A PROOF OF THE TRACE THEOREM OF SOBOLEV SPACES ON LIPSCHITZ DOMAINS

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1 POCEEDINGS OF THE AMEICAN MATHEMATICAL SOCIETY Volume 4, Number, February 996 A POOF OF THE TACE THEOEM OF SOBOLEV SPACES ON LIPSCHITZ DOMAINS ZHONGHAI DING (Communicated by Palle E. T. Jorgensen) Abstract. A complete proof of the trace theorem of Sobolev spaces on Lipschitz domains has not appeared in the literature yet. The purpose of this paper is to give a complete proof of the trace theorem of Sobolev spaces on Lipschitz domains by taking advantage of the intrinsic norm on H s ( Ω). It is proved that the trace operator is a linear bounded operator from H s (Ω) to H s ( Ω) for <s< 3.. Introduction Sobolev spaces play an important role in the study of partial differential equations on smooth nonsmooth domains their boundary value problems. So far the studies of these problems have shown that such spaces are quite appropriate natural. In recent years, boundary value problems on Lipschitz domains the method of layer potentials for their solutions have attracted attention in both pure applied mathematics (see [6], [7], [9] the references therein). On the other h, there are a lot of engineering problems such as point sensor placements at vertex points of the boundary in the mechanical engineering, corrosive engineering spacecraft, point controllers at vertex points of the boundary in the stabilization of structural dynamics, etc., which require very careful local analysis around vertex points of the boundary (see [3] the references therein). Thus Sobolev spaces on Lipschitz domains play a very important role in those studies. Most properties of Sobolev spaces on Lipschitz domains are rigorously proved (see [], [5], [8]). But a complete proof of the trace theorem of Sobolev spaces on Lipschitz domains has not appeared in the literature, to the best of the author s knowledge. On a bounded Lipschitz domain Ω with boundary Ω, we can only define H s ( Ω) in a unique invariant way for s. Thus the trace properties are different from that of Sobolev spaces on smooth domains. For Lipschitz domains, E. Gagliardo [4] (957) proved the trace theorem for H s (Ω) where <s. D. S. Jerison C. E. Kenig [6] (98) stated the trace theorem for the case s = 3 without any proof. M. Costabel [] (988) proved a trace theorem on special Lipschitz domains for the range <s< 3. However, he did not use the natural intrinsic norm on H s ( Ω). It is not obvious that the trace norm in (4.7) of Costabel s paper [] is equivalent to the natural intrinsic trace norm. Because of the needs in studies eceived by the editors September 5, Mathematics Subject Classification. Primary 46E35. Key words phrases. Sobolev spaces, Lipschitz domains, trace theorem. 59 c 996 American Mathematical Society

2 59 ZHONGHAI DING of boundary value problems on Lipschitz domains, optimization stabilization of structural dynamics on Lipschitz domains other contemporary engineering problems, it is necessary to give a complete proof of the trace theorem of Sobolev spaces on Lipschitz domains, which is the purpose of this paper. The idea of our proof is to use Costabel s approach, the interpolation theorem [8] the intrinsic norm of H s ( Ω). In this paper the trace theorem of Sobolev spaces on Lipschitz domains for the range <s< 3 is proved. For s> 3, the trace operator is a bounded linear operator from H s (Ω) to H ( Ω), whose proof is analogous to the proof given in this paper. For s = 3, no proof is available.. Definition of Sobolev spaces on Lipschitz domains Definition (see [9]). A bounded simply connected open subset Ω N is called a Lipschitz domain if P Ω, there exist a rectangular coordinate system (x, s) (x N,s ), a neighborhood V (P) N of P, a function ϕ P : N such that (a) ϕ P (x) ϕ P (y) C P x y, C P < +, x, y N ; (b) V (P ) Ω= { (x, s) N s>ϕ(x) } V(P). The coordinate system (x, s) in Definition may be chosen as a combination of rotation translation of the stard rectangular coordinate system for N.The neighborhood V (P ) may be chosen as a cylinder Z(P, r P ),whichisanopenright circular, doubly truncated cylinder centered at P N with radius r P satisfying the following properties: (a) the bases of Z(P, r P ) are some positive distance from Ω; (b) the s-axis contains the axis of Z(P, r P ); (c) ϕ P may be taken to have compact support in N, supp(ϕ P ) B(0, r P ); (d) P =(0,ϕ P (0)). The pair (Z(P, r P ),ϕ P ) will be called a cylinder coordinate pair at P Ω. By the compactness of Ω, it is possible to cover Ω with a finite number of cylinder coordinate pairs {(Z(P k,r k ),ϕ k )}, k N 0. For the Lipschitz domain Ω a given covering of cylinder coordinate pairs, {(Z(P k,r k ),ϕ k )}, k N 0,thereis anumberm>0such that all φ Pk, k N 0, have Lipschitz norms less than or equal to M. The smallest of such numbers is called the Lipschitz constant for Ω. Definition (see []). Let m 0 be an integer. Denote by H m (Ω), the Sobolev space, the space of all distributions u defined in Ω such that D α u L (Ω), H m (Ω) is equipped with the norm u m,ω = α m Ω α m. D α u dx It is easy to verify that H m (Ω) is a Banach space. If we define the inner product generated from m,ω in a natural way, then H s (Ω) is a Hilbert space. The Sobolev space of non-integer order, H s (Ω), is defined by the real interpolation method (see.

3 A POOF OF THE TACE THEOEM OF SOBOLEV SPACES 593 [], [8]). According to [] [5], when s = m + σ 0 <σ<, H s (Ω) can be equipped with an equivalent intrinsic norm u s,ω = u m,ω + D α u(x) D α u(y) Ω Ω x y N+σ dxdy. α =m Definition 3 (see [5]). Denote by H s ( Ω) (0 <s<), the Sobolev trace space, the space of all distributions u defined on Ω such that u(x) u(x) u(y) dσ x + Ω Ω Ω x y N +s dσ xdσ y < +. H s ( Ω) is equipped with the norm { } u s, Ω = u(x) u(x) u(y) dσ x + Ω Ω Ω x y N +s dσ xdσ y. According to [5] (page 0) the trace spaces defined above are the same as the trace spaces introduced in [] [8]. 3. Proof of the trace theorem Definition 4. For any u C (Ω), define the trace operator γ Ω by γ Ω u(x) =u(x), x Ω. When Ω = N +, denote by γ 0 the trace operator γ N 0 where N 0 = N + = { x N x=(x,0), x N }. The trace theorem of Sobolev spaces on Lipschitz domains is as follows. Theorem. Let Ω be a bounded simply connected Lipschitz domain < s< 3. Then the trace operator γ Ω is a bounded linear operator from Hs (Ω) to H s ( Ω). Before we prove this theorem, we need to establish several lemmas. Definition 5. Ω N is called a special Lipschitz domain if there exists a function ϕ : N such that ϕ(x ) ϕ(y ) M x y, x,y N, Ω= { x N x N >ϕ(x ),x N } Ω = { x N x N =ϕ(x ),x N }. Lemma. Let Ω be a special Lipschitz domain. Define a linear operator T ϕ : L (Ω) L ( N + ), where N + = { x N x N >0 },by (T ϕ u)(x) =u(x,x N +ϕ(x )), a.e.x N +. Then T ϕ is a bounded linear operator from H s (Ω) to H s ( N + ) for 0 s.

4 594 ZHONGHAI DING Proof. From the definition of T ϕ,itiseasytoverifythatt ϕ L(L (Ω),L ( N + )), where L(X, Y ) always denotes the space of all linear bounded operators from X to Y.Ifu H (Ω), then a simple computation yields x i (T ϕ u)(x) = u(x,x N +ϕ(x )) + x i i N, u(x,x N +ϕ(x )) ϕ (x ), x N x i (T ϕ u)(x) = u(x,x N +ϕ(x )). x N x N Since ϕ is uniformly Lipschitz continuous with Lipschitz constant M, T ϕ u H ( N + ) () T ϕ u H ( N + ) C u H (Ω), where C is only dependent of Ω. Thus T ϕ L(H (Ω),H ( N + )). By the interpolation theorem [8], it follows that for 0 s T ϕ L(H s (Ω),H s ( N +)). Any u L ( N + ) can be understood as a function from + to L ( N ), i.e. x N u(,x N ), thus u L ( +,L ( N )). Lemma. Let Ω be a special Lipschitz domain. Let T ϕ be as defined in Lemma s. Then T ϕ is a bounded linear operator from H s (Ω) to Proof. By Lemma, () (3) Let u H (Ω); then we have k x k N H s ( + ; L ( N )) H s ( + ; H ( N )) T ϕ L(H (Ω),H ( + ;L ( N ))) T ϕ L(H (Ω),L ( + ;H ( N ))). (T ϕ u)(x) = k x k u(x,x N +ϕ(x )), k =,, a.e. x N +. N Thus T ϕ u H ( + ; L ( N )) T ϕ u H ( +;L ( N )) u H (Ω). Hence (4) T ϕ L(H (Ω),H ( + ;L ( N ))). Note that for u H (Ω), x i (T ϕ u)(x) = x i u(x,x N +ϕ(x )) + u(x,x N +ϕ(x )) ϕ(x ), x N x i i N, a.e. x N +;

5 A POOF OF THE TACE THEOEM OF SOBOLEV SPACES 595 (T ϕ u)(x) = u(x,x N +ϕ(x )) x N x i x N x i + x N u(x,x N +ϕ(x )) x i ϕ(x ) i N,a.e. x N +. Hence T ϕ u H ( + ; H ( N )), T ϕ u H ( +;H ( N )) C u H (Ω), where C is only dependent on the Lipschitz constant of Ω. Therefore (5) T ϕ L(H (Ω),H ( + ;H ( N ))). By applying the interpolation theorem [8], we obtain from () (4) that T ϕ L(H s (Ω),H s ( + ;L ( N ))), s, from (3) (5) that T ϕ L(H s (Ω),H s ( + ;H ( N ))), s. Therefore we have T ϕ L(H s (Ω),H s ( + ;L ( N )) H s ( + ; H ( N ))). Lemma 3. Let Ω be a special Lipschitz domain 0 s<. Define a linear operator S ϕ : L ( Ω) L ( N ) by S ϕ u(x )=u(x,ϕ(x )), a.e. x N. Then S ϕ is a linear bounded invertible operator from H s ( Ω) to H s ( N ). Proof. Since Ω is a special Lipschitz domain, dσ x = N ϕ (x x ) i i= N ϕ (x x ) i i= + dx + +(N )M, where x =(x,ϕ(x )) Ω. Thus (6) S ϕ u(x ) dx u(x) dσ x +(N )M S ϕ u(x N Ω ) dx, N where M is the Lipschitz constant of Ω. Therefore S ϕ is bounded injective from L ( Ω) to L ( N ). For any f L ( N ), define u(x,x N )=f(x )φ(x N ϕ(x )), a.e. x N, where φ C0 () φ(0) =. By (6), we have γ Ω u L ( Ω) S ϕ (γ Ω u)= f. Hence S ϕ is surjective from L ( Ω) to L ( N ). Thus by (6), S ϕ is a linear bounded invertible operator from L ( Ω) to L ( N ).

6 596 ZHONGHAI DING Notice that x, y Ω, x =(x,ϕ(x )) y =(y,ϕ(y )), x y x y +M x y. For any u H s ( Ω) 0 <s<, u s, Ω = u(x) u(x) u(y) dσ x + Ω Ω Ω x y N +s dσ xdσ y u(x,ϕ(x )) dx N u(x,ϕ(x )) u(y,ϕ(y )) + ( + M ) N +s N N x y N +s dx dy ( + M ) S ϕu N +s s, N. On the other h, u s, Ω = u(x) u(x) u(y) dσ x + Ω Ω Ω x y N +s dσ xdσ y +(N )M N u(x,ϕ(x )) dx + {+(N )M u(x,ϕ(x )) u(y,ϕ(y )) } N N x y N +s dx dy {+(N )M } S ϕ u s, N. Hence (7) S ( + M ) N +s ϕ u s, N u s, Ω +(N )M S ϕ u s, N. 4 Hence S ϕ is bounded injective from H s ( Ω) to H s ( N ). For any f H s ( N ), define u(x,x N )=f(x )φ(x N ϕ(x )), a.e. x N, where φ C0 () φ(0) =. It is easy to check that γ Ω u H s ( Ω) γ Ω u = f. Therefore S ϕ is surjective from H s ( Ω) to H s ( N ). Hence S ϕ is a linear bounded invertible operator from H s ( Ω) to H s ( N ). Theorem. Let Ω be a special Lipschitz domain operator γ Ω : H s (Ω) H s ( Ω) is continuous. <s. Then the trace Proof. For <s, applying the trace theorem on half-space N + have γ 0 is a bounded linear operator from H s ( N + )ontoh s ( N 0 ). [[], [8]], we

7 A POOF OF THE TACE THEOEM OF SOBOLEV SPACES 597 Thus for any u H s (Ω), applying Lemma Lemma 3, we have wherewehaveusedthefactthat γ Ω u H s = ( Ω) S ϕ (S ϕ γ Ω u) H s ( Ω) C S ϕ γ Ω u H s ( N ) = C γ 0 T ϕ u H s ( N ) C T ϕ u Hs ( N + ) C 3 u Hs (Ω), S ϕ γ Ω =γ 0 T ϕ, the C i s always denote constants dependent on Ω. Thus γ Ω is a bounded linearoperatorfromh s (Ω) to H s ( Ω). Lemma 4. Let <s< 3. The trace operator γ 0 is a bounded linear operator from H s ( + ; L ( N )) H s ( + ; H ( N )) onto H s ( N ). Proof. For <s< 3,let W( + )=H s ( + ;L ( N )) H s ( + ; H ( N )), W () =H s (;L ( N )) H s (; H ( N )). Define the extension operator P 0 from W ( + )tow()by { u(x P 0 u(x)=,x N ), if x N > 0, 6u(x, x N ) 8u(x, x N )+3u(x, 3x N ), if x N < 0. It is easy to verify that P 0 is well defined P 0 L(W( + ),W()). For any u W (), define the norm of u by { } u = (( + ξ N ) s +(+ ξ N ) s ( + ξ )) û(ξ) dξ, N where û(ξ) denotes the Fourier transform of u, û(ξ) = e πi x,ξ u(x)dx. N is equivalent to the usual norm on W (). Therefore u W () if only if {( + ξ N ) s +(+ ξ N ) s ( + ξ )}û(ξ) L ( N ), there exists an absolute constant C such that (8) P 0 u C u W (+), for any u W ( + ). Let w(ξ,ξ N )=(+ ξ N ) s +(+ ξ N ) s ( + ξ ).

8 598 ZHONGHAI DING Thus for <s< 3,wehave + w (ξ,ξ N ) dξ N =4 (y s +(+ ξ )y s ) dy =4 ( + ξ ) s + + ξ (y s + y s ) dy 4 ( + ξ ) s + 0 C(s) y s dy + ys ( + ξ ) s, where C(s) =π+ 4 3 s. Therefore for any u W ( +), we have γ 0 u H s ( N ) = (P 0u)(, 0) H s ( N ) Thus γ 0 is bounded from = ( + ξ ) s P 0 u(ξ, 0) dξ N = ( + ξ ) s N ( + ξ ) s { N w (ξ,ξ N ) dξ N P 0 u(ξ,ξ N )dξ N dξ C(s) w (ξ,ξ N ) P 0 u(ξ,ξ N ) dξ N = C(s) P 0 u C u W ( +). H s ( + ; L ( N )) H s ( + ; H ( N )) to H s ( N ). For any f H s ( N ), define û(ξ,x N )= ˆf(ξ )φ(x N ( + ξ )), } w (ξ,ξ N ) (P 0 u)(ξ,ξ N ) dξ N dξ where φ C 0 () φ(0) =. Thus we have γ 0 u(x )=f(x ), x N, û(ξ)= + ξ ˆf(ξ ξ N )ˆφ( + ξ ).

9 A POOF OF THE TACE THEOEM OF SOBOLEV SPACES 599 Hence u = (( + ξ N ) s +(+ ξ N ) s ( + ξ )) û(ξ) dξ N N = ˆf(ξ ) ( + ξ ) { } (( + ξ N ) s +(+ ξ N ) s ( + ξ )) ξ N ˆφ( + ξ ) dξ N dξ ( + ξ ) s ˆf(ξ ) dξ (( + ξ N ) s +(+ ξ N ) s ) ˆφ(ξ N ) dξ N N C ( + ξ ) s ˆf(ξ ) dξ. N Therefore u H s ( + ; L ( N )) H s ( + ; H ( N )). So γ 0 is surjective. Hence the lemma is proved. Theorem 3. Let Ω be a special Lipschitz domain <s< 3. Then the trace operator γ Ω : H s (Ω) H s ( Ω) is bounded. Proof. Notice that S ϕ γ Ω = γ 0 T ϕ. Thus applying Lemma, Lemma 3 Lemma 4, we obtain γ Ω u H s = ( Ω) S ϕ (S ϕ γ Ω u) H s ( Ω) C S ϕ γ Ω u H s ( N ) = C γ 0 T ϕ u H s ( N ) C 3 u Hs (Ω), Thus the theorem is proved. C T ϕ u H s ( +;L ( N )) H s ( +;H ( N )) Now we are ready to prove the trace theorem on general Lipschitz domains. Proof of Theorem. Since Ω is a bounded simply connected Lipschitz domain, there exists a finite number of cylinder coordinate pairs {(Z(P k,r k ),ϕ k )}, k m, such that Ω m k= Z(P k,r k ). Let {φ k } m k= be a unit decomposition of Ω, i.e. (a) supp(φ k ) Z(P k,r k ), k m. (b) m k= φ k(x) =,x Ω. Thus for any u H s (Ω), φ k u H s (Ω Z(P k,r k )), k m, γ Ω u(x) = m (φ k u)(x), x Ω. k=

10 600 ZHONGHAI DING By the zero-extension technique supp(φ k ) Z(P k,r k ), φ k (x)u(x) may be considered as a function on {x N x N >ϕ k (x )}. Hence Theorem Theorem 3 can be applied. Therefore u H s (Ω), applying Theorem Theorem 3, we have γ Ω u H s = m φ k u ( Ω) H s ( Ω) k= m φ k u H s ( (Ω Z(P k,r k ))) k= m C k φ k u H s (Ω Z(P k,r k )) k= C u H s (Ω). Thus the theorem is proved. As a final remark, it must be pointed out that, for s> 3, the trace operator is a bounded linear operator from H s (Ω) to H ( Ω). The proof is similar to the above proof, the detail is omitted. When s = 3, no proof is available. Ackowledgement The author thanks Professors G. Chen H. Boas for their constant interest valuable suggestions in this work. eferences.. A. Adams, Sobolev Spaces, Academic Press, New York, 978. M 56:947. M. Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 9 (988), M 89h: Z. Ding J. Zhou, Constrained LQ problems governed by the potential equation on Lipschitz domain with point observations, J. Math. Pures Appl. 74 (995), E. Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili, en. Sem. Mat. Univ. Padova 7 (957), M :55 5. P. Grisvard, Elliptic problems in nonsmooth domains, Pitman Advanced Publishing Program, Boston, 985. M 86m: D. S. Jerison C. E. Kenig, Boundary value problems on Lipschitz domains, in Studies in Partial Differential Equations (W. Littmann, ed.), MAA Studies in Math., vol. 3, Math. Assoc. of America, 98, pp. 68. M 85f: C. E. Kenig, ecent progress on boundary-value problems on Lipschitz domains, Proc. Sympos. Pure Math., vol. 43, Amer. Math. Soc., Providence, I, 985, pp M 87e: J. L. Lions E. Magenes, Nonhomogeneous boundary value problems applications, vol., Springer-Verlag, New York, 97. M 50: G.Verchota, Layer potentials regularity for the Dirichlet problems for Laplace s equation in Lipschitz domains, J. Funct. Anal. 59 (984), M 86e:35038 Department of Mathematics, Texas A&M University, College Station, Texas Current address: Department of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, Nevada address: dingz@nevada.edu

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