The Fractional Laplacian

Transcription

1 The Fabian Seoanes Correa University of Puerto Rico, Río Piedras Campus February 28, 2017 F. Seoanes Wave Equation 1/ 15

2 Motivation During the last ten years it has been an increasing interest in the study of fractional powers operators. This renewed attention on fractional operators started with the works by L. Caffarelli and L. Silvestre and collaborators [4, 1, 2, 3] on problems involving the fractional Laplacian. F. Seoanes Wave Equation 2/ 15

3 Fourier Transform Schwarts space S (R n ) := { f C (R n ) : m, j N 0 sup (1 + x 2 ) m/2 x R n An important fact is that S (R n ) = L p (R n ) for p < Fourier transform For any ϕ S (R n ), F ϕ(ξ) = 1 (2π) n/2 exp( ix ξ)ϕ(x)dx R n } f (j) (x) < F. Seoanes Wave Equation 3/ 15

4 Fourier Transform We have that F : S S, is an isomorphism and continuous map. By the Riemann- Lebesgue Theorem F : L 1 (R n ) C 0 (R n ) is a continuous, injective but not suprajective The inversion Fourier transform is: F 1 1 ϕ(x) = (2π) n/2 exp( ix ξ)ϕ(ξ)dx R n F 1 (F (ϕ))(x) = ϕ(x) F. Seoanes Wave Equation 4/ 15

5 Fourier Transform Theorem Let f, g S (R n ), a.) F (f )(ξ) = iξf (f )(ξ) b.) F (f g)(ξ) = F (f )(ξ)f (g)(ξ), where (f g)(x) = 1 (2π) n/2 R f (x y)g(y) dy. n c.) F (exp( α x 2 )) = 1 (2α) n/2 exp( ξ 2 /4α) F. Seoanes Wave Equation 5/ 15

6 Fourier Transform Theorem Let f, g S (R n ), a.) F (f )(ξ) = iξf (f )(ξ) b.) F (f g)(ξ) = F (f )(ξ)f (g)(ξ), where (f g)(x) = 1 (2π) n/2 R f (x y)g(y) dy. n c.) F (exp( α x 2 )) = 1 (2α) n/2 exp( ξ 2 /4α) Clearly the Fourier transform is a linear operator over S. F. Seoanes Wave Equation 5/ 15

7 Fourier Transform Heat equation We will solve for u(t, x), t > 0, x R n. t u(t, x) = u, si t > 0 u(0, x) = f (x) (1) Where u = n j=1 2 u represent the Laplace operator. We want xj 2 solutions in S (R n ). (u(t, ) S (R n ), t > 0 fixed). Therefore, F. Seoanes Wave Equation 6/ 15

8 Fourier Transform Heat equation We will solve for u(t, x), t > 0, x R n. t u(t, x) = u, si t > 0 u(0, x) = f (x) (1) Where u = n j=1 2 u represent the Laplace operator. We want xj 2 solutions in S (R n ). (u(t, ) S (R n ), t > 0 fixed). Therefore, df u(t,ξ) dt = ξ 2 F u(t, ξ) F u(0, ξ) = F f (ξ) (2) F. Seoanes Wave Equation 6/ 15

9 Fourier Transform Heat equation We will solve for u(t, x), t > 0, x R n. t u(t, x) = u, si t > 0 u(0, x) = f (x) (1) Where u = n j=1 2 u represent the Laplace operator. We want xj 2 solutions in S (R n ). (u(t, ) S (R n ), t > 0 fixed). Therefore, df u(t,ξ) dt = ξ 2 F u(t, ξ) F u(0, ξ) = F f (ξ) (2) the solution for the ODE is F u(t, ξ) = exp( t ξ 2 )F f (ξ). F. Seoanes Wave Equation 6/ 15

10 Fourier Transform Consider n = 1, for the previous theorem part(c), we have ( ) exp( t ξ 2 1 ) = F 2t exp( x 2 /4t) F. Seoanes Wave Equation 7/ 15

11 Fourier Transform Consider n = 1, for the previous theorem part(c), we have ( ) exp( t ξ 2 1 ) = F 2t exp( x 2 /4t) So that, ( ) 1 F u(t, ξ) = F 2t exp( x 2 /4t) f (ξ) ( ) 1 = F exp( x y 2 /4t)f (y)dy (ξ) 4πt R F. Seoanes Wave Equation 7/ 15

12 Fourier Transform Consider n = 1, for the previous theorem part(c), we have ( ) exp( t ξ 2 1 ) = F 2t exp( x 2 /4t) So that, ( ) 1 F u(t, ξ) = F 2t exp( x 2 /4t) f (ξ) ( ) 1 = F exp( x y 2 /4t)f (y)dy (ξ) 4πt Hence, u(t, x) = 1 4πt R exp( x y 2 /4t)f (y)dy R F. Seoanes Wave Equation 7/ 15

13 Equivalent definitions Example Remember that for f S the fourier transform gives F (( )f )(ξ) = ξ 2 F (ξ), ξ R n Then it is clear how to define the powers of the Laplacian. F. Seoanes Wave Equation 8/ 15

14 Equivalent definitions Example Remember that for f S the fourier transform gives F (( )f )(ξ) = ξ 2 F (ξ), ξ R n Then it is clear how to define the powers of the Laplacian. Definition: For s > 0(our interest is in s (0, 1)) the fractional Laplacian ( ) s acts as F. Seoanes Wave Equation 8/ 15

15 Equivalent definitions Example Remember that for f S the fourier transform gives F (( )f )(ξ) = ξ 2 F (ξ), ξ R n Then it is clear how to define the powers of the Laplacian. Definition: For s > 0(our interest is in s (0, 1)) the fractional Laplacian ( ) s acts as By Fourier transform: lim f = f s 1 ( )s lim f = f s 0 +( )s F. Seoanes Wave Equation 8/ 15

16 Equivalent definitions Example Proposition Let s (0, 1) and f S, then ( ) s f (x) f (y) f (x) = c n,s lim ε 0 R n B ɛ(x) x y n+2s dy where c n,s = π n/2 Γ ( n+2s ) f 2 Γ f ( s) Proposition Let s (0, 1) and f S, then we also have ( ) s f (x) = c n,s 2f (x) f (x + y) f (x y) 2 R n y n+2s dy x R n F. Seoanes Wave Equation 9/ 15

17 Equivalent definitions Example Proposition Let s (0, 1) and f S, then ( ) s f (x) f (y) f (x) = c n,s lim ε 0 R n B ɛ(x) x y n+2s dy where c n,s = π n/2 Γ ( n+2s ) f 2 Γ f ( s) Proposition Let s (0, 1) and f S, then we also have ( ) s f (x) = c n,s 2f (x) f (x + y) f (x y) 2 R n y n+2s dy x R n F. Seoanes Wave Equation 9/ 15

18 Equivalent definitions Example In the theory of stochastic process we are interested in the solution to the fractional diffusion equation Example t u(t, x) = ( ) s u(t, x) u(0, x) = f (x) F. Seoanes Wave Equation 10/ 15

19 W. Arendt, Vector-valued Laplace transforms and Cauchy Problems, Israel. J. Math. 59 (1987), MR 89: W. Arendt, C.J.K. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace transforms and Cauchy Problems, Monographs in Mathematics 96, Birkhäuser, O. El-Mennaoui, Traces de semi-groupes holomorphes singuliers à l origine et comportement asymptotique, Thèse, Besançon, O. El-Mennaoui and V. Keyantuo, On the Schrödinger equation in L p -spaces, Math. Ann. 304 (1996), H. O. Fattorini, Second Order Linear Differential Equations in Banach spaces, North-Holland, Amsterdam, F. Seoanes Wave Equation 11/ 15

20 J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, M. Hieber, Integrated semigroups and differential operators on L p (R N )-spaces, Math. Ann.291 (1991), MR 92g: E. Hille and R.S Phillips, Functional analysis and semigroups, Coll. Publ. 31, American Math. Society, L. Hörmander, Estimates for translation invariant operators in L p spaces, Acta. Math. 104 (1960), pages MR 22: F. Seoanes Wave Equation 12/ 15

21 A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10 (1960), S. Bochner, Diffusion equation and stochastic processes, Proc. Nat. Acad. Sci. U. S. A. 35 (1949), T. Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan 13 (1961), L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math.171 (2008), F. Seoanes Wave Equation 13/ 15

22 L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), L. Silvestre, PhD thesis, The University of Texas at Austin, L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), Valdinoci E., Palatucci G., and Di Nezza E., Hitchhikers guide to the fractional Sobolev spaces (April 2011), available at K. Yosida, Functional analysis, Grund. Math. Wiss. 123, Springer-Verlag 1965 Ḟ. Seoanes Wave Equation 14/ 15

23 Thank You University of Puerto Rico Mathtematics Department F. Seoanes Wave Equation 15/ 15