Electronic Journal of Differential Equations, Vol. 2005(2005), No.??, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) A PROPERTY OF SOBOLEV SPACES ON COPLETE RIEANNIAN ANIFOLDS OGNJEN ILATOVIC Abstract. Let (, g) be a complete Riemannian manifold with metric g and the Riemannian volume form dν. We consider the R k -valued functions T [W 1,2 () L 1 loc ()]k and u [W 1,2 ()] k on, where [W 1,2 ()] k is a Sobolev space on and [W 1,2 ()] k is its dual. We give a sufficient condition for the equality of T, u and the integral of (T u) over, where, is the duality between [W 1,2 ()] k and [W 1,2 ()] k. This is an extension to complete Riemannian manifolds of a result of H. Brézis and F. E. Browder. 1. Introduction and main result The setting. Let (, g) be a C Riemannian manifold without boundary, with metric g = (g jk ) and dim = n. We will assume that is connected, oriented, and complete. By dν we will denote the Riemannian volume element of. In any local coordinates x 1,..., x n, we have dν = det(g jk ) dx 1 dx 2... dx n. By L 2 () we denote the space of real-valued square integrable functions on with the inner product (u, v) = (uv) dν. Unless specified otherwise, in all function spaces below, the functions are realvalued. In what follows, C () denotes the space of smooth functions on, Cc () denotes the space of smooth compactly supported functions on, Ω 1 () denotes the space of smooth 1-forms on, and L 2 (Λ 1 T ) denotes the space of square integrable 1-forms on. By W 1,2 () we denote the completion of Cc () in the norm u 2 W = u 2 dν + du 2 dν, 1,2 where d : C () Ω 1 () is the standard differential. Remark 1.1. It is well known (see, for example, Chapter 2 in [1]) that if (, g) is a complete Riemannian manifold, then W 1,2 () = {u L 2 (): du L 2 (Λ 1 T )}. 2000 athematics Subject Classification. 58J05. Key words and phrases. Complete Riemannian manifold; Sobolev space. c 2005 Texas State University - San arcos. Submitted June 25, 2005. Published??. 1
2 O. ILATOVIC EJDE-2005/?? By W 1,2 () we denote the dual space of W 1,2 (), and by, we will denote the duality between W 1,2 () and W 1,2 (). In what follows, [Cc ()] k, [L 2 ()] k, [L 2 (Λ 1 T )] k and [W 1,2 ()] k denote the space of all ordered k-tuples u = (u 1, u 2,..., u k ) such that u j Cc (), u j L 2 (), u j L 2 (Λ 1 T ), u j W 1,2 (), respectively, for all 1 j k. For u [W 1,2 ()] k, we will use the following notation: du := (du 1, du 2,..., du k ), (1.1) u := (u 2 1 + u 2 2 + + u 2 k) 1/2, (1.2) du := ( du 1 2 + du 2 2 + + du k 2 ) 1/2, (1.3) where du j denotes the length of the cotangent vector du j. The space [W 1,2 ()] k is the completion of [Cc ()] k in the norm u 2 [W 1,2 ()] = u 2 dν + du 2 dν, k where u and du are as in (1.2) and (1.3) respectively. Remark 1.2. As in Remark 1.1, if (, g) is a complete Riemannian manifold, then [W 1,2 ()] k = {u [L 2 ()] k : du [L 2 (Λ 1 T )] k }. Assumption (H1). Assume that (1) u = (u 1, u 2,..., u k ) [W 1,2 ()] k and (2) T = (T 1, T 2,..., T k ), where T 1, T 2,..., T k W 1,2 () L 1 loc (). Here, the notation T j W 1,2 () L 1 loc () means that T j is a.e. defined function belonging to L 1 loc () such that φ T j φ dν, φ Cc (), extends continuously to W 1,2 (). For a.e. x, denote (T u)(x) := T, u := T j (x)u j (x), (1.4) T j, u j, (1.5) where, on the right hand side of (1.5) denotes the duality between W 1,2 () and W 1,2 (). We now state our main result. Theorem 1.3. Assume that (, g) is a complete Riemannian manifold. Assume that u = (u 1, u 2,..., u k ) and T = (T 1, T 2,..., T k ) satisfy the assumption (H1). Assume that there exists a function f L 1 () such that Then (T u) L 1 () and T, u = (T u)(x) f(x), a.e. on. (1.6) (T u)(x) dν(x).
EJDE-2005/?? A PROPERTY OF SOBOLEV SPACES 3 In the following Corollary, by W 1,2 (, C), W 1,2 (, C) and L 1 loc (, C) we denote the complex analogues of spaces W 1,2 (), W 1,2 () and L 1 loc (). By, we denote the Hermitian duality between W 1,2 (, C) and W 1,2 (, C). Corollary 1.4. Assume that (, g) is a complete Riemannian manifold. Assume that T W 1,2 (, C) L 1 loc (, C) and u W 1,2 (, C). Assume that there exists a real-valued function f L 1 () such that Re(T ū) f, a.e. on. Then Re(T ū) L 1 () and Re T, u = Re(T ū) dν. Remark 1.5. Theorem 1.3 and Corollary 1.4 extend the corresponding results of H. Brézis and F. E. Browder [3] from R n to complete Riemannian manifolds. The results of [3] were used, among other applications, in studying self-adjointness and m-accretivity in L 2 (R n, C) of Schrödinger operators with singular potentials; see, for example, H. Brézis and T. Kato [4]. Analogously, Theorem 1.3 and Corollary 1.4 can be used in the study of self-adjoint and m-accretive realizations (in the space L 2 (, C)) of Schrödinger-type operators with singular potentials, where is a complete Riemannian manifold, as well as in the study of partial differential equations on complete Riemannian manifolds. 2. Proof of Theorem 1.3 We will adopt the arguments of H. Brézis and F. E. Browder [3] to the context of a complete Riemannian manifold. In what follows, F : R k R l is a C 1 vectorvalued function F (y) = (F 1 (y), F 2 (y),..., F l (y)). By df (y) we will denote the derivative of F at y = (y 1, y 2,..., y k ). Lemma 2.1. Assume that F C 1 (R k, R l ), F (0) = 0, and for all y R k, df (y) C where C 0 is a constant. Assume that u = (u 1, u 2,..., u k ) [W 1,2 ()] k. Then (F u) [W 1,2 ()] l, and the following holds: where d(f u) = F u j du j, (2.1) F ( F1 = (u), F 2 (u),..., F ) l (u). (2.2) u j (Here the notation F s (u), where 1 s l, denotes the composition of F s and u. The notation d(f u) denotes the ordered l-tuple (d(f 1 u), d(f 2 u),..., d(f l u)), where d(f s u), 1 s l, is the differential of the scalar-valued function F s u on. Proof. Let u [W 1,2 ()] k. By definition of [W 1,2 ()] k, the weak derivatives du j, 1 j k, exist and du j L 2 (). By Lemma 7.5 in [6], it follows that for all
4 O. ILATOVIC EJDE-2005/?? 1 s l, the following holds: where d(f s u) = F s u j F s u j du j, = F s (u). This shows (2.1). Since df is bounded and since du j L 2 (Λ 1 T ), it follows that d(f s u) L 2 (Λ 1 T ) for all 1 s l. Thus d(f u) [L 2 (Λ 1 T )] l. oreover, since u [W 1,2 ()] k and F s u = F s (u) F s (0) C 1 u, where C 1 0 is a constant and u is as in (1.2), it follows that (F s u) L 2 () for all 1 s l. Thus (F u) [L 2 ()] l. Therefore, (F u) [W 1,2 ()] l, and the Lemma is proven. Lemma 2.2. Assume that u, v W 1,2 () L (). Then (uv) W 1,2 () and d(uv) = (du)v + u(dv). (2.3) Proof. By the remark after the equation (7.18) in [6], the equation (2.3) holds if the weak derivatives du, dv exist and if uv L 1 loc () and ((du)v + u(dv)) L1 loc (). By the hypotheses of the Lemma, these conditions are satisfied, and, hence, (2.3) holds. Furthermore, since u, v W 1,2 () L (), we have (uv) L 2 (). By hypotheses of the Lemma and by (2.3) we have d(uv) L 2 (). Thus (uv) W 1,2 (), and the Lemma is proven. In the next lemma, the statement f : R R is a piecewise smooth function means that f is continuous and has piecewise continuous first derivative. Lemma 2.3. Assume that f : R R is a piecewise smooth function with f(0) = 0 and f L (R). Let S denote the set of corner points of f. Assume that u W 1,2 (). Then (f u) W 1,2 () and { f (u) du for all x such that u(x) / S d(f u) = 0 for all x such that u(x) S Proof. By the remark in the second paragraph below the equation (7.24) in [6], the Lemma follows immediately from Theorem 7.8 in [6]. The following Corollary follows immediately from Lemma 2.3. Corollary 2.4. Assume that u W 1,2 (). Then u W 1,2 () and { f (u) du for all x such that u(x) 0 d u = 0 for all x such that u(x) = 0, where f(t) = t, t R.
EJDE-2005/?? A PROPERTY OF SOBOLEV SPACES 5 Remark 2.5. Let f(t) = t, t R. Let c be a real number. By Lemma 7.7 in [6] and by Corollary 2.4, we can write d u = h(u)du a.e. on, where { f (t) for all t 0 h(t) = c otherwise. Lemma 2.6. Assume that u, v W 1,2 () and let Then w W 1,2 () and w(x) := min{u(x), v(x)}. dw max{ du, dv }, a.e. on, where du(x) denotes the norm of the cotangent vector du(x). Proof. We can write w(x) = 1 (u(x) + v(x) u(x) v(x) ). 2 Since u, v W 1,2 (), by Corollary 2.4 we have u v W 1,2 (), and, thus, w W 1,2 (). By Remark 2.5, we have dw(x) = 1 (du(x) + dv(x) (h(u v)) (du(x) dv(x))), 2 a.e. on, (2.4) where h is as in Remark 2.5. Considering dw(x) on sets {x: u(x) > v(x)}, {x: u(x) < v(x)} and {x: u(x) = v(x)}, and using (2.4), we get dw(x) max{ du(x), dv(x) }, a.e. on. This concludes the proof of the Lemma. Lemma 2.7. Let a > 0. Let u = (u 1, u 2,..., u k ) be in [W 1,2 ()] k, let v = (v 1, v 2,..., v k ) be in [W 1,2 () L ()] k, and let ( ) w := ( u 2 + a 2 ) 1/2 min{( u 2 + a 2 ) 1/2 a, ( v 2 + a 2 ) 1/2 a} u, where u is as in (1.2). Then w [W 1,2 () L ()] k and where du is as in (1.3). dw 3 max{ du, dv }, a.e. on, Proof. Let φ = ( u 2 + a 2 ) 1/2 u. Then φ = F u, where F : R k R k is defined by F (y) = ( y 2 + a 2 ) 1/2 y, y R k. Clearly, F C 1 (R k, R k ) and F (0) = 0. It easily checked that the component functions F s (y) = ( y 2 + a 2 ) 1/2 y s satisfy F s = { ( y 2 + a 2 ) 3/2 y s y j for s j ( y 2 + a 2 ) 3/2 ( y 2 yj 2 + a2 ) for s = j. Therefore, for all 1 s, j k, we have F s (y) 1 a,
6 O. ILATOVIC EJDE-2005/?? and, hence, F satisfies the hypotheses of Lemma 2.1. Thus, by Lemma 2.1 we have (F u) = φ [W 1,2 ()] k. We now write the formula for dφ = (dφ 1, dφ 2,..., dφ k ). We have dφ = ( u 2 + a 2 ) 3/2( ( ) ( u 2 + a 2 )du u j du j )u, (2.5) where du is as in (1.1). By (2.5), using triangle inequality and Cauchy-Schwarz inequality, we have dφ ( u 2 + a 2 ) 3/2( ( u 2 + a 2 ) du + ) u j du j u ( u 2 + a 2 ) 3/2 ( ( u 2 + a 2 ) du + u du u ) ( u 2 + a 2 ) 3/2 ( ( u 2 + a 2 ) du + ( u 2 + a 2 ) du ) = 2( u 2 + a 2 ) 1/2 du, a.e. on, (2.6) where du j is the norm of the cotangent vector du j, and u and du are as in (1.2) and (1.3) respectively. Let ψ := min{( u 2 + a 2 ) 1/2 a, ( v 2 + a 2 ) 1/2 a}. Then ( u 2 + a 2 ) 1/2 a = G u and ( v 2 + a 2 ) 1/2 a = G v, where G(y) = ( y 2 + a 2 ) 1/2 a, y R k. Clearly, G C 1 (R k, R) and G(0) = 0, and G = ( y 2 + a 2 ) 1/2 y j, It is easily seen that there exists a constant C 2 0 such that dg(y) C 2 for all y R k. Hence, by Lemma 2.1 we have (G u) W 1,2 () and (G v) W 1,2 (). Thus, by Lemma 2.6 we have ψ W 1,2 (), and dψ max { d(( u 2 + a 2 ) 1/2 a), d(( v 2 + a 2 ) 1/2 a) }, a.e. on. Using triangle inequality and Cauchy-Schwarz inequality, we have d(( u 2 + a 2 ) 1/2 a) = ( u 2 + a 2 ) 1/2( k ) u j du j ( u 2 + a 2 ) 1/2 u du du, where u and du are as in (1.2) and (1.3) respectively. As in (2.7), we obtain d(( v 2 + a 2 ) 1/2 a) dv. (2.7) Therefore, we get dψ max{ du, dv }, a.e. on, (2.8) where dψ is the norm of the cotangent vector dψ, and du and dv are as in (1.3). By definition of φ we have φ [L ()] k and, by definition of ψ we have ψ ( v 2 + a 2 ) 1/2 a.
EJDE-2005/?? A PROPERTY OF SOBOLEV SPACES 7 Thus, ψ v, (2.9) where v is as in (1.2). Since v [L ()] k, we have ψ L (). We have already shown that φ [W 1,2 ()] k and ψ W 1,2 (). By Lemma 2.2 (applied to the components ψφ j, 1 j k, of ψφ) we have w = ψφ [W 1,2 ()] k and By (2.10), (2.6) and (2.8), we have a.e. on : dw = (dψ)φ + ψ(dφ) d(ψφ) = (dψ)φ + ψ(dφ). (2.10) dψ φ + ψ dφ (max{ du, dv }) φ + 2( u 2 + a 2 ) 1/2 du ψ max{ du, dv } + 2 du 3 max{ du, dv }, where the third inequality holds since φ 1 and ψ ( u 2 + a 2 ) 1/2 1. This concludes the proof of the Lemma. Lemma 2.8. Let T = (T 1, T 2,..., T k ) and u = (u 1, u 2,..., u k ) be as in the hypotheses of Theorem 1.3. Additionally, assume that u has compact support and u [L ()] k. Then the conclusion of Theorem 1.3 holds. Proof. Since the vector-valued function u = (u 1, u 2,..., u k ) [W 1,2 ()] k is compactly supported, it follows that the functions u j are compactly supported. Thus, using a partition of unity we can assume that u j is supported in a coordinate neighborhood V j. Thus we can use the Friedrichs mollifiers. Let ρ j > 0 and (u j ) ρj := J ρj u, where J ρj denotes the Friedrichs mollifying operator as in Section 5.12 of [2]. Then (u j ) ρ j Cc (), and, as ρ j 0+, we have (u j ) ρ j u j in W 1,2 (); see, for example, Lemma 5.13 in [2]. Thus T j, (u j ) ρj T j, u j, as ρ j 0+, (2.11) where, is as on the right hand side of (1.5). Since (u j ) ρj Cc () and T j L 1 loc (), we have T j, (u j ) ρj = (T j (u j ) ρj ) dν. (2.12) Next, we will show that lim ρ j 0+ (T j (u j ) ρ j ) dν = (T j u j ) dν. (2.13) Since u j L () is compactly supported, by properties of Friedrichs mollifiers (see, for example, the proof of Theorem 1.2.1 in [5]) it follows that (i) there exists a compact set K j containing the supports of u j and u ρ j j 0 < ρ j < 1, and (ii) the following inequality holds for all ρ j > 0: for all u ρj j L u j L. (2.14)
8 O. ILATOVIC EJDE-2005/?? Since (u j ) ρ j u j in L 2 () as ρ j 0+, after passing to a subsequence we have By (2.14) we have (u j ) ρj u j a.e. on, as ρ j 0 +. (2.15) T j (x)(u j ) ρj (x) T j (x) u j L, a.e. on. (2.16) Since T j L 1 loc (), it follows that T j L 1 (K j ). By (2.15), (2.16) and since T j L 1 (K j ), using dominated convergence theorem, we have lim ρ j 0+ (T j (u j ) ρ j ) dν = lim (T j (u j ) ρ j ) dν = (T j u j ) dν = (T j u j ) dν, ρ j 0+ K j K j and (2.13) is proven. Now, using (2.11), (2.12), (2.13) and the notations (1.4) and (1.5), we get T, u = T j, u j = = = lim T j, (u j ) ρ j ρ j 0+ lim ρ j 0+ This concludes the proof of the Lemma. (T j u j ) dν = (T j (u j ) ρj ) dν (T u) dν. (2.17) Proof of Theorem 1.3. Let u [W 1,2 ()] k. By definition of [W 1,2 ()] k in Section 1, there exists a sequence v m [Cc ()] k such that v m u in [W 1,2 ()] k, as m +. In particular, v m u in [L 2 ()] k, and, hence, we can extract a subsequence, again denoted by v m, such that v m u a.e. on. Define a sequence λ m by λ m := ( u 2 + 1 ) 1/2 {( min m 2 u 2 + 1 ) 1/2 1 ( m 2 m, v m 2 + 1 ) 1/2 1 } m 2, m where v m is the chosen subsequence of v m such that v m u a.e. on, as m +. Clearly, 0 λ m 1. Define w m := λ m u. (2.18) We know that u [W 1,2 ()] k and v m [Cc ()] k. Thus, by Lemma 2.7, for all m = 1, 2, 3,..., we have w m [W 1,2 () L ()] k, and d(w m ) 3 max{ du, d(v m ) }, (2.19) where du is as in (1.2). Furthermore, for all m = 1, 2, 3,..., we have w m (x) u(x), (2.20) where is as in (1.2). Since u [L 2 ()] k, by (2.20) it follows that {w m } is a bounded sequence in [L 2 ()] k. Since v m u in [W 1,2 ()] k, it follows that the sequence {v m } is bounded in [W 1,2 ()] k. In particular, the sequence {d(v m )} is bounded in
EJDE-2005/?? A PROPERTY OF SOBOLEV SPACES 9 [L 2 (Λ 1 T )] k. Hence, by (2.19) it follows that {d(w m )} is a bounded sequence in [L 2 (Λ 1 T )] k. Therefore, {w m } is a bounded sequence in [W 1,2 ()] k. By Lemma V.1.4 in [7] it follows that there exists a subsequence of {w m }, which we again denote by {w m }, such that w m converges weakly to some z [W 1,2 ()] k. This means that for every continuous linear functional A [W 1,2 ()] k, we have Since A(w m ) A(z), as m +. [W 1,2 ()] k [L 2 ()] k [W 1,2 ()] k, it follows that w m z in weakly [L 2 ()] k. We will now show that, as m +, w m u in [L 2 ()] k. By definition of w m in (2.18) it follows that w m u a.e. on. Since u [L 2 ()] k, using (2.20) and dominated convergence theorem we get w m u in [L 2 ()] k, as m +. In particular, w m u weakly in [L 2 ()] k. Therefore, by the uniqueness of the weak limit (see, for example, the beginning of Section III.1.6 in [7]), we have z = u. Therefore, w m u weakly in [W 1,2 ()] k. Thus, since T [W 1,2 ()] k, we have T, w m T, u, as m +. (2.21) By the definition of λ m and (2.18) it follows that w m (x) v m (x). (2.22) Since v m [Cc ()] k, by (2.22) it follows that the functions w m have compact support. We have shown earlier that w m [W 1,2 () L ()] k. Thus, by Lemma 2.8, the following equality holds: T, w m = (T w m ) dν. (2.23) Let f be as in the hypotheses of the Theorem. Then T w m = T (λ m u) = λ m (T u) λ m f f. (2.24) By (2.24) it follows that T w m + f 0. Consider the sequence T w m + f. Since f L 1 () and (T w m ) L 1 (), by Fatou s lemma we get lim inf (T m + wm + f ) dν lim inf (T w m + f ) dν. (2.25) m + Since w m u a.e. on as m +, we have T w m T u a.e. on as m +. Thus, by (2.25) we have (T u + f ) dν f dν + lim inf (T w m ) dν, m + and, hence, by (2.23) and (2.21) we have (T u + f ) dν f dν + lim inf (T w m ) dν m + = f dν + lim inf T, m + wm = f dν + T, u.
10 O. ILATOVIC EJDE-2005/?? Since f L 1 (), we have (T u + f ) L 1 (), and, hence, (T u) L 1 (). We have T w m = λ m (T u) T u, and by definition of w m, we get, as m +, T w m T u, a.e. on. Using dominated convergence theorem, we get (T w m ) dν = lim m + By (2.26), (2.23) and (2.21), we get T, u = This concludes the proof of the Theorem. (T u) dν. (T u) dν (2.26) Proof of Corollary 1.4. Let T 1 = Re T and T 2 = Im T. Let u 1 = Re u and u 2 = Im u. Then Re T, u = T 1, u 1 + T 2, u 2 and Re(T ū) = T 1 u 1 +T 2 u 2. Thus, Corollary 1.4 follows from Theorem 1.3. References [1] T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer-Verlag, Berlin, 1998. [2]. Braverman, O. ilatovic,. Shubin, Essential self-adjointness of Schrödinger type operators on manifolds, Russian ath. Surveys, 57(4) (2002), 641 692. [3] H. Brézis, F. E. Browder, Sur une propriété des espaces de Sobolev, C. R. Acad. Sci. Paris Sr. A-B, 287, no. 3, (1978), A113 A115. (French). [4] H. Brézis, T. Kato, Remarks on the Schrödinger operator with singular complex potentials, J. ath. Pures Appl., 58(9) (1979), 137 151. [5] G. Friedlander,. Joshi, Introduction to the Theory of Distributions, Cambridge University Press, 1998. [6] D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1998. [7] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1980. Ognjen ilatovic Department of athematics and Statistics, University of North Florida, Jacksonville, FL 32224, USA E-mail address: omilatov@unf.edu