the first Green identity (3.11) gives, Ǎ D u, v = ˇf m, v, ˇf m, v = f, v + T + ( f, u), γ + v N v H 2 s 0 (, D ), (3.17) where, evidently, ˇf m [H 2 s 0 (, D )]. This leads to the following weak setting. The mixed (Dirichlet-Neumann) problem: for ˇf m [H 2 s 0 (, D )] and φ 0 H s 1 2 ( D ), find u H s () such that Ǎ D u, v = ˇf m, v v H 2 s 0 (, D ), (3.18) γ + u = φ 0 on D. (3.19) The Neumann and the mixed problems are formulated in terms of the aggregate right hand sides ˇf and ˇf m, respectively, prescribed on their own, i.e., without necessary splitting them into the right hand side inside the domain and the part related with the prescribed co-normal derivative. If a right hand side extension f and an associated non-zero generalized co-normal derivative T + ( f, u) are prescribed instead, then ˇf and ˇf m can be expressed through them by relations (3.12), (3.17). Thus the co-normal derivative does not enter, in fact, the weak settings of the Dirichlet, Neumann or mixed problem, implying that the non-uniqueness of T + ( f, u) for a given function u H s (), 1 2 < s < 3 2, does not influence the BVP weak settings, (cf. [2, Section 2.2, item 4] for s = 1). On the other hand, for a given u H s () the aggregate right hand sides ˇf and ˇf m are uniquely determined by (3.16), (3.18), as are, of course, f and φ 0 by (3.14), (3.15)/(3.19). Note that one can take v = w to make the settings (3.14)-(3.15), (3.16) and (3.18)-(3.19) look closer to the usual variational formulations, cf. e.g. [12]. 3.3 Canonical co-normal derivative As we have seen above, for an arbitrary u H s (), 1 2 < s < 3 2, the co-normal derivative T + ( f, u) is generally non-uniquely determined by u. An exception is T + (Ǎu, u) 0 but such co-normal derivative evidently differs from the strong co-normal derivative T + c u, given by (3.8) for sufficiently smooth u. Another 19

20 one way of making generalized co-normal derivative unique in u H 1 () was presented in [7, Lemma 5.1.1] and is in fact associated with an extension of Au H 1 () to f H 1 (), such that f is orthogonal in H 1 (R n ) to H 1 H 1 (R n ). However it appears (see Lemma A.1), that even for infinitely smooth functions f such extension f does not generally belong to L 2 (R n ), which implies that the so-defined conormal derivative operator τ from [7, Lemma 5.1.1] is not a bounded extension of the strong co-normal derivative operator. Nevertheless, it is still possible to point out some subspaces of H s 1 (), 2 < s < 3 2, where a unique definition of the co-normal derivative by u is possible and leads to the strong co-normal derivative for sufficiently smooth u. We define below one such sufficiently wide subspace. DEFINITION 3.3. Let s R and A : H s () D () be a linear operator. For t 1 2, we introduce a space H s,t (; A ) := {g : g H s (), A g = f g, fg H t ()} equipped with the graphic norm, g 2 H s,t (;A ) := g 2 H s () + f g 2 Ht (). The distribution f g H t (), t 1 2, in the above definition is an extension of the distribution A g H t (), and the extension is unique (if it does exist), since otherwise the difference between any two extensions belongs to H t but Ht = {0} for t 1 2 due to the Theorem The uniqueness implies that the norm g H s,t (;A ) is well defined. Note that another subspace of such kind, where A g belongs to L p () instead of H t (), was presented in [6, p. 59]. A particular case, H s,0 (; A ), was extensively employed in [4]. If s 1 s 2 and t 1 t 2, then we have the embedding, H s 2,t 2 (; A ) H s 1,t 1 (; A ). REMARK 3.4. If s R, 1 2 < t < 1 2, and A : H s () H t () is a linear continuous operator, then H s,t (; A ) = H s () by Theorem LEMMA 3.5. Let s R. If a linear operator A : H s () D () is continuous, then the space H s,t (; A ) is complete for any t 1 2. Proof. Let {g k } be a Cauchy sequence in H s,t (; A ). Then there exists a Cauchy sequence { f k } in H t () such that f k = A g k. Since H s () and H t () are complete, there exist elements g 0 H s () and f 0 H t () such that g k g 0 H s () 0, f k f 0 0 as k. On the other hand, continuity Ht () of A implies that A (g k g 0 ), ϕ 0 for any ϕ D(). Taking into account that A g k = f k, we obtain f 0 A g 0, ϕ f 0 f k, ϕ + f k A g 0, ϕ f 0 f k Ht () ϕ H t () + A (g k g 0 ), ϕ 0, k ϕ D(), i.e., A g 0 = f 0 H t (), which implies A g 0 is extendable to f 0 H t () and thus g 0 H s,t (; A ). 20

21 We will further use the space H s,t (; A ) for the case when the operator A is the operator A from (3.2) or the operator A formally adjoined to it (see Section 4). DEFINITION 3.6. Let s R, t 1 2. The operator à mapping functions u Hs,t (; A) to the extension of the distribution Au H t () to H t () will be called the canonical extension of the operator A. REMARK 3.7. If s R, t 1 2, then Ãu Ht () u H s,t (;A) by definition of the space H s,t (; A), i.e., the linear operator à : Hs,t (; A) H t () is continuous. Moreover, if 1 2 < t < 1 2, then by Theorem 2.16 and uniqueness of the extension of H t () to H t (), we have the representation à := Ẽt A. As in [17, Definition 3] for scalar PDE, let us define the canonical co-normal derivative operator. This extends [6, Theorem ] and [4, Lemma 3.2] where co-normal derivative operators acting on functions from Hp 1,0 (; ) and H 1,0 (; A), respectively, were defined. DEFINITION 3.8. For u H s, 1 2 (; A), 1 2 < s < 3 2, we define the canonical co-normal derivative as T + u := T + (Ãu, u) Hs 3 2 (), i.e., T + u, w := Ě(u, γ 1w) Ãu, γ 1w = Ǎu Ãu, γ 1w w H 3 2 s (), where γ 1 : H s 1 2 () H s () is a bounded right inverse to the trace operator. Theorem 3.2 for the generalized co-normal derivative and Definition 3.3 imply the following statement. THEOREM 3.9. Under hypotheses of Definition 3.8, the canonical co-normal derivative T + u is independent of the operator γ 1, the operator T + : H s, 1 2 (; A) H s 3 2 () is continuous, and the first Green identity holds in the following form, T + u, γ + v = T + (Ãu, u), γ+ v = Ě(u, v) Ãu, v = Ǎu Ãu, v v H 2 s (). Thus unlike the generalized co-normal derivative, the canonical co-normal derivative is uniquely defined by the function u and the operator A only, uniquely fixing an extension of the latter on the boundary. Definitions 3.1 and 3.8 imply that the generalized co-normal derivative of u H s, 1 2 (; A), 1 2 < s < 3 2, for any other extension f H s 2 () of the distribution Au H 1 2 () can be expressed as T + ( f, u), w = T + u, w + Ãu f, γ 1 w w H 3 2 s (). Note that the distributions Ǎu f, Ǎu Ãu and à f belong to H 2 s H 2 s (), while Ãu = Ǎu = f = Au H s 2 (). since Ãu, Ǎu, f belong to 21

22 T + u, w = Ě(u, γ 1 w) Ã u, γ 1w = Ě(u, γ 1 w) Ãu, γ 1 w = T + u, w JMAA (accepted, to appear) Since by Theorem 3.9 the canonical co-normal derivative does not depend on the extension operator γ 1, the latter can be always chosen such that γ 1 w has a support only near the boundary, which means that the co-normal derivative T + u is determined by the behavior of u near the boundary. We can formalize this in the following statement. THEOREM Let and be bounded or unbounded open Lipschitz domains,, u H s, 1 2 (; A), u H s, 1 2 ( ; A), 1 2 < s < 3 2, while T + u and T + u be the canonical co-normal derivatives on and respectively. Then T + u = r T + u. Proof. By the definition of the restriction operator r and Definition 3.8 we have, T + u, w := Ě (u, γ 1w) Ã u, γ 1w w H 3 2 s ( ) : r \ w = 0, where γ 1 : Hs 1 2 ( ) H s ( ) is a bounded right inverse to the trace operator. Since γγ 1 w = 0 on \, we can extend γ 1 w by zero on \ to γ 1 w. The operator γ 1 : H s 1 2 () H s () is continuous, and we arrive at w H 3 2 s (), Theorem 3.10 can be considered as an alternative definition of the canonical co-normal derivative, where the domain can be chosen arbitrarily small, and particularly can be take bounded when is unbounded (with compact boundary). Note that similar reasoning holds also for the generalized co-normal derivative. To give conditions when the canonical co-normal derivative T + u coincides with the strong co-normal derivative T c + u, if the latter does exist in the trace sense, we prove in Lemma 3.12 below that D() is dense in H s,t (; A). The proof is based on the following local regularity theorem well known for the case of infinitely smooth coefficients, see e.g. [20, 1, 12]. THEOREM Let be an open set in R n, s 1 R, function u H s 1 loc ()m, m 1, satisfy strongly elliptic system (3.1) in with f H s 2 loc ()m, s 2 > s 1 2, and infinitely smooth coefficients. Then u H s 2+2 loc () m. Now we are in the position to prove the density theorem THEOREM If is a bounded Lipschitz domain, s R, 1 2 t < 1 2 and the operator A is strongly elliptic on, then D() is dense in H s,t (; A). 22

23 Proof. We modify appropriately the proof from [6, Lemma ] given for another space of such kind associated with the Laplace operator. For every continuous linear functional l on H s,t (; A) there exist distributions h H s () and g H t () such that l(u) = h, u + g, Ãu. To prove the lemma claim, it suffice to show that any l, which vanishes on D(), will vanish on any u H s,t (; A). Indeed, if l(ϕ) = 0 for any ϕ D(), then h, ϕ + g, Ãϕ = 0. (3.20) Let us consider the case 1 2 < t < 1 2 first and extend g outside to g = Ẽ t g H t (). Equation (3.20) gives by Theorem 2.16, h, ϕ + g, Aϕ = h, ϕ + g, Aϕ = h, ϕ + Ẽ t g, Aϕ = h, ϕ + g, Ẽt Aϕ = h, ϕ + g, Ãϕ = 0 for any ϕ D( ) on some domain, where the operator A is still strongly elliptic. This means A g = h in (3.21) in the sense of distributions, where A is the operator formally adjoint to A. If t s 2, then evidently g H 2 s (). If t > s 2, then (3.21) and Theorem 3.11 imply g H 2 s loc ( ) and consequently g H 2 s (). In the case t = 1 2, one can extend g H 1 2 () outside by zero to g H 1 2 ϵ (), 0 < ϵ, and prove as in the previous paragraph that g H 2 s (). If 1 2 < t < 1 2 or [t = 1 2, s 3 2 ] then for any u Hs,t (; A), we have, l(u) = A g, u + g, Ãu = g, Au + g, Au = 0. Thus l is identically zero. On the other hand, if t = 1 2, s > 3 2, let { g k} D() be a sequence converging, as k, to g in H () = H 1 2 (), cf. Theorem 2.12, and thus to g in H 2 s (). Then for any u H s, 1 2 (; A), we have, l(u) = A g, { u + g, Ãu = lim A g } k, u + g k, Ãu k which completes the proof. = lim k { g k, Au + g k, Au } = 0, 23

24 LEMMA Let u H s, 1 2 (; A), 1 2 < s < 3 2, and {u k} D() be a sequence such that Then T c + u k T + u H s 3 0 as k. 2 () u k u H s, as k. (3.22) (;A) Proof. Using the definition of T + u and the classical first Green identity for u k, we have for any w H 3 2 s (), T + u T c + u k, w = Ě(u u k, γ 1 w) Ã(u u k), γ 1 w C u u k H s, 1 2 (;A) w H 3 2 s (). This implies T c + u k T + u H s 3 u u k 2 () H s, 1 0 as k. 2 (;A) Note that a sequence satisfying (3.22) does always exist for bounded Lipschitz domains by Theorem The following statement gives the equivalence of the classical co-normal derivative (in the trace sense) and the canonical co-normal derivative, for functions from H s (), s > 3 2. COROLLARY If u H s (), s > 3 2, then T + u = T + c u L 2 (). Proof. If u H s (), 3 2 < s < 5 2, then γ+ [ j u] H s 3 2 (), T + c u L 2 () and u H s,s 2 (; A) H s, 1 2 (; A) H 1, 1 2 (; A) by Remark 3.4. Let {u k } D() be a sequence such that u k u H s () 0 and thus Then u k u H 1, 1 2 (;A) u k u H s, 1 2 (;A) C u k u H s () 0, k. T + u T + c u H 1 2 () T + u T + c u k H 1 2 () + T + c (u k u) H 1 2 (), where the first norm in the right hand side vanishes as k by Lemma 3.13, while for the second norm we have, T + c (u k u) H 1 2 () a ij γ + [ j (u k u)]n j L2 () i, C 1 a L () γ + (u k u) L2 () C 2 a L () u k u H s () 0, k. For s 5 2 the corollary follows by imbedding. 24

25 For a Lipschitz domain, the membership u H s,t loc (; A) with 1 2 < s < 3 2, 1 2 < t < 1 2 implies by Theorem 3.11 that u H t+2 loc (). Thus u Ht+2 loc ( 1) for any Lipschitz subdomain 1 of such that 1. On 1 then T + u = T + c u L 2 ( 1 ) by Corollary LEMMA Let and { k } be Lipschitz domains such that k and k as k (cf. Definition 2.1). If u H s,t loc (; A) for some s ( 1 2, 3 2 ) and t ( 1 2, 1 2 ), then T + u, v + = lim k T + c u, v + k for any v H 2 s ( + ). Proof. By Theorem 3.10 it suffice to consider only a bounded domain. Let k := \ k be the layer between and k. By Theorem 3.11, u H t+2 loc (), which by Corollary 3.14 implies T + u = T + c u L 2 ( k ) on k. Then T + u, v + T + c u, v + k = T + u, v + k = Ě k (u, v) à k u, v k = Ě k (u, v) Au, ṽ k k, (3.23) where à u = Ẽt k r k k Au H t ( k ) and ṽ = Ẽ t k kr k v H t ( k ) are the unique extensions of r Au k H t ( k ) and r v H2 s ( k k ) H t ( k ), respectively. By (3.6) and Theorem 2.16 we have for the first term in the right hand side of (3.23), Ě (u, v) C k i, a ij L ( k ) j u H s 1 ( k ) i v H 1 s ( k ) + C b j L ( k ) j u H s 1 ( k ) v H 1 s ( k ) + C c L ( k ) u H s 1 ( k ) v H 1 s ( k ), where C does not depend on k for sufficiently large k. Then for 1 2 < s 1, Ě (u, v) C k C i, a ij L () j u H s 1 ( k ) i v H 1 s ()+ b j L () j u H s 1 ( k ) v H 1 s () + C c L () u H s 1 ( k ) v H 1 s () {C 1 u H s 1 ( k ) + C 2 u H s 1 ( k ) } v H 2 s () 0, k by Lemma 2.17 since the Lebesgue measure of k tends to zero. For 1 < s < 3 2 similarly, Ě (u, v) C k C i, a ij L () j u H s 1 () i v H 1 s ( k ) + b j L () j u H s 1 () v H 1 s ( k ) + C c L () u H s 1 () v H 1 s ( k ) {C 3 v H 1 s ( k ) + C 4 v H 1 s ( k ) } u H s () 0, k. 25