Pólya-Szegö s Principle for Nonlocal Functionals

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1 International Journal of Mathematical Analysis Vol. 12, 218, no. 5, HIKARI Ltd, Pólya-Szegö s Principle for Nonlocal Functionals Tiziano Granucci ISIS Leonardo da Vinci, Via del Terzolle 77, Firenze, Italy Copyright c 218 Tiziano Granucci. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we apply some properties of the spherically symmetric rearrangement to the study of nonlocal functionals. The main result is a Polya-Szegö s principle for nonlocal seminorms. As a corollary we obtain an alternative proof of the Pólya-Szegö s principle. Keywords: Pólya-Szegö s principle; nonlocal functionals 1 Introduction In this paper we apply some properties of the spherically symmetric rearrangement to the study of nonlocal functionals. The main result is a Polya-Szegö s principle for nonlocal seminorms. As a corollary we obtain an alternative proof of the Pólya-Szegö s principle, reefer to [7, 8, 9]. Let f L p ( ),with p 1, we denote with µ f (λ) L N ({ x : f(x) > λ }) (1) the Lebesgue measure of the set { x : f(x) > λ }. Let Ω be a L N -measurable set then we define [ ] L N 1 N (Ω) R Ω ϖ N (2) where B 1 () { x : x 1 }, ϖ N L N (B 1 ()) π N 2 Γ( N +1) and Γ ( N + 1) 2 2 t N 2 e t dt.

2 246 Tiziano Granucci Moreover we define Ω { x : x < R Ω } (3) the spherically symmetric rearrangement of the set Ω. If f is a positive function, since f (x) 1 {f(y)>λ} (x) dλ (4) we define f (x) 1 {f(y)>λ} (x) dλ (5) the decreasing symmetric rearrangement of f, where {f (y) > λ} is the symmetric rearrangement set of {f (y) > λ}. Theorem 1 (Pólya-Szegö s principle) If f W 1,p ( ), with p 1, and f > then f p f p (6) where f p [ f p dx ] 1 p. For any p 1 and s (, 1) we denote by W ( s,p ) the fractional Sobolev space W ( { s,p ) u L ( p ) u (x) u (y) : L ( } p ) x y N p +s The space W s,p ( ) is a intermediate Banach space between L p ( ) and W 1,p ( ) endowed with the norm where u W s,p ( ) [ u p L p ( ) + [u] s,p [u] s,p u (x) u (y) p ] 1 p x y N+sp dxdy In recent years, nonlocal functional have appeared in a number of applications in the modeling of discontinuos physical, biological and social quantities. J. Bourgain, H. Brezis and P. Mironescu, refer to [1, 2, 3, 4, 6], investigated

3 Pólya-Szegö s principle for nonlocal functionals 247 the asympotic behaviour of the seminorm [u] s,p, precisely if Ω is a subdomain of, p 1 and u W 1,p (Ω) then lim s 1 (1 s) Ω Ω u (x) u (y) p x y N+sp dxdy K p,n Ω u p dx (7) where K p,n 1 w x p dh N 1 and w S N 1. Similar result hold for p S N 1 BV spaces, for magnetic Sobolev space and Orlicz Sobolev spaces, refer to [5]. The main result of the paper is the following theorem. Theorem 2 (Pólya-Szegö s principle for nonlocal seminorms) If f W s,p ( ), with < s < 1, p 1, and f > then [f ] s,p [f] s,p. (8) 2 Proof of Theorem Properties of rearrangements The symmetric decreasing rearrangement f of a Borel function f has the following properties 1. f is no-negative; 2. f (x) f (y) if x y and f (x) > f (y) if x y ; 3. { x : f (x) > t } { x : f (x) > t } and { x : f (x) > t } { x : f (x) > t } for every t > ; 4. if Φ : R + R + is a nondecreasing function then (Φ (f (x))) Φ (f (x)) for every x ; 5. if f (x) g (x) for every x then f (x) g (x) for every x ; 6. moreover we have the following inequality f (x) 1 {g(y) s} (x) dx f (x) 1 {g (y) s} (x) dx

4 248 Tiziano Granucci 2.2 Lemma Lemma 3 Let Φ : R [, ) be a convex function on R. If Φ (), Φ C 2 (R\ {}), then Φ (f (y + h) f (y)) dy Φ (f (y + h) f (y)) dy for every h. Proof. Let us consider f (x + h) f (x+h) ds 1 {f (y+h)>s} (x) ds since it follows {f (x + h) > s} {f (x + h) > s} f (x+h) f (x + h) ds 1 {f(x+h)>s} (x) ds { { Φ (t) if t Φ (t) if t We define Φ + (t) and Φ if t < (t) if t >, it follows Φ (t) Φ + (t) + Φ (t), moreover Φ + and Φ are convex function. Let us consider Φ + (f (y + h) f (y)) dy then Φ + (f (y + h) f (y)) dy ds Φ + (f (y + h) s) 1 {f(z)<s} (y) dsdy Φ + (f (y + h) s) 1 {f(z)<s} (y) dy where Φ +(s) is the derivative of Φ + (s). Using the property (6) of symmetric decreasing rearrangement we have ( Φ+ (f (y + h) s) 1 {f(z)<s} (y) dy Φ + (f (y + h) s)) 1{f (z)<s} (y) dy then by the property (4) of symmetric decreasing rearrangement it follows ( ) Φ+ (f (y + h) s) 1 {f(z)<s} (y) dy Φ + (f (y + h) s) 1 {f (z)<s} (y) dy

5 Pólya-Szegö s principle for nonlocal functionals 249 and Φ + (f (y + h) f (y)) dy Φ + (f (y + h) f (y)) dy. The same results hold for Φ and therefore for Φ. 2.3 Proof of theorem 2 Let us consider f (y + z) f (y) p dy then by Lemma 3 we have f (y + z) f (y) p dy f (y + z) f (y) p dy Let us consider then we get and it follows [f] s,p f(x) f(y) p x y N+sp dxdy f (x) f (y) p x y N+... dxdy f(y+z) f(y) p dz dz f (y+z) f (y) p dzdy f (y + z) f (y) p dy f (y + z) f (y) p dy f (x) f (y) p x y N+sp [f] s,p [f ] s,p. dzdy dxdy 3 Application: the proof of Theorem 1 If f C 1 c Using (7) it follows ( R N ) and f then by Theorem 2 we get [f] s,p [f ] s,p. f p lim (1 s) [f] s 1 s,p lim (1 s) [f ] s,p f p s 1 ( and, since f C ) c 1 R N is dense in W ( 1,p ), it follows the Polya-Szegö s principle (Theorem 1).

6 25 Tiziano Granucci References [1] J. Bourgain, H. Brezis, P. Mironescu, Another Look at Sobolev Spaces, Optimal Control and Partial Differential Equations. A Volume in Honor of Professor Alain Bensoussan s 6th Birthday (eds. J. L. Menaldi, E. Rofman and A. Sulem), IOS Press, Amsterdam, 21, [2] J. Bourgain, H. Brezis, P. Mironescu, Limiting embedding theorems for W s,p when s and applications, J. Anal. Math., 87 (22), [3] J. Bourgain, H. M. Nguyen, A new characterization of Sobolev spaces, Comptes Rendus Mathematique, 343 (26), [4] H. Brezis, How to recognize constant functions. Connections with Sobolev spaces, Russian Math. Surveys, 57 (22), [5] G. Leoni, D. Sprctor, Characterization of Sobolev and BV spaces, J. Funct. Anal., 261 (211), [6] V. Maz ya, T. Shaposhnikova, On Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (22), [7] G. Pólya, G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, Vol. 27, Princeton University Press, N. J., [8] G. Talenti, Best constant in Sobolev inequality, Ann. Math. Pura Appl., Series 4, 11 (1976), [9] G. Talenti, Nonlinear elliptic equations, rearrangements of funtions and Orlicz spaces, Ann. Math. Pura Appl. Series 4, 12 (1979), Received: April 26, 218; Published: May 11, 218