LOGARITHMIC SOBOLEV INEQUALITY REVISITED

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1 LOGARITHMIC SOBOLEV IEQUALITY REVISITED HOAI-MIH GUYE AD MARCO SQUASSIA Abstract. We provide a new characterization of the logarithmic Sobolev inequality. 1. Introduction The classical Sobolev inequality translates information about the derivatives of a function into information about the size of the function itself. Precisely for a function u with square summable gradient in dimension one obtains that u is /( -summable that is a gain in summability which depends on and which tends to deteriorate as. On the other hand since the middle fifties people have started looking at possible replacements of the Sobolev inequality in order to provide an improvement in the summability independent of the dimension which can be done in terms of integrability properties of u log u. This was firstly done by Stam [3] who proved the logarithmic Sobolev inequality with Gauss measure dg u u log dg 1 u dg dg = e π x dx. R π R u dg The formula was originally discovered in quantum field theory in order to handle estimates which are uniform in the space dimension for systems with a large number of variables. A different proof and further insight was obtained by Gross in [17]. See also the work of Adams and Clarke [1] for an elementary proof of the previous inequality. These properties are widely used in statistical mechanics quantum field theory and differential geometry. A variant of the logarithmic Sobolev inequality with Gauss measure is given by the following one parameter family of euclidean inequalities [18 Theorem 8.1] R u log u u dx + (1 + log a u a π R u dx. for any u H 1 (R and a > 0. A version of this inequality for fractional Sobolev spaces H s (R can be found in [13]. Recently some new characterization of the Sobolev spaces were provided in [ 19 1] (see also [ ] in terms of the following family of nonlocal functionals I δ (u := { u(y u(x >δ} δ dxdy δ > 0 x y + where u is a measurable function on R. In particular if 3 and I δ (u < for some δ > 0 then in [1] it was proved that (1.1 u /( dx C I δ (u /( { u >λ δ} 010 Mathematics Subject Classification. 6E35 8D0 8B10 9A50. Key words and phrases. onlocal functionals logarithmic Sobolev inequality entropy. The second author is member of Gruppo azionale per l Analisi Matematica la Probabilità e le loro Applicazioni (GAMPA of the Istituto azionale di Alta Matematica (IdAM. 1

2 H.-M. GUYE AD M. SQUASSIA for some positive constants C and λ. This is a sort of nonlocal improvement of the classical Sobolev inequality and it is also possible to show that in the singular limit δ 0 one recovers the classical Sobolev result since I δ converges to the Dirichlet energy up to a normalization constant. The aim of this note is to remark that in this context also a logarithmic type estimate holds. Thus we have the summability gain independent of can be controlled in terms of I δ (u. More precisely we have the following Theorem 1.1. Let u L (R ( 3. There is a positive constant C such that R u u log u u dx + log u ( log C δ u + C I δ (u for all δ > 0. In particular if u L (R is such that I δ (u < for some δ > 0 then (1. u log u dx < +. R Proof. By a simple normalization argument we may reduce the assertion to proving that (1.3 u log u dx ( R log C δ + C I δ (u for all δ > 0 for any u L (R such that u = 1. Considering the normalized outer measure µ(e := u (xdx µ(r = 1 E and using Jensen s inequality for concave nonlinearities and with measure µ we have ( ( log u (1. dx = log u dµ log u dµ R R R = u log u dx. R On the other hand applying (1.1 we derive that for all δ > 0 u log u dx log R (D δ + C I δ (u for some positive constant D which implies (1.3. Here we used the fact that u dx λ since R u dx = 1. { u λ δ} δ Defining a notion of entropy as typical in statistical mechanics: f f Ent µ (f := log dµ + f 1µ f 1µ log f 1µ f 0 f 1µ := R the conclusion of the previous results reads as u L (R δ > 0 : I δ (u < + = Ent L (u < +. Remark 1. (Logarithmic LS. If u H 1 (R then the results of [19] show that (1.5 lim I δ (u = Q u dx δ 0 R fdµ

3 LOGARITHMIC SOBOLEV IEQUALITY REVISITED 3 for some constant Q > 0. Hence passing to the limit as δ 0 in the inequality of Theorem 1.1 one recovers classical forms of the logarithmic inequality. The logarithmic Schrödinger equation (1.6 i t φ + φ + φ log φ = 0 φ : [0 R C 3 admits applications to quantum mechanics quantum optics transport and diffusion phenomena theory of superfluidity and Bose-Einstein condensation (see [5] and [10 1]. The standing waves solutions of (1.6 solve the following semi-linear elliptic problem (1.7 u + ωu = u log u u H 1 (R. These equations were recently investigated in [1 ]. From a variational point of view the search of solutions to (1.7 can be associated with the study of critical points (in a nonsmooth sense of the lower semi-continuous functional J : H 1 (R R {+ } defined by J(u = 1 u dx + ω + 1 R R u dx 1 R u log u dx which is well defined by the logarithmic Sobolev inequality. Due to Theorem 1.1 and (1.5 one could handle a kind of nonlocal approximations of (1.7 formally defined for δ > 0 by I δ (u + ωu = u log u which are associated with the energy functional J δ : H 1 (R R {+ } defined by J δ (u = I δ (u + ω + 1 u dx 1 u log u dx. R R Since there holds I δ (u C R u dx for all δ > 0 and u H 1 (R (cf. [19 Theorem ] the energy functional J δ is well defined for every δ > 0. Remark 1.3 (Magnetic case. If A : R R is locally bounded and u : R C we set x+y Ψ u (x y := e i(x y A( u(y x y R. It was observed in [15] that the following Diamagnetic inequality holds In turn by defining we have u(x u(y Ψ u (x x Ψ u (x y for a.e. x y R. I A δ (u := { Ψ u(xy Ψ u(xx >δ} δ dxdy x y + (1.8 I δ ( u I A δ (u for all δ > 0 and all measurable u : R C. Then Theorem 1.1 yields the following Magnetic logarithmic Sobolev inequality. For u L (R there is a positive constant C such that R u u log u u dx + log u ( log C δ u + C Iδ A (u. otice that since I δ ( u u as δ 0 [19] and IA δ (u u iau as δ 0 [] from inequality (1.8 one recovers u u iau which follows from the well-know diamagnetic inequality for the gradients u u iau see [18]. As a companion to Theorem 1.1 we also have the following

4 H.-M. GUYE AD M. SQUASSIA Theorem 1.. Let u L (R ( 3. Assume that there exists a non-decreasing function F : R + R + such that F (ts t β F (s for any s t 0 and some β > 0 and F ( u(x u(y (1.9 x y + dxdy < +. Then there exists a positive constant C F such that R u u log u u dx + log u β (C log F u β + C F In particular condition (1. holds. { u >λ F (1/} F ( u(x u(y x y + dxdy Proof. Consider the statement when u = 1. In light of inequality (1. since by [1 Proposition 6] there exists C > 0 and λ > 0 such that ( 1 (1.10 u /( F ( u(x u(y /( dx C F (1/ x y + dxdy by arguing as in the previous proof we get ( u log u log (D F + D F R where we used the fact that u { u λ F (1/} F ( u(x u(y /( x y + dxdy dx λ F (1/ since R u dx = 1. Then we get u log u (C R log F + C F In the general case using the sub-homogeneity condition on F yields R u u log u u ( log C F + C F u β which yields the desired conclusion. { u(y u(x >δ} F ( u(x u(y x y + dxdy. F ( u(x u(y x y + dxdy Remark 1.5 (L p (R -version. If p > 1 and > p one has a variant of (1. namely ( (1.11 log u p p dx p u p log u p dx. R p R Then by arguing as in the proofs of Theorems 1.1 and 1. with δ p F ( u(x u(y (1.1 u dxdy u x y +p x y +p dxdy in place of I δ (u and (1.9 respectively it is possible to get corresponding log-sobolev inequalities as for the case p = via the results of [1]. In particular if u L p (R and the functionals in (1.1 are finite at u for some δ > 0 then R u p log u p dx < +. The Euclidean logarithmic Sobolev inequalities for the p-case have been intensively studied see e.g. the work of Del Pino and Dolbeault [16] and the references therein.

5 LOGARITHMIC SOBOLEV IEQUALITY REVISITED 5 References [1] R.A. Adams F.H. Clarke Gross s logarithmic Sobolev inequality: a simple proof Amer. J. Math. 101 ( [] J. Bourgain H. Brezis P. Mironescu Another look at Sobolev spaces in Optimal Control and Partial Differential Equations. A Volume in Honor of Professor Alain Bensoussan s 60th Birthday (eds. J. L. Menaldi E. Rofman and A. Sulem IOS Press Amsterdam [3] J. Bourgain H. Brezis P. Mironescu Limiting embedding theorems for W sp when s 1 and applications J. Anal. Math. 87 ( [] J. Bourgain and H-M. guyen A new characterization of Sobolev spaces C. R. Acad. Sci. Paris 33 ( [5] H. Brezis How to recognize constant functions. Connections with Sobolev spaces Russian Mathematical Surveys 57 ( [6] H. Brezis ew approximations of the total variation and filters in imaging Rend Accad. Lincei 6 ( [7] H. Brezis H-M. guyen on-local functionals related to the total variation and connections with Image Processing preprint. 1 [8] H. Brezis H-M. guyen The BBM formula revisited Atti Accad. az. Lincei Rend. Lincei Mat. Appl. 7 ( [9] H. Brezis H-M. guyen Two subtle convex nonlocal approximations of the BV-norm onlinear Anal. 137 ( [10] T. Cazenave Stable solutions of the logarithmic Schrödinger equation onlinear Anal. 7 ( [11] T. Cazenave An introduction to nonlinear Schrödinger equations Textos de Métodos Matemáticos 6 Universidade Federal do Rio de Janeiro [1] T. Cazenave A. Haraux Équations d évolution avec non linéarité logarithmique Ann. Fac. Sci. Toulouse Math. ( [13] A. Cotsiolis. Tavoularis On logarithmic Sobolev inequalities for higher order fractional derivatives Comptes Rendus Acad. Sci. Paris 30 ( [1] P. d Avenia E. Montefusco M. Squassina On the logarithmic Schrödinger equation Commun. Contemp. Math. 16 ( pp. 3 [15] P. d Avenia M. Squassina Ground states for fractional magnetic operators ESAIM COCV to appear. 3 [16] M. Del Pino J. Dolbeault The optimal euclidean L p -Sobolev logarithmic inequality J. Funct. Anal. 197 ( [17] L. Gross Logarithmic Sobolev Inequalities Amer. J. Math. 97 ( [18] E. Lieb and M. Loss Analysis Graduate Studies in Mathematics 1 AMS [19] H-M. guyen Some new characterizations of Sobolev spaces J. Funct. Anal. 37 ( [0] H-M. guyen Further characterizations of Sobolev spaces J. Eur. Math. Soc. 10 ( [1] H-M. guyen Some inequalities related to Sobolev norms Calc. Var. PDEs 1 ( [] H-M. guyen A. Pinamonti M. Squassina E. Vecchi A new characterization of magnetic Sobolev spaces in preparation. 3 [3] A.J. Stam Some inequalities satisfied by the quantities of information of Fisher and Shannon Information and Control ( [] W.C. Troy Uniqueness of positive ground state solutions of the logarithmic Schrödinger equation Arch. Ration. Mech. Anal. ( [5] K.G. Zloshchastiev Logarithmic nonlinearity in theories of quantum gravity: origin of time and observational consequences Grav. Cosmol. 16 ( (H.-M. guyen Department of Mathematics EPFL SB CAMA Station 8 CH-1015 Lausanne Switzerland address: hoai-minh.nguyen@epfl.ch (M. Squassina Dipartimento di Matematica e Fisica Università Cattolica del Sacro Cuore Via dei Musei 1 I-511 Brescia Italy address: marco.squassina@unicatt.it