The Fractional Laplacian - UPR-RPemmy.uprrp.edu/lmedina/courses/spring2017/LaTeX/oralexam.pdf · problems involving the fractional Laplacian. F. Seoanes Wave ... = exp( t j˘j2)Ff(˘)

$Page 1: The Fractional Laplacian - UPR-RPemmy.uprrp.edu/lmedina/courses/spring2017/LaTeX/oralexam.pdf · problems involving the fractional Laplacian. F. Seoanes Wave ... = exp( t j˘j2)Ff(˘)$
IntroductionPreliminaries

Fractional LaplacianReference

The Fractional Laplacian

Fabian Seoanes Correa

University of Puerto Rico, Rıo Piedras Campus

February 28, 2017

F. Seoanes Wave Equation 1/ 15



Motivation

During the last ten years it has been an increasing interest inthe study of fractional powers operators. This renewedattention on fractional operators started with the works by L.Caffarelli and L. Silvestre and collaborators [4, 1, 2, 3] onproblems involving the fractional Laplacian.




Fourier Transform

Schwarts space

S (Rn) :=

{f ∈ C∞(Rn) : ∀m, j ∈ N0 sup

x∈Rn(1 + |x |2)m/2

∣∣∣f (j)(x)∣∣∣ <∞}

An important fact is that S (Rn) = Lp(Rn) for ≤ p <∞

Fourier transform

For any ϕ ∈ S (Rn),

Fϕ(ξ) =1

(2π)n/2

∫Rn

exp(−ix · ξ)ϕ(x)dx




Fourier Transform

We have that F : S → S , is an isomorphism andcontinuous map.

By the Riemann- Lebesgue Theorem F : L1(Rn)→ C0(Rn) isa continuous, injective but not suprajective

The inversion Fourier transform is:

F−1ϕ(x) =1

(2π)n/2

∫Rn

exp(−ix · ξ)ϕ(ξ)dx

F−1(F (ϕ))(x) = ϕ(x)




Fourier Transform

Theorem

Let f , g ∈ S (Rn),

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

∫Rn f (x − y)g(y) dy .

c.) F (exp(−α |x |2)) = 1(2α)n/2 exp(− |ξ|2 /4α)

Clearly the Fourier transform is a linear operator over S .




Fourier Transform

Theorem

Let f , g ∈ S (Rn),

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

∫Rn f (x − y)g(y) dy .

c.) F (exp(−α |x |2)) = 1(2α)n/2 exp(− |ξ|2 /4α)

Clearly the Fourier transform is a linear operator over S .




Fourier Transform

Heat equation

We will solve for u(t, x), t > 0, x ∈ Rn.∂tu(t, x) = ∆u , si t > 0

u(0, x) = f (x)(1)

Where ∆u =∑n

j=1∂2u∂x2

jrepresent the Laplace operator. We want

solutions in S (Rn). (u(t, ·) ∈ S (Rn), t > 0 fixed). Therefore,

dFu(t,ξ)

dt = − |ξ|2 Fu(t, ξ)

Fu(0, ξ) = F f (ξ)

(2)

the solution for the ODE is Fu(t, ξ) = exp(−t |ξ|2)F f (ξ).




Fourier Transform

Heat equation


u(0, x) = f (x)(1)

Where ∆u =∑n

j=1∂2u∂x2



dFu(t,ξ)

dt = − |ξ|2 Fu(t, ξ)

Fu(0, ξ) = F f (ξ)

(2)





Fourier Transform

Heat equation


u(0, x) = f (x)(1)

Where ∆u =∑n

j=1∂2u∂x2



dFu(t,ξ)

dt = − |ξ|2 Fu(t, ξ)

Fu(0, ξ) = F f (ξ)

(2)





Fourier Transform

Consider n = 1, for the previous theorem part(c), we have

exp(−t |ξ|2) = F

(√1

2texp(− |x |2 /4t)

)

So that,

Fu(t, ξ) = F

(√1

2texp(− |x |2 /4t) ∗ f

)(ξ)

= F

(1√4πt

∫R

exp(− |x − y |2 /4t)f (y)dy

)(ξ)

Hence, u(t, x) = 1√4πt

∫R exp(− |x − y |2 /4t)f (y)dy




Fourier Transform


exp(−t |ξ|2) = F

(√1

2texp(− |x |2 /4t)

)

So that,

Fu(t, ξ) = F

(√1

2texp(− |x |2 /4t) ∗ f

)(ξ)

= F

(1√4πt

∫R

exp(− |x − y |2 /4t)f (y)dy

)(ξ)


∫R exp(− |x − y |2 /4t)f (y)dy




Fourier Transform


exp(−t |ξ|2) = F

(√1

2texp(− |x |2 /4t)

)

So that,

Fu(t, ξ) = F

(√1

2texp(− |x |2 /4t) ∗ f

)(ξ)

= F

(1√4πt

∫R

exp(− |x − y |2 /4t)f (y)dy

)(ξ)


∫R exp(− |x − y |2 /4t)f (y)dy




Equivalent definitionsExample

Remember that for f ∈ S the fourier transform gives

F ((−∆)f )(ξ) = |ξ|2 F (ξ), ξ ∈ Rn

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:

lims→1−

(−∆)s f = −∆f lims→0+

(−∆)s f = f






F ((−∆)f )(ξ) = |ξ|2 F (ξ), ξ ∈ Rn


Definition:



lims→1−

(−∆)s f = −∆f lims→0+

(−∆)s f = f






F ((−∆)f )(ξ) = |ξ|2 F (ξ), ξ ∈ Rn


Definition:



lims→1−

(−∆)s f = −∆f lims→0+

(−∆)s f = f





Proposition

Let s ∈ (0, 1) and f ∈ S , then

(−∆)s f (x) = cn,s limε→0

∫Rn−Bε(x)

f (x)− f (y)

|x − y |n+2sdy

where

cn,s = π−n/2 Γf

(n+2s

2

)Γf (−s)

Proposition

Let s ∈ (0, 1) and f ∈ S , then we also have

(−∆)s f (x) =cn,s

2

∫Rn

2f (x)− f (x + y)− f (x − y)

|y |n+2sdy

∀x ∈ Rn

Proof.

See [4].





Proposition

Let s ∈ (0, 1) and f ∈ S , then

(−∆)s f (x) = cn,s limε→0

∫Rn−Bε(x)

f (x)− f (y)

|x − y |n+2sdy

where

cn,s = π−n/2 Γf

(n+2s

2

)Γf (−s)

Proposition

Let s ∈ (0, 1) and f ∈ S , then we also have

(−∆)s f (x) =cn,s

2

∫Rn

2f (x)− f (x + y)− f (x − y)

|y |n+2sdy

∀x ∈ Rn

Proof.

See [4].





In the theory of stochastic process we are interested in the solutionto the fractional diffusion equation

Example ∂tu(t, x) = −(−∆)su(t, x)

u(0, x) = f (x)




Reference

W. Arendt, Vector-valued Laplace transforms and CauchyProblems, Israel. J. Math. 59 (1987), 327-352. MR 89:47064.

W. Arendt, C.J.K. Batty, M. Hieber, and F. Neubrander,Vector-valued Laplace transforms and Cauchy Problems,Monographs in Mathematics 96, Birkhauser, 2001.

O. El-Mennaoui, Traces de semi-groupes holomorphessinguliers a l’origine et comportement asymptotique, These,Besancon, 1992.

O. El-Mennaoui and V. Keyantuo, On the Schrodingerequation in Lp-spaces, Math. Ann. 304 (1996), 293-302.

H. O. Fattorini, Second Order Linear Differential Equations inBanach spaces, North-Holland, Amsterdam, 1985.




Reference

J. A. Goldstein, Semigroups of Linear Operators andApplications, Oxford Univ. Press, 1985.

M. Hieber, Integrated semigroups and differential operators onLp(RN)-spaces, Math. Ann.291 (1991), 1-16. MR 92g:47052.

E. Hille and R.S Phillips, Functional analysis and semigroups,Coll. Publ. 31, American Math. Society, 1957.

L. Hormander, Estimates for translation invariant operators inLp spaces, Acta. Math. 104 (1960), pages 93-140. MR22:12389.




Reference

A. V. Balakrishnan, Fractional powers of closed operators andthe semigroups generated by them, Pacific J. Math. 10 (1960),419-437

S. Bochner, Diffusion equation and stochastic processes, Proc.Nat. Acad. Sci. U. S. A. 35 (1949), 368-370.

T. Kato, Fractional powers of dissipative operators, J. Math.Soc. Japan 13 (1961), 246-274.

L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimatesfor the solution and the free boundary of the obstacle problemfor the fractional Laplacian, Invent. Math.171 (2008), 425-461.




Reference

L. A. Caffarelli and L. Silvestre, An extension problem relatedto the fractional Laplacian, Comm. Partial DifferentialEquations 32 (2007), 1245-1260.

L. Silvestre, PhD thesis, The University of Texas at Austin,2005.

L. Silvestre, Regularity of the obstacle problem for a fractionalpower of the Laplace operator, Comm. Pure Appl. Math. 60(2007), 67-112.

Valdinoci E., Palatucci G., and Di Nezza E., Hitchhikers guideto the fractional Sobolev spaces (April 2011), available athttp://arxiv.org/abs/1104.4345.

K. Yosida, Functional analysis, Grund. Math. Wiss. 123,Springer-Verlag 1965.




Thank You

University of Puerto RicoMathtematics Departmentwww.math.uprrp.edu


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The Fractional Laplacian - UPR-RPemmy.uprrp.edu/lmedina/courses/spring2017/LaTeX/oralexam.pdf · problems involving the fractional Laplacian. F. Seoanes Wave ... = exp( t j˘j2)Ff(˘)