Four definitions for the fractional Laplacian

116 Pages·2017·1.32 MB·English

Four definitions for the fractional Laplacian
N. Accomazzo (UPV/EHU), S. Baena (UB), A. Becerra Tom´e (US), J.
Mart´ınez (BCAM), A. Rodr´ıguez (UCM), I. Soler (UM)
VIII Escuela-Taller de Ana´lisis Funcional
Basque Center for Applied Mathematics (BCAM)
bpc 1
Nap time, March 2 , 2018
1
International Women’s Day
Group 3 Fractional Laplacian VIII Escuela-Taller 1 / 40
Basic example of fractional operator : fractional Laplacian
Laplace fractional operator: several points of view
Functional analysis: M. Riesz, S. Bochner, W. Feller, E. Hille, R. S. Phillips,
A. V. Balakrishnan, T. Kato, Mart´ınez–Carracedo y Sanz–Alix, K. Yosida
Potencial theory for fractional laplacian: N. S. Landkof
L´evy’s processes: K. Bogdan e.a.
Partial Derivative Ecuations: L. Ca↵arelli y L. Silvestre
Scattering theory: C. R. Graham y M. Zworski, S-Y. A. Chang y M.d.M.
Gonz´alez
Kato’s square root (solved by P. Auscher e.a.)
Group 3 Fractional Laplacian VIII Escuela-Taller 2 / 40
Laplace fractional operator: several points of view
Functional analysis: M. Riesz, S. Bochner, W. Feller, E. Hille, R. S. Phillips,
A. V. Balakrishnan, T. Kato, Mart´ınez–Carracedo y Sanz–Alix, K. Yosida
Potencial theory for fractional laplacian: N. S. Landkof
L´evy’s processes: K. Bogdan e.a.
Partial Derivative Ecuations: L. Ca↵arelli y L. Silvestre
Scattering theory: C. R. Graham y M. Zworski, S-Y. A. Chang y M.d.M.
Gonz´alez
Kato’s square root (solved by P. Auscher e.a.)
Basic example of fractional operator : fractional Laplacian
Group 3 Fractional Laplacian VIII Escuela-Taller 2 / 40
A pointwise definition of the
fractional Laplacian
Group 3 Fractional Laplacian VIII Escuela-Taller 3 / 40
n • n
S (R ) is the space C (R ) of functions that
2 p/2 a
kf k = sup sup (1 + |x| ) |∂ f (x)| < • p 2 N [ {0}
p
|a|p x2Rn
This space endowed with the metric topology
• kf gk
p p
d(f , g) = Â 2
1+ kf gk
p=0 p
The working space
n
We are going to work with the space S (R ) of L. Schwartz’ rapidly decreasing
functions.
Group 3 Fractional Laplacian VIII Escuela-Taller 4 / 40
This space endowed with the metric topology
• kf gk
p p
d(f , g) = Â 2
1+ kf gk
p=0 p
The working space
n
We are going to work with the space S (R ) of L. Schwartz’ rapidly decreasing
functions.
n • n
S (R ) is the space C (R ) of functions that
2 p/2 a
kf k = sup sup (1 + |x| ) |∂ f (x)| < • p 2 N [ {0}
p
|a|p x2Rn
Group 3 Fractional Laplacian VIII Escuela-Taller 4 / 40
The working space
n
We are going to work with the space S (R ) of L. Schwartz’ rapidly decreasing
functions.
n • n
S (R ) is the space C (R ) of functions that
2 p/2 a
kf k = sup sup (1 + |x| ) |∂ f (x)| < • p 2 N [ {0}
p
|a|p x2Rn
This space endowed with the metric topology
• kf gk
p p
d(f , g) = Â 2
1+ kf gk
p=0 p
Group 3 Fractional Laplacian VIII Escuela-Taller 4 / 40
If we introduce the spherical and solid averaging operators
ˆ
x+y
f (x + y) + f (x y) 1
My f (x) = Ay f (x) = f (t) dt
2 2y x y
00
then we can reformulate f (x) like this
00 f (x) My f (x) f (x) Ay f (x)
f (x) = 2 l´ım = 6 l´ım
y!0 y2 y!0 y2
First definition motivation
2
Let f 2 C (a, b), then for every x 2 (a, b) o n e h a s
00 2f (x) f (x + y) f (x y)
f (x) = l´ım
y!0 y2
Group 3 Fractional Laplacian VIII Escuela-Taller 5 / 40
00
then we can reformulate f (x) like this
00 f (x) My f (x) f (x) Ay f (x)
f (x) = 2 l´ım = 6 l´ım
y!0 y2 y!0 y2
First definition motivation
2
Let f 2 C (a, b), then for every x 2 (a, b) o n e h a s
00 2f (x) f (x + y) f (x y)
f (x) = l´ım
y!0 y2
If we introduce the spherical and solid averaging operators
ˆ
x+y
f (x + y) + f (x y) 1
My f (x) = Ay f (x) = f (t) dt
2 2y x y
Group 3 Fractional Laplacian VIII Escuela-Taller 5 / 40
First definition motivation
2
Let f 2 C (a, b), then for every x 2 (a, b) o n e h a s
00 2f (x) f (x + y) f (x y)
f (x) = l´ım
y!0 y2
If we introduce the spherical and solid averaging operators
ˆ
x+y
f (x + y) + f (x y) 1
My f (x) = Ay f (x) = f (t) dt
2 2y x y
00
then we can reformulate f (x) like this
00 f (x) My f (x) f (x) Ay f (x)
f (x) = 2 l´ım = 6 l´ım
y!0 y2 y!0 y2
Group 3 Fractional Laplacian VIII Escuela-Taller 5 / 40