COMPOSITION IN FRACTIONAL SOBOLEV SPACES. 1. Introduction. A classical result about composition in Sobolev spaces asserts

Transcription

1 DISCRETE AND CONTINUOUS Website: ttp://mat.smsu.edu/journal DYNAMICAL SYSTEMS Volume 7, Number, April 001 pp COMPOSITION IN FRACTIONAL SOBOLEV SPACES HAIM BREZIS (1)() AND PETRU MIRONESCU (3) 1. Introduction. A classical result about composition in Sobolev spaces asserts tat if u W k,p (Ω) L (Ω) and Φ C k (R), ten Φ u W k,p (Ω). Here Ω denotes a smoot bounded domain in R N, k 1 is an integer and 1 p <. Tis result was first proved in [13] wit te elp of te Gagliardo-Nirenberg inequality [14]. In particular if u W k,p (Ω) wit kp > N and Φ C k (R) ten Φ u W k,p since W k,p L by te Sobolev embedding teorem. Wen kp = N te situation is more delicate since W k,p is not contained in L. However te following result still olds (see [],[3]) Teorem 1. Assume u W k,p (Ω) were k 1 is an integer, 1 p <, and kp = N. (1) Let Φ C k (R) wit Ten D j Φ L (R) j k. () Φ u W k,p (Ω) Te proof is based on te following Lemma 1. Assume u W k,p (Ω) W 1,kp (Ω) were k 1 is an integer and 1 p <. Assume Φ C k (R) satisfies (). Ten Φ u W k,p (Ω). Proof of Teorem 1. Since u W k,p we ave by te Sobolev embedding wit u W k 1,p L q 1 q = 1 p k 1 N. AMS Subject Classification: 46E39 Keywords and prases: Fractional Sobolev spaces, Besov spaces, composition of functions. Acknowledgment: Te first autor (H.B.) is partially supported by a European Grant ERB FMRX CT He is also a member of te Institut Universitaire de France. Part of tis work was done wen te second autor (P.M.) was visiting Rutgers University; e tanks te Matematics Department for its invitation and ospitality. We tank S. Klainerman for useful discussions. 41

2 4 HAIM BREZIS AND PETRU MIRONESCU Applying assumption (1) we find q = N = kp and tus u W 1,kp. We deduce from Lemma 1 tat Φ u W k,p. Proof of Lemma 1. Note tat if u W k,p L wit k 1 integer and 1 p < ten u W 1,kp by te Gagliardo - Nirenberg inequality [14]. Tus, Lemma 1 is a generalization of te standard result about composition. In fact, it is proved exactly in te same way as in te standard case (wen u W k,p L ). Wen k = te conclusion is trivial. Assume, for example tat, k = 3, ten W 3,p W 1,3p W,3p/ by te Gagliardo - Nirenberg inequality. Ten D 3 (Φ u) = Φ (u)d 3 u + 3Φ (u)d udu + Φ (u)(du) 3, and tus Φ u W 3,p since D u p Du p ( D u 3p/ ) /3( Du 3p) 1/3 A simular argument olds for any k 4. C u p/ W 3,p u 3p/ W 1,3p. Starting in te mid-60 s a number of autors considered composition in various classes of Sobolev spaces W s,p, were s > 0 is a real number and 1 p <. Te most commonly used are te Bessel potential spaces L s,p (R N ) = {f = G s g; g L p (R N )} were Ĝs = (1+ ξ ) s/ and te Besov spaces Bp s,p (R N ) (wo s definition is recalled below wen s is not an integer). It is well-known (see e.g. [1],[19] and [0]) tat if k is an integer, L k,p coincides wit te standard Sobolev space W k,p ; also if p =, te Bessel potential L s, and te Besov spaces B s, coincide for every s non-integer and tey are usually denoted by H s. Wen p te spaces L s,p and are distinct. B s,p p Te first result about composition in fractional Sobolev spaces seems to be due to Mizoata [1] for H s,s > N/. In 1970 Peetre [15] considered Bp s,p L using interpolation tecniques; a very simple direct argument for te same class, L, was given by M. Escobedo [10] (see te proof of Lemma below). B s,p p Starting in 1980 tecniques of dyadic analysis and Littlewood-Paley decomposition à la Bony [5] were introduced. For example, Y. Meyer [11] considered composition in L s,p for sp > N; see also [16],[4],[9] for H s wit s > N/ or for H s L, any s > 0. We refer to [17],[6],[7],[18] and teir bibliograpies for oter directions of researc concerning composition in Sobolev spaces. In wat follow we denote by W s,p (Ω) te restriction of Bp s,p (R N ) to Ω wen s is not an integer. Our main result is te following Teorem. Assume u W s,p (Ω) were s > 1 is a real number, 1 < p <, and sp = N. (3) Let Φ C k (R), were k = [s] + 1, be suc tat D j Φ L (R) j k. (4)

3 COMPOSITION IN FRACTIONAL SOBOLEV SPACES 43 Ten Φ u W s,p (Ω). Te proof of Teorem relies on a variant of Lemma 1 for fractional Sobolev spaces. Lemma. Let u W s,p (Ω), were s > 1 is a real number and 1 < p <. Assume, in addition, tat u W σ,q for some σ (0, 1) wit q = sp/σ. (5) Let Φ C k (R), were k = [s] + 1, be suc tat (4) olds. Ten Φ u W s,p Proof of Teorem. By te Sobolev embedding teorem we ave wit r < s and In view of assumption (3) we find In particular, for all σ (0, 1) wit W s,p W r,q 1 q = 1 (s r) p N. q = N/r. u W σ,q q = N σ = sp σ. Tus we may apply Lemma and conclude tat Φ u W s,p. Remark 1. Teorem is known to be true wen te Sobolev spaces W s,p are replaced by te Bessel potential spaces L s,p wit sp = N; see D. Adams and M. Frazier [3]. Even toug te two results are closely related it does not seem possible to deduce one from te oter. Teir argument relies on a variant of Lemma for Bessel potential spaces: Let u L s,p L 1,sp were s > 1 is a real number and 1 < p <. Let Φ be as in Lemma. Ten Φ u L s,p. Remark. Te assumption in Lemma, u W s,p W σ,q, wit q = sp/σ for some σ (0, 1), is weaker tan te assumption u W s,p L but it is stronger tan te assumption u W 1,sp ; tis is a consequence of Gagliardo - Nirenberg type inequalities (see e.g. te proof of Lemma D.1 in te Appendix D of [8]). It is terefore natural to raise te following: Open Problem. Is te conclusion of Lemma valid if one assumes only u W s,p W 1,sp were s > 1 is a (non-integer) real number? Before giving te proof of Lemma we recall some properties of W s,p wen s is not an integer.

4 44 HAIM BREZIS AND PETRU MIRONESCU Wen 0 < σ < 1 and 1 < p < te standard definition of W σ,p is W σ,p (Ω) = {f L p (Ω); f(x) f(y) p dxdy < }. x y N+σp If s > 1 is not an integer write s = [s] + σ were [s] denotes te integer part of s and 0 < σ < 1. Ten W s,p (Ω) = {f W [s],p (Ω), D α f W σ,p for α = [s]}. Tere is a very useful caracterization of W s,p in terms of finite differences (see Triebel [0], p.110). Here it is more convenient to work wit functions defined on all of R N and to consider teir restrictions to Ω. Set so tat (δ u)(x) = u(x + ) u(x), R N, (δ u)(x) = u(x + ) u(x + ) + u(x), etc... Given s > 1 not integer, fix any integer M > s. Ten W s,p = {f L p ; M f(x) p dxd < }. Proof of Lemma. It suffices to consider te case were s is not an integer. For simplicity we treat just te case were 1 < s <. Te same argument extends to general s >, s noninteger, using te same type of computations as in Escobedo [10]. Te key observation is tat δ (Φ u) can be expressed in terms of δ u and δ u. Tis is te purpose of our next computation. Set Since Φ L (R) we ave and since Φ L (R) we also ave Combining (6) and (7) we find X = u(x + ) Y = u(x + ) Z = u(x). Φ(X) Φ(Y ) = Φ (Y )(X Y ) + 0( X Y ) (6) Φ(X) Φ(Y ) = Φ (Y )(X Y ) + 0( X Y ). (7) Φ(X) Φ(Y ) = Φ (Y )(X Y ) + 0( X Y a ) for any 1 a ( we will coose a specific value of a later) Similarly Φ(Z) Φ(Y ) = Φ (Y )(Z Y ) + 0( Z Y a )

5 COMPOSITION IN FRACTIONAL SOBOLEV SPACES 45 Since one finds δ (Φ u)(x) = (Φ(X) Φ(Y )) + (Φ(Z) Φ(Y )), δ (Φ u)(x) C( δ u(x) + δ u(x + ) a + δ u(x) a ). (8) Tis yields (Φ u)(x) p dxd C u(x) p u(x) ap dxd + C dxd. (9) Te first integral on te rigt-and side of (9) is finite since u W s,p. To andle te second integral we argue as follows. From te assumption u W s,p W σ,q wit σ (0, 1) and q given by (5) we know tat u(x) p dxd < and From (10) and Hölder s inequality we derive tat u(x) r u(x) q dxd <. (10) dxd < (11) for all r [p, q], i.e., u W τ,r wit τ = sp/r. We now coose and r = ap [p, q]. It follows tat wic is te desired in equality. a = min{, s/σ}, so tat a [1, ] u(x) ap dxd <, Remark 3. Tere could be anoter natural proof of Teorem by induction on [s]. One migt attempt to prove tat D(Φ u) = Φ (u)du W s 1,p. Note tat u W (s 1),N/(s 1) and tus (by induction) we would ave Φ (u) W (s 1),N/(s 1). On te oter and Du W s 1,p. In order to conclude we need a lemma about products, but we are not aware of any suc tool. Remark 4. Wen s (or equivalently p) is a rational number, and Φ C wit D j Φ L j, tere is a simple proof of Teorem based on trace teory and Teorem 1. Assume for simplicity tat Ω = R N. Suppose tat s is not an integer, but tat s 1 = s + 1/p is an integer. Ten u is te trace of some function u 1 W s1,p (R N+1 ). Ten s 1 p = N + 1 and by Teorem 1 we deduce tat Φ u 1 W s1,p (R N+1 ). Taking traces we find Φ u W s,p (R N ). If s 1 is not an integer we keep extending u 1 to iger dimensions and stop at te first integer k suc tat s k = s + k/p is an integer ( tis is possible since p is rational and s + k/p = (N + k)/p becomes an integer for some integer k). We ave an extension

6 46 HAIM BREZIS AND PETRU MIRONESCU u k W s k,p (R N+k ) of u. Ten Φ u k W s k,p (R N+k ) by Teorem 1. Taking back traces yields u W s,p. References 1. R. Adams, Sobolev Spaces, Acad. Press, D.R. Adams, On te existence of capacitary strong type estimates in R n, Ark. Mat., 14, (1976), D.R. Adams and M. Frazier, Composition operators on potential spaces, Proc. Amer. Mat. Soc., 114, (199), S. Alinac and P. Gérard, Opérateurs pseudo-différentiels et téorème de Nas-Moser, Interéditions, J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles nonlinéaires, Ann. Sc. Ec. Norm Sup. 14, (1981), G. Bourdaud, Le calcul fonctionnel dans les espaces de Sobolev, Invent. Mat., 104, (1991), G. Bourdaud and Y. Meyer, Fonctions qui opèrent sur les espaces de Sobolev, J. Funct. Anal., 97, (1991), J. Bourgain, H.Brezis and P.Mironescu, Lifting in Sobolev spaces, J. d Analyse 80, (000), J.-Y. Cemin, Fluides parfaits incompressibles 30, Astérisque, M. Escobedo, Some remarks on te density of regular mappings in Sobolev classes of S M - valued functions, Rev. Mat. Univ. Complut. Madrid, 1, (1988), Y.Meyer, Remarques sur un téorème de J.-M. Bony,, Suppl. Rend. Circ. Mat. Palermo, Series II 1, (1981), S. Mizoata, Lectures on te Caucy problem, Tata Inst., Bombay, J. Moser, A rapidly convergent iteration metod and non-linear differential equations, Ann. Sc. Norm. Sup. Pisa 0, (1966), L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Sup. Pisa, 13, (1959), J. Peetre, Interpolation of Lipscitz operators and metric spaces, Matematica (Cluj) 1, (1970), J. Rauc and M. Reed, Nonlinear microlocal analysis of semilinear yperbolic systems in one space dimension, Duke Mat J. 49, (198), T. Runst, Mapping properties of non-linear operators in spaces of Triebel-Lizorkin and Besov type, Analysis Matematica, 1, (1986), T. Runst and W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, Walter de Gruyter, Berlin and New York, H. Triebel, Interpolation teory, function spaces, differential operators, Nort-Holland, Amsterdam, H. Triebel, Teory of function spaces, Birkäuser, Basel and Boston, Revised version received November 000. (1) ANALYSE NUMÉRIQUE UNIVERSITÉ P. ET M. CURIE, B.C PL. JUSSIEU 755 PARIS CEDEX 05 () RUTGERS UNIVERSITY DEPT. OF MATH., HILL CENTER, BUSCH CAMPUS 110 FRELINGHUYSEN RD, PISCATAWAY, NJ address: brezis@ccr.jussieu.fr; brezis@mat.rutgers.edu (3) DEPARTEMENT DE MATHÉMATIQUES UNIVERSITÉ PARIS-SUD ORSAY address: : Petru.Mironescu@mat.u-psud.fr