Mouhamed Fall, (AIMS)

PDE from the south

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Mouhamed Fall (African Institute for Mathematical Sciences) - Regularity results for elliptic equations involving regional fractional Laplacian

For more information go to http://mat.ufcg.edu.br/pdefromthesouth/

00:00

in the rational fractional laplacian

00:03

so i will introduce in the next slides

00:06

so the object but still

00:09

just to say that this is a very hot

00:12

topic

00:13

and there are several open problems

00:16

that i will tell about uh

00:20

and it's interesting uh because it

00:23

provides somehow a

00:24

natural uh so natural

00:29

so uh limits uh for the

00:32

for the classical laplacian and for

00:34

classical elliptic regularity theory

00:37

and i think so uh it is important to

00:41

to study these questions not only for

00:44

applications but also for the analytic

00:46

point of view

00:49

what is important and interesting to see

00:51

here is that in fact

00:53

many arguments that are used in the

00:56

classical world cannot walk anymore and

00:58

somehow one has to

01:01

to redefine things and and somehow to

01:04

to find out new ideas in order to

01:08

overcome

01:08

certain difficulties that are involved

01:11

in this

01:12

problem uh so

01:15

let me just uh introduce quickly uh

01:21

uh the first object of study which

01:24

is uh made the fractional application

01:27

that

01:28

maybe many of you know uh so which is an

01:31

integral operator and it is non-local

01:34

in a sense uh if you look at maybe

01:38

if you even if you have a function w

01:41

which is in it is compactly supported

01:45

so you know that just looking at the

01:48

integral equation so the the fractional

01:50

laplacian when it

01:52

when it acts on w this function does not

01:55

necessarily have a compact support

01:58

so which means you cannot you cannot

02:01

only local information so it cannot tell

02:04

you

02:05

much on this operator

02:10

and maybe this is a pv integral so the

02:12

pv integral is defined

02:14

as a limit and maybe uh so

02:18

i think some of you knows also deep

02:21

but we can discuss this anyone who have

02:24

questions can just

02:26

can just stop me uh because it is a

02:28

lecture and since

02:29

it's online maybe uh with slides

02:32

but it's we cannot write everything

02:36

so please stop me and ask questions

02:39

anytime so the

02:43

the constant so that is in front of this

02:45

which

02:46

is a constant that simply uh

02:50

that simply

02:54

tells about

02:58

about when this

03:01

that when we look at the fourier

03:02

transform of this operator

03:05

in this case we have a nice formula we

03:07

will come

03:08

back we will come back to this later on

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and in fact

03:11

using fourier transform or exactly these

03:14

integral equations

03:17

with this term so we can see we with the

03:20

gamma

03:21

function 31 minus s so if s

03:24

is converging to one also so we we

03:27

recover

03:28

the fractional approach and so we

03:31

recover the fractional

03:32

blush and so this is uh something that

03:35

is important and somehow this

03:37

operator provides a limit for the

03:40

classical laplace so the next

03:46

uh here now is what is somehow

03:50

known as the regional fractional

03:52

laplacian or somehow the omega original

03:54

fraction of laplacian

03:56

so maybe if you look at here also we

03:58

feel that nothing changes

04:00

so except that the domain of integration

04:03

is

04:04

now regional

04:07

it's not on the whole space but it's

04:09

rational so this is

04:11

the reason why we call the name of

04:14

regional so in a sense if you look at

04:18

the process for the fractional

04:19

application

04:21

so you have a levy process

04:25

it's generated by levis process which is

04:27

symmetric

04:29

uh stable but

04:32

for the regional fractional laplacian it

04:34

is generated by a process

04:37

which is obtained of course from the

04:39

from the one of the fractional laplacian

04:41

but still that levy process

04:42

is not is not is not symmetric

04:49

but still it is an interesting process

04:53

because it allows somehow to understand

04:57

particles that are sensory to jump

05:00

outside omega so this is the ratio this

05:04

is the reason why the

05:05

public the probabilistic colleagues

05:09

they they they call it sensory

05:11

fractional laplacian or maybe

05:13

sensory process because the process is

05:15

somehow sensory to jump

05:17

outside the domain omega

05:20

and in that case we can somehow define

05:24

different problems in in the case when

05:27

we are interested

05:28

in phenomenon in which we have uh

05:31

particles that should not allow that

05:34

that are not allowed to jump outside

05:36

omega but are

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killed on the boundary or

05:41

reflected when they reach the bond

05:44

and in the brownian case we all know

05:47

that these are

05:47

just the kind of directly conditioned

05:51

in norman condition and so let me just

05:54

go on directly to the to those

05:59

definitions so

06:04

i will start maybe perhaps with perhaps

06:06

in uh

06:07

elementary things that but maybe uh

06:10

just to introduce the notion that we

06:12

want to introduce

06:14

so let me just uh uh

06:17

okay so this is the aim also okay so

06:20

sorry so this is the aim

06:21

of the of the lecture is to give the an

06:24

overview

06:25

of the boundary regularity so for this

06:27

problem that i just mentioned before

06:29

uh so boundary regularity for problems

06:32

involving

06:33

the rational fractional laplacian and

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where

06:36

in the case where we have uh directly

06:40

or neumann boundary conditions so in a

06:42

sense in the situation where

06:46

we have kill the process when they reach

06:48

the boundary or reflected

06:51

so in the brownian case

06:56

this is a

07:01

well known and introduced that

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i just consider so omega an

07:08

open little set and i think maybe many

07:10

of you are already familiar with these

07:12

spaces

07:13

and i introduced the sovereign spaces h1

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and the h1 0 space

07:18

and so which is just the better function

07:21

in h1

07:22

such that so u is zero at the boundary

07:25

and in this case so one is interested in

07:28

the

07:29

in the most famous problems uh the

07:31

simple problems

07:33

uh which are just in the director

07:36

problem

07:37

uh which is uh

07:40

which is just a weak solution for the

07:42

laplacian with a

07:44

solution which are which are zero at the

07:46

boundary

07:48

so these are somehow uh called the

07:51

the direction so the the operator that

07:54

comes with the direct

07:54

boundary conditions comes with just when

07:57

you kill brownian motion

07:59

uh at the boundary which is very

08:02

possible

08:03

but in the case of the fractional

08:04

appliance and we will see that later on

08:06

in in some cases uh the process may not

08:10

even touch the boundary

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and so this is also a future

08:16

that we will see later on and in the

08:19

reflected case

08:21

uh this is the usual norman problem

08:24

where uh so uh we have the fractional

08:27

we have the laplacian of u equal f but u

08:31

is in h1 so in a sense in three

08:34

so we we have a few test functions

08:37

compared to four

08:39

but still in this situation we will see

08:42

that

08:44

so the so that the solutions they have

08:47

they do not have any problem in terms of

08:50

boundary regularity

08:52

but this will not be the case when we

08:54

are when we when we

08:55

when we when we consider the levy

08:57

process uh

08:59

and we will see we will see later on

09:02

that

09:03

this this these two problems will be

09:06

will have

09:06

will be very different in fact

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okay so uh just

09:16

uh just to consider simply so the weak

09:20

formulation of these problems which

09:22

are well known so somehow here uh

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so the the main difference is that in

09:28

fact one is just considering different

09:30

test functions

09:32

uh and and somehow

09:38

and somehow we we in the in the in the

09:41

norman case

09:42

one is looking at functions which are

09:45

continuous and smooth up to the boundary

09:48

well sometimes you can't remove c c

09:50

infinity c the c can be removed of

09:52

course

09:56

and this is just to allow a lot of uh

09:59

to obtain a free boundary on you so you

10:03

do not

10:03

we do not restrict any any we do not

10:06

prescribe any boundary condition on you

10:09

and this will bring a kind of free

10:11

boundary so which is

10:13

a normal derivative but v in the in the

10:16

case of the killed brownian motion or

10:18

maybe the dirichlet problem

10:20

uh one sees that the boundary condition

10:22

is already in the space

10:25

and and somehow uh so uh the

10:29

the simplest uh

10:32

and somehow here the the the important

10:36

point here that one always consider is

10:39

the ghost green formula so

10:41

which allows to say that if you have a

10:44

classical solution

10:47

then

10:51

if you have a classical solution then

10:54

you can make your integration by part

10:57

and then your and then your weak

11:00

solutions

11:01

will become classical solutions

11:08

and this would give somehow that the

11:10

solution

11:11

of the equation that you are looking for

11:13

in a pointwise sense

11:16

but then the question that comes is

11:18

clear now

11:19

so from weak solutions would you can you

11:22

prove that your solution is regular

11:24

enough

11:26

to become classical and also what can

11:29

you see

11:30

at the boundary that allows you also to

11:32

make a full integration by parts

11:35

so that your weak solution

11:39

will be a classical solution and solve

11:42

the equation that you want

11:46

so in the in this in the i will just

11:49

tell

11:50

the classical results for the

11:54

for this for the local operators

11:59

so the the restriction on the test

12:02

functions if you look at this

12:04

we we see immediately that from the

12:06

ghost green formula if i have

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if i have a classical solution then with

12:11

the restriction on the space of test

12:13

functions

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then i will have just simply uh

12:17

a directly problem or maybe a normal

12:20

boundary condition

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because i i live free i live

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i consider a lot of test functions

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uh and so for example this is the number

12:34

and boundary condition somehow that will

12:36

come

12:37

becomes zero

12:41

so this is uh this is the this is a very

12:44

classical uh results that maybe can be

12:48

found also most of these can be found in

12:50

many textbooks

12:53

in particular the book of brazil the

12:56

most the famous book of praises i would

12:58

say

12:59

which contains all the argument

13:03

and motivation of course so

13:06

what we what we would like

13:09

now to show let me just show that what

13:11

is

13:12

known for this uh for this uh for this

13:15

uh

13:16

what is the regularity for the weak

13:18

solutions

13:20

the known regularity are

13:23

somehow oh sorry let me talk about

13:25

existence because

13:26

it will be important to parallel all of

13:28

this in the non-local theory

13:30

you know when you in the non-local

13:32

theory so there are things

13:34

so i'm writing this because in the

13:36

non-local case you you

13:37

you you may have sometimes

13:40

these inequalities or maybe this type of

13:44

regularity but sometimes

13:45

you may not and so this is the reason

13:49

why i prefer to write

13:51

all of this just to introduce what will

13:53

come next

13:56

so the usual price inequality maybe you

13:58

all know that the gradient of the

14:00

function

14:00

is bounded by the square norm of the

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function the l2 norm

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if i'm if i have if i'm zero at the

14:06

boundary

14:07

and use a constant which depends somehow

14:09

in general on the measure of omega

14:12

and which of course blows up when the

14:15

measure of the omega is becoming smaller

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and smaller

14:19

but at the same time we also have

14:23

the five car inequality which you

14:25

sometimes call it the park every finger

14:29

one card every single inequality in

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which case that

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the function should be perpendicular to

14:35

one to the constant function one

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and this is somehow uh important for the

14:45

fractional laplace for the first sorry

14:46

for the for the for the

14:48

for the existence of solution in the

14:49

normal problem

14:51

and of course

14:55

the constant function is in general a

14:57

solution so

14:58

which means you have to kill it from

15:00

killing it in which you wait

15:02

which means you have to be perpendicular

15:05

to the constant function

15:07

in the l2 sense which is exactly

15:10

this

15:14

and yet these

15:18

inequalities will tell me that

15:22

in fact i will be quality

15:28

when i'm looking at the minimization

15:30

problem

15:32

i'm coercive and so

15:35

i can apply lux milligram or maybe

15:38

it is representation theorem or maybe

15:41

any kind of

15:42

argument that that

15:45

that allows me

15:48

there are many many many arguments

15:52

about anyone who is interested maybe can

15:54

just

15:55

look at the book of praises

16:06

my writing are very bad i hope you can

16:08

read this but

16:09

it's

16:13

very strange so let me

16:17

so the book of praises could be a very

16:20

nice to understand this so you know uh

16:23

but all these arguments are very

16:24

abstract so in fact

16:26

in general you only need a bilinear form

16:28

and then

16:29

which is coercive continues and then you

16:31

have solutions

16:35

so so and and this is in general what

16:37

happens for the for the

16:39

for for many things so even if you

16:40

replace the the the gradient

16:42

is a non-degrading square noun with

16:44

something else

16:45

in general if you have this type of

16:47

inequalities which allows you to get

16:50

it allows you uh to get a coercivity

16:52

then you get existence of minimizer so

16:54

which means even in the non-local theory

16:56

so it's

16:57

it's still the same and and it's not it

17:00

doesn't cost too much somehow to

17:02

obtain to to obtain existence of

17:05

solutions and things like that

17:16

so so the classical regular theory so

17:20

let me

17:21

just uh consider uh

17:24

here uh the the the dirichlet problem so

17:27

here which is the solute

17:29

these are weak solutions

17:32

so if i have variational solutions for

17:35

both the deeply entertainment problem

17:38

i can immediately obtain

17:41

many interesting arguments some

17:44

sorry so the first is that in fact i can

17:48

have strong solutions

17:50

so in some textbooks they use the word

17:52

strong solutions because

17:54

when i have f l p then i can know that

17:57

the fractional laplacian of u

18:00

or the fractional laplacian of v which

18:02

is the solution to the

18:03

directly ornament problem they they are

18:07

point wide defined

18:08

in fact they are in lp

18:11

so and in other text

18:14

books also they call this the calderon

18:16

sigmund theory

18:19

uh so and you see it's it's not it's w2p

18:23

and and when i consider w2p so which

18:26

means i have interesting things

18:28

up to the boundary and in fact

18:32

a simple and a simple

18:36

implication i would say also is that

18:39

if if p is large then in fact i am

18:43

in c 2 minus n over p

18:47

up to the boundary so which means

18:50

also my even my even my

18:54

gradient is bounded and somehow up to

18:56

the boundary and even holder continues

18:59

this is a result that can be obtained uh

19:03

so maybe by using uh different

19:07

uh different arguments maybe blow up

19:10

arguments or maybe

19:11

uh perturbative arguments

19:15

uh like cathedrally or many

19:18

many many arguments can be used in order

19:20

to prove

19:21

this without even without knowing this

19:30

and of course it's also if you have

19:33

for instance if omega is is the full

19:36

space then one can also prove

19:38

it by simply by simply uh

19:42

using the risk kernel

19:47

but in a sense it comes directly by

19:49

using the more

19:50

embedding uh the more embedding of

19:54

the w2p in c2 minus n

19:57

over p and and so yeah

20:02

it's the images for classical theory and

20:06

now

20:06

uh another interesting point is

20:11

the shouter theory

20:14

uh oops so the shoulder theory in this

20:18

case also for both solutions

20:20

we know that if if my if i have folder

20:23

continues my right hand side

20:25

holder continues and omega is

20:28

folded continuous

20:32

c to alpha then my solution are also

20:35

c-12

20:39

and this is important

20:44

and so the book for all these can be so

20:47

for all these results

20:48

of the book the the the one of the best

20:51

textbook is the gilbert schrodinger

20:56

and so gilbert rodinger so they use

21:01

freddie mercury in order to prove these

21:04

maximum principles and this slide but

21:06

but still

21:08

you can use foreign which works

21:10

perfectly

21:11

for the laplace operator but when you

21:14

come in the case of the fractional

21:16

laplacian so there is a serious problem

21:19

and the federal material will not work

21:21

uh the reason is that in a sense you

21:24

know uh

21:25

you cannot have you cannot have this

21:28

result so it's impossible

21:30

that you that you can obtain this

21:34

this is the this is the main problem uh

21:37

so

21:38

this is the reason why in a sense one

21:40

should

21:41

somehow find out new techniques in order

21:44

to

21:44

uh in order to obtain this regularity

21:46

result

21:48

so the productivity is very interesting

21:51

and also works

21:52

nicely when you replace laplacian with a

21:54

divergence type

21:55

operator or even or a non-diverse type

21:58

operator

22:01

but it's not clear so the perturbative

22:04

argument or maybe the compactness

22:06

argument

22:07

also in the spirit of kafir really

22:10

also works perfectly uh so this is and a

22:13

blow up argument

22:14

also are nice enough but sometimes

22:18

it works sometimes they do not work for

22:20

like further for

22:21

for for uh for

22:25

for for the problems for the non-local

22:27

problems that will be

22:38

considered

22:40

so the question now uh we

22:43

are asking is maybe okay

22:47

so here we simply consider

22:51

weak solutions by looking at the h1

22:54

somehow energy space the semi normal of

22:56

the energy space

22:59

but then if i look at the if i replace

23:02

now the h1

23:04

sub ellipse space by the hs subway space

23:07

for s more than one

23:09

so what will be what will happen

23:13

and you will see that even if i dig if i

23:16

if i

23:17

if i consider h s for s smaller than one

23:20

this is the reason why i didn't

23:21

put h1 omega and hs omega then even for

23:25

s smaller than one there are many many

23:27

spaces

23:28

there are many spaces that approximate

23:30

h1 0

23:31

and 81 omega but which are both

23:34

fractional

23:34

and they have interesting similarities

23:38

so this is the reason why i just arrived

23:40

h1 in hs but instead of h1 omega and hs

23:43

omega and we will see

23:45

we will see that we will see the

23:47

different spaces

23:48

that can comes into play and also those

23:52

those bases will define as we should

23:56

expect uh the natural boundary condition

24:00

whether it will be dirichlet neumann or

24:03

things like that

24:13

so the

24:16

the fractional sublist basis so the

24:18

first

24:19

so the the the the fraction of civil

24:22

species so let me just

24:23

look at uh i just look at this what we

24:26

call sometimes people call the gallium

24:28

norms defined by legally i don't know

24:31

but if only guys in iran

24:33

and and even not an iron one can define

24:35

them by

24:36

fourier transform maybe uh

24:40

say the vessel species and things

24:43

but still if you if it's restricted on s

24:46

between zero and one

24:47

uh it's better it's simpler maybe to use

24:50

the gallery

24:51

at the seminar because uh the use of

24:54

these

24:55

seminars will allow us to see the

24:57

operator under consideration

25:04

so the reference for this is

25:07

the textbook of griswa

25:11

which contains all these which contains

25:14

many of these results even sometimes

25:16

omega may not be considered lipschitz

25:18

but even continuous also

25:21

and the results that i'm going to say

25:26

below

25:29

so hs0 similar like h10 so is the

25:32

smallest space that contains uh

25:35

c infinity c in h s omega and

25:38

[Music]

25:39

and so well one can one can think maybe

25:42

that it has the same a

25:44

property than h1 but it's not the case

25:47

uh we will see that it's not it doesn't

25:50

have the same properties like

25:51

h1 but even if it is defined by the

25:55

closure of the infinite respect to the

25:57

hs noun

25:58

is it's an interesting space and in fact

26:02

and sometimes comparable to the hs

26:05

space omega

26:13

another interesting space

26:16

is also the space calligraphic age

26:22

which is also the space hs0

26:27

calligraphic hs which is the space of

26:30

functions

26:31

u in hsrn but still

26:34

u is the zero in the exterior of omega

26:37

and this space is

26:39

undoubted with respect to the to the

26:41

norm

26:47

so in a sense it's very interesting so

26:50

so the norm

26:53

hs and somehow the fourier transform

26:57

notation can works perfectly in this

27:00

case

27:03

so the space telegraphic h

27:06

is is is if it is different from the

27:10

space hs0

27:14

and whenever you have a space

27:18

such kind of space uh these are kind of

27:21

dirty space they somehow define

27:23

operators and all the operators are we

27:26

will see

27:27

uh so they are different

27:32

in a in in a certain sense

27:35

uh and previously that late

27:39

so there are interesting properties so

27:41

one of the first properties is the

27:42

density

27:44

so if i remember if i i think even omega

27:47

lip sheets is even too much but even

27:49

omega is continuous i think you have

27:51

synthetic see

27:53

omega bar is dancing in hs

27:57

of course by definition you don't need

28:01

to talk about

28:02

we don't need that but so we have the

28:04

density of c infinity c

28:05

omega bar in h s

28:11

we also have a density of infinity c

28:16

in h as 0 with respect to the h

28:20

as norm of course

28:24

with respect to the hf norm

28:29

which is true by weakening the

28:32

regularity of omega continuity is enough

28:43

and also we have the link between the

28:46

space hs0

28:49

and also the space

28:59

so if i if i wanted to be uh

29:02

if i wanted to to consider a function

29:05

for for which

29:06

if i extend by 0 i will be in

29:09

calligraphic edge because you know

29:10

the calligraphic hs0 are functions which

29:13

are defined

29:14

on rn so

29:17

this is the reason why we consider the

29:19

extension

29:22

to the function that is defined in rn

29:25

but

29:25

if i consider a function u which is in

29:28

hs0

29:28

and if i extend it by zero it is not

29:31

necessarily in hs0

29:32

unless i ask this type of

29:35

rb term to be finite

29:45

this type of hard term so these are

29:48

scale invariant term in the sense that

29:52

it scales exactly

29:56

like like the seminar

30:01

and so it is

30:04

it is not in fact negligible but

30:08

if you look at this carefully then one

30:10

can see that in fact

30:13

the constant function say the function

30:15

one

30:17

for s smaller than one half belongs to

30:20

hs0

30:23

so so if i if i look at this so

30:27

if 2s is smaller than 1

30:36

then the constant function one belongs

30:39

to

30:40

hs0

30:47

and also satisfies

30:57

which is

31:02

so as a consequence so if i

31:05

extend the characteristic function then

31:08

belongs to hs0 my

31:12

calligraphic edge

31:20

so this is uh a nice way somehow to find

31:24

out whether certain functions

31:25

also belongs to calligraphic hs omega

31:30

but if 2s is larger than 1 this is

31:32

completely this is false

31:37

and and so the the link between the two

31:40

spaces so one can

31:42

see that by just

31:46

just looking at the double integral

31:49

uh so if i look at the double integral

31:52

because h is zero so this is given by

31:54

the hs norm

31:55

in rm so if i look at the double

31:58

integral

31:59

so sometimes if i write so in ram

32:04

integral n

32:08

then i can separate it so with the

32:10

seminar

32:11

in omega squared

32:15

plus the integral

32:18

in omega integral in r omega

32:26

and so this is the so this is this

32:28

feminine

32:29

square

32:38

i don't know why i something is the

32:41

electricity

32:43

i think my hand has some electricity

32:46

okay so let me so the feminine squared

32:49

is given just

32:50

by this and so if i

32:53

if i look at this so i will have some

32:55

something like u squared

32:57

because uh somehow x and and i have the

33:00

x minus y to the power n

33:02

plus 2 s so the dx is for omega

33:06

and and and this guy is exactly

33:10

when i look at this so this term so the

33:13

integral

33:15

over rn minus omega

33:24

so this is the term that is somehow

33:27

almost equal to the distance function to

33:29

the x

33:30

to the minus two s and this is the

33:33

reason why

33:34

uh one always have to find out this

33:39

this deferred term

33:47

another interesting fact

34:00

hs0 and hs

34:03

are the same for s between zero and one

34:06

half

34:08

so for s between strictly smaller than

34:11

one half

34:13

it's not that difficult to prove

34:16

but for f equal one half which is

34:18

critical it's

34:20

not easy to prove and so uh

34:24

for this reason so uh

34:27

one can look at the book of grief and

34:30

also but it was

34:32

where he quoted also the works of jack

34:35

louise

34:39

and so this is already important to see

34:41

to to show that in fact if

34:43

i look at the operator's problems

34:47

variation

34:50

that comes from these spaces that

34:54

there is something interesting that if s

34:56

between zero and one half i have an

34:58

intersection of the directly and

34:59

one problem yeah

35:03

it is somehow very trendy but the

35:05

probabilistic people

35:06

they found out this and they they have a

35:09

nice interpretation

35:11

which you will see uh if you will see

35:13

very soon

35:18

so another property is that in fact we

35:21

also

35:22

have a thresh theorem that

35:25

if s is larger than one half

35:28

then i have

35:33

the embedding so i have a trace so hf

35:37

will have a trace

35:38

and of course in fact this is compact

35:40

yeah so this is a compact

35:42

this is the compact embedding because i

35:45

can be

35:46

because i can be uh in hs

35:49

minus one half and this was

35:53

stated in many books but never proved

35:57

uh but being proved it

36:00

in 1996 with a

36:04

very clever argument i think it was

36:07

important in fact to

36:08

to to to write a proof instead of

36:12

just quoting different papers in which

36:15

maybe the proof are not very clear

36:21

okay so now i think we have all the

36:23

spaces that we need

36:27

in terms of if you if you are if you are

36:30

if you are

36:31

if you are interested in the in the in

36:34

the fractional in the problems

36:36

arising problems involving the

36:38

fractional laplacian or maybe

36:40

regional fractional laplacian

36:56

so

37:00

what is what we always need uh

37:03

in general is uh is that we always we

37:07

the sub-level inequalities which also

37:10

gives the provide embedding

37:12

and and why i tell about this non-linear

37:14

inequality

37:15

so in general this non-linear inequality

37:18

allows

37:20

to run a digital

37:23

iteration in order to obtain

37:26

a priori l infinity bond and approve

37:29

holder

37:31

continuity and harmonic inequality so

37:34

one in general does not even need to get

37:37

to star

37:37

s but so the scale invariant inequality

37:40

but

37:40

simply simply a non-linear inequality

37:44

is in general enough

37:49

but the two star as norm is just

38:02

so the true stars norm it just allows to

38:06

uh to obtain uh scale invariant

38:09

inequality which is sometimes

38:10

interesting it is interesting for some

38:12

applications but when it turns

38:14

and it comes to regularity estimates one

38:17

does only need

38:18

a non-linear inequality so 2 star can be

38:20

any p but we still larger than 2.

38:26

and of course if 2s is larger than 1

38:28

then

38:29

we can replace the seminar with the full

38:31

norm

38:32

so in this case

38:36

in this case that we also have of

38:39

the the sub inequality for the problem

38:43

inequality for role p

38:45

between one and infinity and which also

38:48

provides an only

38:58

quantity

39:04

and also in the case of buffer domain

39:09

so here uh it's not

39:13

it's not always true that you can just

39:15

consider the seminar

39:17

but in most cases you need to consider

39:19

the full norm

39:25

in some cases the seminar does not bound

39:29

from below so we do not have only

39:32

the inequality even if we

39:36

even if we consider p which is the scale

39:38

invariant now

39:42

but if you add the l2 norm we always

39:44

have a non-linear inequality

39:46

which is in general enough or what one

39:50

what one needs in the

39:53

in the in the in the in the upreary

39:55

balance for

39:56

four weeks solution

40:00

uh so uh and so the so maybe uh

40:04

there are domain there are uh domains uh

40:07

so here in a sense one needs some kind

40:09

of extension

40:10

uh property domain which have extension

40:13

property

40:13

in order to use in order to use

40:17

this inequality uh but if omega sleep

40:21

sheets

40:22

it's fine uh but there are open sets in

40:25

which

40:26

one does not have the the inequality

40:30

uh that one does not have the sublime

40:33

inequality

40:34

so which means one has to be very

40:36

careful

40:38

uh when one applies or maybe think that

40:40

this

40:41

is an immediate consequence of the of

40:43

the

40:45

of the one of the inequality that

40:46

happens in rm

40:49

so uh i think there are interesting uh

40:53

books uh but maybe uh the lecture note

40:57

of uh of uh valdenocchi

41:01

and uh palatochi maybe this

41:06

they i think it contains interesting

41:10

interesting results so let me

41:14

[Music]

41:18

uh

41:22

so the hitch so the hitchhiker's guide

41:24

so the heat shield guide the contains

41:27

in the in the section five extension

41:30

of couple of spaces dominance

41:34

in which you can obtain the cyber life

41:37

inequality

41:39

but if omega is leave you are

41:42

completely safe so i i write here n

41:46

minus 2 s plus which means that if if n

41:49

is smaller than 2s then then i can go up

41:52

to infinity so this

41:53

is just

41:56

or this is defined before n

41:59

larger than 2 s but if n is smaller than

42:01

2s i go up to infinity

42:13

okay so now i now would like to

42:16

introduce the problems that are

42:17

involving uh

42:19

involving the fractional laplacian so

42:22

let me just introduce this energy

42:24

uh so uh sometimes one start is the

42:28

problem

42:28

and then look at the energy but i do it

42:30

in the other way around

42:32

i look at the energy and then i

42:36

i try to look at the irregular glass

42:37

equation of the energy

42:39

but this is not the correct way in

42:42

general the correct way

42:44

you have a problem and then you define

42:46

your energy the energy that you want to

42:48

minimize

42:49

it is sometimes it's very it's very easy

42:52

to obtain the energy

42:54

that has to be minimized and give you

42:55

the earlier equation of the problem you

42:57

want

42:58

but sometimes it's not immediate at all

43:02

but for the problem we are considering

43:04

this this is

43:07

it's not that difficult and so

43:10

here uh so one so this

43:15

this this nice uh so this what is under

43:18

consideration in

43:19

in so if you look at in r2 and i look at

43:22

omega here omega here

43:24

so uh what i'm just looking at is the

43:27

energy in

43:28

the double energy

43:31

in the spaces

43:38

and so this is the

43:44

so the double energy in the space and

43:46

somehow if you

43:47

when when you look at this energy you

43:49

can prescribe your directly a problem

43:52

with the non-zero digital problem of

43:54

course if you like you can also

43:56

prescribe a kind of exterior

43:58

normal normal boundary condition

44:16

so the fractional legitimate approach so

44:18

if you wanted to fraction of particular

44:20

problems so here you see

44:22

i look at the calligraphic it is zero

44:28

if i look at the calligraphic edge is

44:30

zero uh so this is exactly uh

44:34

the semi norm in hs rn

44:51

of course square one one over four

44:55

and

44:58

and so uh because uh if you remember

45:01

very well the calligraphic hs0 which is

45:03

different from hs0 is something that is

45:06

defined on the whole space

45:07

it's a function that are defined on the

45:09

whole space rn but even if they are zero

45:11

at the bottom

45:17

in the case now if i leave

45:22

the problem if i leave the boundary free

45:27

which means i do not prescribe anything

45:30

in the exterior

45:33

then in that case i minimize among

45:37

among functions which are in hsrn

45:42

then both problems will become are

45:45

different but in a sense we we can see

45:48

this as the classical

45:49

directory problem and the other one as a

45:52

classical name and problem

45:55

and in fact these are the case because

45:57

if you look at the energy and if you

45:58

send

45:59

s to one recover the classical directly

46:03

problem which you introduced

46:04

in the first slides and if and also the

46:07

norman problem

46:08

as we introduce it in the first slides

46:17

but let us just look at the

46:23

the earlier so the derivative the first

46:26

derivative of the energy

46:28

that we are considering so here if you

46:31

look at this energy

46:32

then we simply see two terms that comes

46:36

here one is

46:40

the integral in rn so which is so the

46:43

integral in

46:44

rn of ux minus uy

46:47

which is the the fractional laplacian

46:52

and another

47:01

which comes the fractional application

47:03

and another term so this is somehow

47:06

a kind of one can see say it's a kind of

47:08

counterpart of the

47:10

of the of the of the ghost green formula

47:14

but this just comes by using the

47:16

symmetry

47:17

of the kernel k x y is one over mod

47:21

x minus y to the power plus 2f

47:27

and somehow this leaves

47:31

this says that at least if i look at the

47:35

minimizers for problem 1 over 12

47:39

in this case i minimize and phi

47:42

is infinity c function in omega

47:46

which which means this is 0

47:51

and therefore this term will disappear

47:53

however

47:54

if i let phi to vary in rn

47:59

then i will obtain a kind of nominal

48:01

boundary condition

48:02

which was introduced by

48:06

by dipiero rostoton serra ross autumn

48:10

sorry

48:10

and involved notch

48:22

and so if i the consequence of this

48:29

is precisely uh that

48:32

we have the person problem for

48:36

let me do we have a personal problem

48:39

which is the brings so the the dirichlet

48:42

energy so if i minimize

48:44

so my solution will become

48:51

become

48:54

uh the director problem so with exterior

48:57

normal boundary condition

49:00

and also a kind of nerman boundary

49:03

condition

49:05

in the x area

49:08

so these are the problems that the

49:10

energy that i am considering in the

49:13

in the in so which is a non-local energy

49:17

gives to me so a directly problem

49:20

exterior directly problem and a kind of

49:23

exterior normal problem

49:26

and both problems so if s is converging

49:29

to one

49:37

then we recover the

49:42

classical

49:45

problem

49:49

local problems

49:52

which means the

49:56

direct problem for the

49:59

for the laplacian and the norman problem

50:02

for the laplacian

50:04

so uh so this was introduced

50:07

by uh

50:15

cross buttons

50:20

and welding notes

50:25

kind of exterior normal boundary

50:27

condition

50:28

and uh

50:32

and it was since then so it attracted

50:34

many people

50:35

and in fact there are interesting

50:37

probabilistic interpretations

50:40

it was recently

50:43

i think it was one dry czech who also

50:46

construct processes

50:48

related to this type of normal voluntary

50:50

condition in which case

50:53

which in which case somehow if particles

50:56

are allowed to jump in the exterior of

50:58

omega

50:59

but whenever they jump so they they they

51:01

are reflected again

51:02

inside omega

51:06

uh in the dirichlet case this is very

51:09

well known

51:10

that in this case so the process are

51:13

killed

51:13

when they leave omega so in this case

51:18

you can think about the particles will

51:20

come are allowed to jump in the exterior

51:22

of omega but whenever they are there so

51:24

they are killed

51:27

let me just say quickly that in the

51:30

the problems that we want to consider in

51:33

this talk which are the regional case

51:36

there particles will not be allowed to

51:39

jump into exterior

51:41

they can only jump inside omega but they

51:44

will never jump

51:45

in the exterior and this is the they

51:48

they so

51:49

somehow they they are sensory to jump

51:51

outside and somehow they jump in

51:53

original

51:53

part direction and the consideration

51:56

because the mission

51:57

is killed the measure is killed

52:02

and is stopped omega

52:20

so uh let me quickly discuss

52:23

a few interesting properties

52:27

for the fractional laplacian

52:32

is that if you look at the fractional

52:34

appliance and so it's an it's an it's a

52:37

pseudo differential operator

52:41

with a beautiful symbol

52:44

not differentiable at zero but still

52:48

nice to use and of course uh maybe you

52:51

you you see maybe simply from here that

52:54

if s is converging to one then you get

52:56

the symbol of the laplace operator

53:05

and we have interesting mapping

53:06

properties that

53:08

these are very important in general so

53:10

for instance

53:13

for instance i mentioned in the

53:14

beginning that

53:16

sometimes one can use the um

53:20

for example in the book of childbirth

53:21

truth that one can use pradel mateuri to

53:24

obtain also shout out theory

53:26

so this is these are interesting

53:28

properties that in fact

53:30

allows you to do that that your operator

53:32

has some interesting mapping properties

53:39

so for the fight for so just to mention

53:42

that for the original fractional

53:43

laplacian this is not true

53:45

this is not true

53:48

so for the distributional domain of the

53:51

operator which is very important

53:53

is that in fact so you so it's defined

53:56

by the space l1 s

53:58

which is the space of functions for

54:00

which you have integrability

54:02

uh far away uh at infinity

54:06

so and this is uh this allows

54:10

in a sense to say that in fact

54:14

the fractional laplacian acts on

54:17

temporary distribution

54:22

and this comes just from these estimates

54:28

in a sense if we simply recall

54:36

so if you recall the

54:41

oh sometimes the why this is not writing

54:44

properly

54:45

okay so let me just stop okay so uh if

54:48

you recall

54:49

uh the if you recall the the the

54:52

fractional laplace and then we see that

54:54

if x

54:55

is compactly supported then then far

54:57

away

54:58

uh we always have this term so in a

55:01

sense this is not compactly supported

55:03

but anytime you have u in l 1 s then

55:07

the fractional laplacian applied to u

55:10

is a distribution

55:26

so the fractional reflection applied to

55:28

you will become a distribution

55:31

uh just because of this estimate

55:34

well some in some way

55:38

several people they try to enlarge

55:41

this distributional domain

55:46

the distributional domain is in a sense

55:48

maybe i would say the

55:49

the the largest domain in which may when

55:52

you you can apply the fractional

55:54

application on those functions

55:56

and there somehow they define a kind of

55:59

equivalent class uh

56:00

so it's a paper of dpiro uh valdinochi

56:04

and and savine by considering functions

56:08

with polynomial growth

56:10

so when you look at functions with

56:12

polynomial growth of course you know

56:13

that

56:14

if the polynomial the growth of the

56:16

polynomial is larger than

56:17

2s then you cannot define the fractional

56:19

laplacian acting on those functions

56:22

but still they find a way to define it

56:25

and in that case so you have an

56:27

equivalent class of solution so in a

56:29

sense

56:30

when you act a function when you act to

56:33

you

56:34

on functions you have this type of

56:35

growth that when you make the action of

56:37

the fractional application of those one

56:39

you have sometimes of this kind of

56:40

growth you have many

56:43

you can have many solutions in fact for

56:45

for the problems

56:46

if you wanted to find solutions into

56:48

this polynomial

56:50

but uh so but it's an interesting fact

56:53

that was used

56:54

recently by ross orton and

56:59

in order to obtain the the optimal and

57:03

full boundary regularity for the

57:05

directly problem in the fractional

57:07

application so this is a paper that

57:08

appeared

57:10

presently in advances of mathematics in

57:14

2020

57:18

so another important factor is that in

57:21

fact

57:22

when you have a convul when you have a

57:23

molecular a nice modifier

57:26

then you you can commute

57:29

you can commute the fractional flash and

57:33

also with

57:34

this modifier

57:38

so recall that the fraction and so you

57:41

uh this so this

57:42

if functions that are here are temporary

57:44

distributions

57:45

and therefore they have free transform

57:48

in other words so this can be proved

57:50

using fourier transform

57:52

or simply using the symmetry of the

57:55

integral

57:56

uh so it's a it's a very nice thing to

57:59

have

57:59

in a sense maybe that you see if you if

58:02

you have

58:02

functions which are somehow harmonic or

58:05

things like that

58:06

so you do not need to consider

58:10

distributional solutions but you can

58:12

simply consider c infinity solutions

58:14

you prove what you want to prove and

58:16

then you pass to the limit

58:19

you also have the cattle inequality

58:23

so you know the cattle inequality is

58:27

is is sometimes useful also if you want

58:30

to obtain so decay or maybe bounce of

58:32

solutions

58:34

uh and maybe one can prove

58:38

the more general results maybe that

58:42

let me try to write

59:06

so phi is leap sheets and convex

59:18

so why is lipstick and convex uh then

59:21

one can also have

59:23

the more general inequality uh which is

59:26

known as the cattle inequality

59:30

rather than only absolute value and

59:33

uh why this is interesting

59:36

uh of course this is interesting because

59:39

somehow you can think about

59:41

uh considering phi of t

59:44

equal t to the power alpha

59:47

and therefore with alpha larger than one

59:57

and in this case one can immediately

60:01

think that with a non-linear inequality

60:05

then i can run the dejorgi

60:08

the dejorji mother iteration

60:12

in order to obtain in order to obtain

60:16

the apiary all their estimate

60:21

and hardnack inequalities

60:27

yeah so this is these are these are the

60:29

in general the main ingredients that one

60:31

makes

60:34

a non-linear survey of inequalities and

60:37

this kind of

60:38

in this kind of and this kind of

60:42

cattle inequality

60:45

uh so what all these result that i am

60:48

saying also

60:49

uh you know i just consider the

60:51

fractional laplacian but it works for

60:53

many many many other problems

60:55

like like replacing the kernel of the

60:58

fractional laplacian but with the

61:00

translation in one and kerma

61:06

only about but anytime you consider a

61:09

translation invariant kernel

61:10

then in general you have all these

61:13

all these properties and therefore one

61:16

can just

61:17

consider general non-local operators and

61:20

this

61:20

and one can simply apply uh

61:24

use the kind of the geology iteration in

61:26

order to get upright bounds

61:27

because uh yeah so you have cutter

61:30

inequality and a non-linear symbolic

61:36

inequality

61:45

so uh so these are these are

61:49

the questions well i was also personally

61:52

interested

61:53

in is is that maybe to find out

61:57

explicit solutions that comes with the

61:59

fractional laplacian

62:01

and these are sometimes sometimes they

62:04

are very rare

62:06

very difficult to get

62:10

and so what i was first

62:13

was looking uh some years ago was that

62:16

are there

62:16

uh functions which are

62:22

uh so which are harmonic in the whole

62:24

space

62:25

with respect to the fractional

62:27

application so these are in the

62:29

distributional sense

62:30

but as i said before you may even

62:32

consider use infinity because the

62:34

convolution works perfectly

62:39

because the fact that the operator

62:40

commits with convolution

62:46

and and in a sense when i was so trying

62:48

to to to understand this then i find out

62:50

that in fact

62:51

as long as any time that i stay in the

62:54

distributional domain of the fractional

62:56

laplacian then

62:57

there is no there is no uh no

63:00

no harmonic function which is not

63:02

undefined function

63:04

uh so and afterwards so there are

63:09

so i used a proof so by proving a kind

63:12

of non-local

63:14

kind of non-local cauchy estimates

63:17

uh so from which i use them to prove

63:20

this but

63:21

a nice proof or maybe a simple proof

63:23

also comes

63:25

can be used also which is uh which

63:28

uses precisely the fourier transform

63:31

uh the fourier transform because u is a

63:34

temporary distribution because it is in

63:36

l1s and therefore you can simply

63:40

the courier transform and invert the

63:42

operator

63:44

in order to obtain that

63:47

so so the idea could be somehow that

63:52

if i can make the fourier transform then

63:55

i would say

63:56

so the

64:10

wow

64:21

so if i look at the fourier transform

64:23

then i will see that one

64:24

theta to the power 2 s you had is 0 but

64:28

mod 0 to the power 2 s is different from

64:30

0 everywhere

64:31

except except at 0

64:35

so which means

64:39

so the support of you heart

64:44

is in zero so this is formal proof

64:50

and from which this implies that in fact

64:54

u is a polynomial

65:17

so this is a simple proof that can be

65:20

used by fluid transform but of course to

65:22

make the fourier transform

65:23

in this direction so one has to do a

65:25

little bit as

65:26

little works but this is exactly the

65:29

proof

65:30

and you know uh why so this is not

65:32

correct for the fractional lab for the

65:34

photo for the laplacian of course for

65:35

the laplacian the distributional domain

65:37

is l1 lock

65:40

an l1 lock is not a temporary

65:42

distribution therefore

65:44

you cannot you cannot use this argument

65:50

so in a thought so the argument that is

65:53

used here is just because the fractional

65:55

laplacian

65:55

acts on temporary distributions

66:10

so another rigidity result that uh

66:14

that were proved by

66:18

that we proved to uh

66:22

by uh blue mental and great tour and

66:26

race of the icon they completed the the

66:29

first

66:30

the mean uh the the expectation of the

66:34

first exit time of the ball

66:36

or the for the for the stable process

66:40

generating that generates the fractional

66:42

application

66:43

and in this case they obtained that in

66:46

fact

66:47

the expectation of the first exit time

66:50

is given precisely

66:52

by this function u of x equal 1 minus

66:56

mod x squared to the power s

67:00

and which this function solves exactly

67:04

uh that this is constant

67:06

one uh in b1 the constant

67:10

c is chosen is such the way that you

67:12

have one here and you

67:14

zero in the exterior and this function

67:17

also provides

67:18

also the boundary behavior of all

67:19

solutions because

67:21

it's like the distance function to the

67:23

ball

67:24

one minus mod x

67:28

so 1

67:38

to the power s so which is the distance

67:39

function to the boundary and this is the

67:42

boundary behavior

67:44

for solutions which means one cannot

67:46

expect

67:47

in general better than this for solution

68:01

so another rigidity result that was

68:04

also approved by uh when heron cheng

68:08

and also is that in fact on the half

68:12

space

68:14

one can also classify all possible

68:17

solutions

68:19

is in the sense that if you is harmonic

68:22

is harmonic in the heart space and zero

68:25

at the boundary then

68:27

then it is proportional to it is

68:29

proportional to

68:30

x n to the power s which

68:36

is

68:38

well so you know for example if one is

68:42

looking at

68:43

the case of the laplacian this is

68:45

somehow

68:46

very it is simply uh maybe one can make

68:50

simply an odd reflection

68:54

and maybe use use

68:57

the the rigidity result on the whole

69:00

space

69:00

with spherical harmonics when you have a

69:03

growth which is not

69:04

big enough because the this growth makes

69:07

a lot of restriction

69:10

so l1f provides growth in this growth

69:13

restricts

69:14

that when you when you make use the

69:15

spherical harmonics then you have

69:17

polynomials and all these polynomials

69:19

will be killed except those that grows

69:23

below 100 l1s which means functions that

69:26

grows below 2s

69:29

and this is the reason why you have all

69:31

these restrictions

69:32

and this is the reason also why

69:35

the construct you cannot have many many

69:38

functions

69:38

however these restrictions are also

69:42

kind of very bad because

69:45

if you want to prove holder regularity

69:48

maybe or maybe the shout out theory you

69:50

need

69:51

you need ck plus alpha solution somehow

69:54

you need

69:55

you need solutions

69:58

which are infinity somehow but if you

70:01

want solutions which are c infinity but

70:03

that means those solutions will have a

70:07

type of expansion

70:09

of of polynomials of any order

70:12

and these functions are not in l 1 s so

70:15

this is really the reason why

70:17

it's on general difficult uh when you

70:20

look at con when you consider

70:22

the fractional laplacian operator

70:23

because anytime sometimes when you blow

70:25

up

70:26

you go out of the definition the domain

70:29

of definition of this

70:30

of the laplacian of the fractional

70:32

application which is dangerous

70:35

and one way to overcome this is

70:39

to differentiate all the time the

70:41

solutions or

70:42

maybe uh to use this result

70:45

of dpro uh serving and valid energy

70:49

on on functions with with polynomial

70:56

growth

71:10

so

71:12

in the case now of the neumann boundary

71:14

condition

71:15

so this is also in a classical

71:19

thing that comes also for the neumann

71:22

problem

71:23

uh for the number one problem in general

71:25

so only

71:26

i find functions which are tangent are

71:29

found functions will depend on the

71:31

original

71:31

on the horizontal directions solve would

71:34

solve the equations

71:39

and here we have the same

71:45

for the for the normal boundary

71:46

condition

71:50

exterior normal boundary condition and

71:52

we have all the horizontal

71:54

horizontal front functions here you can

71:58

see

71:58

x prime which also solves this equation

72:02

and one can check

72:04

that this is true and so a natural open

72:07

question here is that

72:09

are these the only uh the only function

72:13

is it true that these are the only

72:15

functions

72:16

that solve this problem so these

72:20

open questions is the converse

72:24

is an open it's an interesting open

72:26

question

72:27

because uh whenever once you know that

72:30

then you can carry on a nice argument

72:34

and somehow close up open problems that

72:37

i will tell

72:39

that i will explain later which

72:42

means a kind of full regularity boundary

72:45

regularity for exterior normal problem

72:48

so converse so this u is a solution so

72:51

the converse is

72:54

is open

73:04

crazy great

73:11

so the converse

73:16

i don't know what is going on

73:21

so the convert in any case is open and

73:23

uh so what is uh

73:25

what i mean here by the converse is that

73:27

suppose

73:28

that you have a solution we solve

73:32

18 then is it true that u

73:35

is of the form a x prime plus b

73:45

and there is no result about this even

73:47

in one dimension nothing so in one

73:49

dimension one could

73:50

expect one should expect constant

73:52

functions but nothing is known

73:54

about this

74:09

so further properties maybe that somehow

74:11

allows a

74:14

and which we will use we will see that

74:16

how they are important also we will see

74:18

also how they are important for the

74:20

regional fractional application

74:22

is all of these results so because many

74:25

of the things that comes for the

74:26

fractional laplacian it comes from the

74:27

region of the canal

74:29

but we have more hours to discuss this

74:32

in the coming lectures

74:34

so the wing maximum principle is

74:38

rather

74:41

easy to prove uh

74:45

that if you have uh just a function u

74:49

so which is an hs0 hs0 also can be

74:52

decreased

74:53

by uh by by

74:56

by more by can be weakened

75:00

so here so can be weakened

75:04

by considering something more

75:08

maybe can be weakened by considering

75:10

something u

75:11

is classical solution but maybe what is

75:14

simpler maybe you is the continuous

75:18

a or lower semi-continuous in omega

75:21

and in in l1

75:35

sorry for this i don't know what is

75:37

happening for the pen but anyway

75:39

so if if u is in l1 s and it is

75:41

continuous

75:42

then you also have the

75:46

you also have the solution that

75:48

continues up to the boundary of course

75:50

or lower semi continue when you also

75:53

have the weak maximum principle

76:03

and the strong maximum principle or

76:05

maybe the hope lemma

76:06

is it comes also is in the in the sense

76:09

that if

76:10

if i have if i have a solution also

76:12

which is continuous i can also weaken h

76:14

as zero and either u is equal to zero or

76:18

a

76:18

or i'm i'm larger than the distance

76:21

function to the boundary to the power s

76:24

which is the optimal regularity that we

76:27

that we mentioned before

76:35

and this is somehow hopefully about but

76:36

not here that uh

76:39

we have the hope dilemma but in the

76:41

classical theory the hopeful lemma in

76:43

fact it

76:43

tells us how that whether the function

76:45

is increasing or decreasing from the

76:47

boundary

76:48

but but you know this half lemma does

76:51

not tell that

76:52

and of course in order to reach the

76:54

gradient or maybe to see

76:57

somehow the monotonicity of you one need

77:00

further estimate

77:06

which is sometimes important to have uh

77:09

and and this is and this is an

77:11

interesting open problem in my opinion

77:13

that

77:13

even if i have you which is c1

77:17

sorry which is which is somehow

77:20

continuous up to the boundary

77:22

and c1 lock locally

77:25

c1 then if do i know the whether the

77:28

solution

77:29

is somehow monotone uh

77:32

from the boundary or not so this is uh

77:35

this is something that is i think

77:37

very important so in some cases

77:40

it's true and this is uh

77:43

this is proved uh recently uh

77:46

in the paper but for for for i would say

77:49

again functions

77:50

and this is true this is proved in the

77:52

recent paper

77:54

uh with uh with steven yaro and myself

77:58

where the wicked gradient is well

78:01

defined

78:02

and have the same sign like u of a

78:04

distance function

78:06

and in this case we have even if the

78:08

gradient blows up

78:09

at the boundary but you still have the

78:11

sign so it converts to plus infinity

78:14

but you still have a sign and this tells

78:15

about the monotonicity of the solutions

78:31

so uh i'll just stop here after this

78:34

slide

78:35

and then we start maybe uh the next

78:40

selector with the with the with the

78:42

original fracture laplacian

78:44

and perhaps will become more technical

78:46

in the proofs so these are

78:48

so these results are

78:51

known of course since many years ago

78:55

but it's very is there are modern proofs

78:58

also

78:58

in in many papers of sylvester and

79:01

in papers of russia and sarah

79:05

so which which provides interior holder

79:08

regularity

79:09

and also this is also known

79:14

since many years that in fact we have

79:17

strong solutions in the interior

79:23

but the boundary is in general very

79:25

strange

79:28

and so what is known in the

79:31

in the in the with the optimal regime so

79:34

this is recently

79:35

approved by albert angelo and uh

79:39

and and enclosed autumn is in fact that

79:42

in fact you can have if if omega so the

79:46

the optimality is respect to omega of

79:47

course because if omega is infinity

79:50

this was known from grub

79:54

and somehow if f is c alpha then you

79:58

have

79:59

the function over the different function

80:01

you see s plus

80:02

alpha and so sometimes

80:05

you can think about this that does that

80:07

this is you multiply delta

80:09

s with the cs plus alpha and you can

80:11

think about you almost

80:13

have like two s plus alpha

80:16

so uh this is so if this was open in

80:19

fact

80:20

the optimal regularity that is needed

80:23

for the boundary in order to get this

80:24

and this was resolved recently

80:26

uh by uh by by russell and sarah and

80:30

aberdeen

80:40

so this is a

80:43

result somehow of grub

80:46

uh sorry so this is sp so

80:51

this is uh

80:54

let me consider this this is not part so

80:56

this is hsp

80:58

i'm sorry here

81:03

so this is the lp theory for the

81:06

fractional application

81:07

and we are very different from w to sp

81:11

in fact there is you cannot have

81:14

uh the solution cannot be in wtft there

81:17

are contact examples

81:19

in fact the the explicit solutions for

81:21

the ball

81:22

show that you can never be you can never

81:24

have w to sp and all these results are

81:29

optimal

81:29

but except that in this case uh so the

81:33

the the regulator assumption on the

81:35

boundary

81:36

uh the optimality is still open so this

81:38

was proved by

81:40

grub uh first for omega is infinity and

81:43

group

81:43

and helmut abell for in the in the case

81:47

where

81:47

uh where you know the omega is

81:51

his holder uh is he see one better but

81:54

for some better which is not optimal

82:00

so the question that may be that we'll

82:04

ask

82:04

is what also happened in the norman case

82:06

in the norman case

82:08

what is known this is uh very recent

82:13

and so for now the only known regularity

82:16

in the norman case is that in fact the

82:18

solution is 2s minus 1 plus

82:20

alpha for some alpha positive

82:25

all the rest is open

82:28

and this is a result uh that that

82:32

is that was proved sorry

82:35

that was proved by

82:41

and several in different orders so the

82:44

the only result that you know is 2 s

82:45

minus 1 plus alpha

82:48

and of course one should expect more

82:50

than this

82:51

in my opinion because but this will come

82:55

when one can prove the rigidity result

83:00

namely the classification of the

83:03

solutions in the heart space

83:05

then i think that would be a very good

83:06

step in order to see

83:09

that one should in fact obtain

83:13

more than 2 s minus 1 plus alpha this is

83:16

really not optimal

83:22

and so the the natural questions that we

83:25

should

83:25

that we are going to ask and that we

83:28

will discuss

83:30

are uh whether

83:33

these results are true

83:38

for the original fractional laplacians

83:40

or

83:43

as we will see you will see that in fact

83:45

there are

83:46

almost no results very very few maybe

83:49

two papers

83:50

dealing with the boundary regulator for

83:52

original fractional laplacian

83:54

and these are the papers these are the

83:56

results that i'm going to talk about

83:59

in the in the coming in the coming

84:01

lectures

84:02

uh and perhaps we will be a bit more

84:05

technical in the

84:06

and then and and talk about a little bit

84:08

the proofs

84:10

not deeply of course but we'll just

84:12

sketch some proofs

84:14

but in any case what we want is will be

84:17

that to discuss about what will happen

84:20

in the case

84:21

where we replace the fractional

84:23

laplacian

84:24

with the original fractional laplacian

84:27

so

84:27

uh i will probably stop here for today

84:32

and maybe leave the remaining

84:35

time for discussions

84:41

okay thank you professor faux for the

84:44

very interesting

84:45

lecture uh

84:48

some questions or comments

84:56

may i ask one one question again hey

84:59

there is this this no my question is

85:01

about w2

85:03

calculus regularity i mean lovely just

85:06

just

85:06

w2 something so my question is there is

85:09

this paper by

85:10

by who you from from columbia

85:13

and i think the paper the the the the

85:16

results some sort of

85:17

fun falling estimate for

85:20

for non-local problems so my question is

85:24

is definitely is there any clear

85:27

uh bottleneck or clear obstacle

85:31

to move from this w to

85:34

delta to something like w two

85:38

b or w to d or something like this

85:41

as as in the usual in the usual uh w to

85:45

p

85:45

calculate theory

85:49

yes in fact there is a kind of w

85:52

i would say tp

85:55

regularity of course uh

85:58

up so tp omega itself uh

86:02

but with t with t so so one one

86:05

one is now what knows is for example if

86:08

f is in

86:08

l2 uh then you are in somehow h

86:14

or maybe w is the same because

86:18

it's

86:21

s plus one half

86:25

minus epsilon two

86:32

so one s plus one half minus epsilon

86:36

omega so in a sense and this is the

86:40

one-half minus epsilon is kind of

86:43

optimal

86:44

because uh because of the yeah because

86:47

of the

86:48

explicit solution so that you can get

86:50

when you consider a ball

86:54

okay let's see did i answer question

86:58

yes yes okay

87:01

thank you some more questions or

87:04

comments

87:06

uh i have one is maybe it's naive but

87:10

uh the the solution

87:15

you talk about uh weak solution

87:19

but all the results uh hold

87:22

or viscose solution all the the results

87:25

that you talked

87:27

yes of course many of them are true for

87:30

viscosity solutions for instance

87:32

i think even ross orton sarah for them

87:35

they use

87:36

this projective solutions graph doesn't

87:39

use viscosity solutions

87:41

so it depends it depends on on the

87:44

others

87:45

but in fact i think one interesting

87:47

question which is some

87:48

studied i think by uh by bukur

87:52

and quarter if to go uh sometimes

87:56

when you have this positive solution is

87:58

it resolution and things

87:59

like that so moving from this positive

88:01

or weak solutions

88:03

but for example but the proof

88:08

the results in fact in general works for

88:10

both of them

88:11

but it depends for some from other to

88:13

author someday just completely consider

88:15

viscosity solution

88:17

okay okay thank you a summary comment or

88:21

question

88:25

okay then then uh we met again

88:29

a wednesday wednesday uh at night

88:32

thank you professor paul see you later

88:35

at nine twelve in our case about nine

88:38

plus three

88:40

yeah yeah

88:43

my