Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Dirich

let

pro

ble

ms

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singular

convectio

nte

rms

and

applica

tions

LU

CIO

BO

CCARD

O

Dip

artimen

todiM

atematica

-U

niversita

diRom

a1

boccard

o@m

at.unirom

a1.it

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Dirich

let

pro

ble

ms

with

singular

convectio

nte

rms

and

applica

tions

LU

CIO

BO

CCARD

O

Dip

artimen

todiM

atematica

-U

niversita

diRom

a1

boccard

o@m

at.unirom

a1.it

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

SH

ALO

M

boker

tov

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

SH

ALO

M

Haifa

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Haifa

21.4

.2009

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Tech

nio

n

Nonlin

ear

PD

Eand

Boundary

Valu

ePro

ble

mw

ithM

easu

reD

ata

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Tech

nio

n

Nonlin

ear

PD

Eand

Boundary

Valu

ePro

ble

mw

ithM

easu

reD

ata

25/02/10 1

0:4

6V

ero

n's

Fest

Pagin

a 1

di 2

file:/

//U

sers

/lu

cio

boccard

o/D

eskto

p/H

AIF

A-slid

es/-V

ero

n's

%20Fest.h

tml

Poster

Invited

Speak

ers

Particip

ants

Subjects

Pro

gram

Abstracts

Reg

istration

Trav

el

Acco

mm

odatio

n

More

info

rmatio

n

Photo

s

Ce

nte

r fo

r M

ath

em

atic

al S

cie

nc

es

RE

SE

AR

CH

WO

RK

SH

OP

OF

TH

E IS

RA

EL

SC

IEN

CE

FO

UN

DA

TIO

N O

N

No

nlin

ear P

DE

an

d B

ou

nd

ary

Va

lue P

rob

lems

with

Mea

sure

Da

ta

Tech

nio

n - Israel In

stitute o

f Tech

nolo

gy

Haifa, Israel

1-5

Ma

rch 2

01

0

Th

e aim

of th

e w

ork

sho

p is to

brin

g to

geth

er researchers w

ork

ing

on

partial d

ifferential eq

uatio

ns, an

d to

exch

ang

e n

ew resu

lts, and

dev

elop

men

ts in n

on

linear P

DE

s and

bo

un

dary

valu

e pro

blem

s with

measu

red

ata.F

or th

e m

ain su

bjects o

f the w

ork

sho

p click

here

.

The c

onfe

rence is

held

on th

e o

ccasio

n o

f

the 6

0th

birth

day

of

Marie-F

ran

çoise B

idau

t-Véro

nan

d L

au

rent V

éron

Partial fu

nd

ing

for th

e wo

rksh

op

was o

btain

ed fro

m

Cen

ter for M

ath

ematica

l Scien

ces at th

e T

echnio

n

The Isra

el Scien

ce F

oundatio

n

And

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

BO

NA

NN

IVERSA

IRE,M

ARIE

-FRA

NCE

et

LAU

REN

T!

Nonlin

ear

PD

Eand

Boundary

Valu

ePro

ble

mw

ithM

easu

reD

ata

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

BO

NA

NN

IVERSA

IRE,M

ARIE

-FRA

NCE

et

LAU

REN

T!

Nonlin

ear

PD

Eand

Boundary

Valu

ePro

ble

mw

ithM

easu

reD

ata

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

BO

NA

NN

IVERSA

IRE,M

ARIE

-FRA

NCE

et

LAU

REN

T!

Nonlin

ear

PD

Eand

Boundary

Valu

ePro

ble

mw

ithM

easu

reD

ata

Unfortu

nately,

Ido

not

have

joint

pap

ersw

ithyou

!

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Unfo

rtunate

ly,Ido

not

have

join

tpapers

with

you,but

...

Thanks

toth

eorg

anize

rsRESEARCH WORKSHOP OFTHE ISRAEL SCIENCE FOUNDATION ON

Technion - Israel Institute of TechnologyHaifa, Israel

1-5 March 2010

Conference on the occasion ofMarie-Françoise Bidaut-Véronand Laurent Véron's60 th birthday

Partial funding for the workshop was obtained fromCenter for Mathematical Sciences at the TechnionThe Israel Science FoundationLaboratoire de Mathématiques et Physique Théorique de l'Université de Tours

Website: http://www.math.technion.ac.il/~pincho/Veron/index.html

Nonlinear PDE andBoundary Value Problems

with Measure DataNonlinear PDE and

Boundary Value Problems with Measure Data

Organizing Committee: Emmanuel Lesigne, Moshe Marcus, Yehuda PinchoverScientific Committee:

Guy Barles, Moshe Marcus, Yehuda Pinchover

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Unfo

rtunate

ly,Ido

not

have

join

tpapers

with

you,but

...

Thanks

toth

eorg

anize

rs

Toda

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Haifa

,3.3

.2010

Mosh

e,Laure

nt

(Corto

na)

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Haifa

,3.3

.2010

Mosh

e,Laure

nt

(Corto

na)(?

where

isth

em

ainorgan

izer?)

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Haifa

,3.3

.2010

Main

org

anize

r

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Haifa

,3.3

.2010

Mosh

e,Laure

nt

(Tours)

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Haifa

,3.3

.2010

Lum

iny

2001-Luminyconferen

ceonQuasilin

earEllip

ticandParabolic

Equatio

nsandSystem

s,Electro

nic

JournalofDifferen

tialEquatio

ns,

Conferen

ce08,2002,pp47–52.

http

://ejd

e.math.sw

t.eduorhttp

://ejd

e.math.unt.ed

uftp

ejde.m

ath.sw

t.edu(lo

gin:ftp

)

Arem

arkonsomenonlinearellip

ticproblem

s∗

LucioBoccardo

Abstract

Weshallproveanexisten

ceresultofW

1,p

0(Ω)solutionsforthebound-

aryvalueproblem

−diva(x,u,∇u)=FinΩ

u=0on∂Ω

(0.1)

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Haifa

,3.3

.2010

Do

not

forg

et

our

”get-to

geth

er”

inRom

a

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Haifa

,3.3

.2010

Do

not

forg

et

our

”get-to

geth

er”

inRom

a

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Follo

win

gG

uid

oSta

mpacch

ia

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Follo

win

gG

uid

oSta

mpacch

ia

Nonlin

ear

PD

Eand

Boundary

Valu

ePro

ble

mw

ithM

easu

reD

ata

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

.

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Follo

win

gG

uid

oSta

mpacch

ia

Nonlin

ear

PD

Eand

Boundary

Valu

ePro

ble

mw

ithM

easu

reD

ata

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

.

Ωbou

nded

,op

enset

inR

N,

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Follo

win

gG

uid

oSta

mpacch

ia

Nonlin

ear

PD

Eand

Boundary

Valu

ePro

ble

mw

ithM

easu

reD

ata

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

.

Ωbou

nded

,op

enset

inR

N,

0<

α,

α|ξ| 2

≤M

(x)ξξ,

|M(x

)|≤

β,

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Follo

win

gG

uid

oSta

mpacch

ia

Nonlin

ear

PD

Eand

Boundary

Valu

ePro

ble

mw

ithM

easu

reD

ata

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

.

Ωbou

nded

,op

enset

inR

N,

0<

α,

α|ξ| 2

≤M

(x)ξξ,

|M(x

)|≤

β,

f∈

Lm(Ω

),2N

N+

2≤

m≤∞

,

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Follo

win

gG

uid

oSta

mpacch

ia

Nonlin

ear

PD

Eand

Boundary

Valu

ePro

ble

mw

ithM

easu

reD

ata

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

.

Ωbou

nded

,op

enset

inR

N,

0<

α,

α|ξ| 2

≤M

(x)ξξ,

|M(x

)|≤

β,

f∈

Lm(Ω

),2N

N+

2≤

m≤∞

,

E∈

(LN(Ω

))N

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Follo

win

gG

uid

oSta

mpacch

ia

Nonlin

ear

PD

Eand

Boundary

Valu

ePro

ble

mw

ithM

easu

reD

ata

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

.

Ωbou

nded

,op

enset

inR

N,

0<

α,

α|ξ| 2

≤M

(x)ξξ,

|M(x

)|≤

β,

f∈

Lm(Ω

),2N

N+

2≤

m≤∞

,

E∈

(LN(Ω

))N

!No

coercivity!

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Follo

win

gG

uid

oSta

mpacch

ia

Nonlin

ear

PD

Eand

Boundary

Valu

ePro

ble

mw

ithM

easu

reD

ata

!No

coerciv

ity!

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

.∫Ω

M(x

)∇u∇

u=

∫Ω

uE

(x)∇

u+

∫Ω

f(x

)u

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Follo

win

gG

uid

oSta

mpacch

ia

Nonlin

ear

PD

Eand

Boundary

Valu

ePro

ble

mw

ithM

easu

reD

ata

!No

coerciv

ity!

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

.∫Ω

M(x

)∇u∇

u=

∫Ω

uE

(x)∇

u+

∫Ω

f(x

)u

α

∫Ω

|∇u| 2≤

[∫Ω

|u| 2

∗]

12∗[∫Ω

|E(x

)| N]

1N[∫Ω

|∇u| 2

]12+

∫Ω

f(x

)u

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Follo

win

gG

uid

oSta

mpacch

ia

Nonlin

ear

PD

Eand

Boundary

Valu

ePro

ble

mw

ithM

easu

reD

ata

!No

coerciv

ity!

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

.∫Ω

M(x

)∇u∇

u=

∫Ω

uE

(x)∇

u+

∫Ω

f(x

)u

α

∫Ω

|∇u| 2≤

[∫Ω

|u| 2

∗]

12∗[∫Ω

|E(x

)| N]

1N[∫Ω

|∇u| 2

]12+

∫Ω

f(x

)u

α−

1S

[∫Ω

|E(x

)| N]

1N

∫Ω

|∇u| 2≤

∫Ω

f(x

)u

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Follo

win

gG

uid

oSta

mpacch

ia

Nonlin

ear

PD

Eand

Boundary

Valu

ePro

ble

mw

ithM

easu

reD

ata

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

.

•Ω

bou

nded

,op

enset

inR

N,

•0

<α,

α|ξ| 2

≤M

(x)ξξ,

|M(x

)|≤

β,

•f∈

Lm(Ω

),2N

N+

2≤

m≤∞

,

•E∈

(LN(Ω

))N

α>

1S

[∫Ω

|E(x

)| N]

1N

⇒∃

u

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Follo

win

gG

uid

oSta

mpacch

ia

Nonlin

ear

PD

Eand

Boundary

Valu

ePro

ble

mw

ithM

easu

reD

ata

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

.

•Ω

bou

nded

,op

enset

inR

N,

•0

<α,

α|ξ| 2

≤M

(x)ξξ,

|M(x

)|≤

β,

•f∈

Lm(Ω

),2N

N+

2≤

m≤∞

,

•E∈

(LN(Ω

))N

α>

1S

[∫Ω

|E(x

)| N]

1N

⇒∃

u

G.Stam

pacch

ia:∃

u:

−div

(M(x

)∇u))

+λ

2u=−

div

(uE

(x))

+f(x

)in

Ω.

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Follo

win

gG

uid

oSta

mpacch

ia

Asim

ilar(?

??)

nonlin

ear

pro

ble

m

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Follo

win

gG

uid

oSta

mpacch

ia

Asim

ilar(?

??)

nonlin

ear

pro

ble

m

−

div

(M(x

)∇u))

=−

div

( Φ(u

))+

f(x

):

Ω,

u=

0:

∂Ω

.

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Follo

win

gG

uid

oSta

mpacch

ia

Asim

ilar(?

??)

nonlin

ear

pro

ble

m

−

div

(M(x

)∇u))

=−

div

( Φ(u

))+

f(x

):

Ω,

u=

0:

∂Ω

.

Form

al(u

seu

as

test

functio

n)

α

∫Ω

|∇u| 2≤

∫Ω

[Φ(u

)]∇u

︸︷︷

︸

=0:

div

.th.

+

∫Ω

|f||u|

⇒α

∫Ω

|∇u| 2≤

∫Ω

|f||u|

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Follo

win

gG

uid

oSta

mpacch

ia

div

(E)

=0

div

(E)

=0⇒

−

div

(M(x

)∇u))

=−

div

(uE

)+

f(x

):

Ω,

u=

0:

∂Ω

.

Form

al

α

∫Ω

|∇u| 2≤

∫Ω

|f||u|

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Appro

x.

pro

ble

ms

−

div

(M(x

)∇u))

=−

div

(uE

)+

f(x

):

Ω,

u=

0:

∂Ω

.

un∈

W1,2

0(Ω

):

−div

(M(x

)∇u

n )=−

div

(u

n

1+

1n |un |

E

1+

1n |E|

)

+f(x

)

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Appro

x.

pro

ble

ms

−

div

(M(x

)∇u))

=−

div

(uE

)+

f(x

):

Ω,

u=

0:

∂Ω

.

un∈

W1,2

0(Ω

):

−div

(M(x

)∇u

n )=−

div

(u

n

1+

1n |un |

E

1+

1n |E|

)

+f(x

)

•N

onlin

ear+

Sch

auder→

linear!

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Appro

x.

pro

ble

ms

−div

(M(x

)∇u

n )=−

div

(u

n

1+

1n |un |

E

1+

1n |E|

)

+f

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Appro

x.

pro

ble

ms

−div

(M(x

)∇u

n )=−

div

(u

n

1+

1n |un |

E

1+

1n |E|

)

+f

use

un

1+|u

n | ⇒α

∫Ω

|∇u

n | 2

(1+|u

n |)2≤

∫Ω

|un |

1+|u

n | |En |

|∇u

n |

(1+|u

n |) +

∫Ω

|f|

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Appro

x.

pro

ble

ms

−div

(M(x

)∇u

n )=−

div

(u

n

1+

1n |un |

E

1+

1n |E|

)

+f

use

un

1+|u

n | ⇒α

∫Ω

|∇u

n | 2

(1+|u

n |)2≤

∫Ω

|un |

1+|u

n | |En |

|∇u

n |

(1+|u

n |) +

∫Ω

|f|

α2

∫Ω

|∇u

n | 2

(1+|u

n |)2≤

12α

∫Ω

|En | 2

+

∫Ω

|f|,

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Appro

x.

pro

ble

ms

−div

(M(x

)∇u

n )=−

div

(u

n

1+

1n |un |

E

1+

1n |E|

)

+f

use

un

1+|u

n | ⇒α

∫Ω

|∇u

n | 2

(1+|u

n |)2≤

∫Ω

|un |

1+|u

n | |En |

|∇u

n |

(1+|u

n |) +

∫Ω

|f|

α2

∫Ω

|∇u

n | 2

(1+|u

n |)2≤

12α

∫Ω

|En | 2

+

∫Ω

|f|,

∫Ω

|∇log

(1+

un )| 2

≤1α2

∫Ω

|E| 2

+2α

∫Ω

|f|

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Appro

x.

pro

ble

ms

−div

(M(x

)∇u

n )=−

div

(u

n

1+

1n |un |

E

1+

1n |E|

)

+f

use

un

1+|u

n | ⇒α

∫Ω

|∇u

n | 2

(1+|u

n |)2≤

∫Ω

|un |

1+|u

n | |En |

|∇u

n |

(1+|u

n |) +

∫Ω

|f|

α2

∫Ω

|∇u

n | 2

(1+|u

n |)2≤

12α

∫Ω

|En | 2

+

∫Ω

|f|,

∫Ω

|∇log

(1+

un )| 2

≤1α2

∫Ω

|E| 2

+2α

∫Ω

|f|

[∫Ω

|log(1

+|u

n |)| 2∗]

22∗

≤1

S2α

2

∫Ω

|E| 2

+2

S2α

∫Ω

|f|

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Appro

x.

pro

ble

ms

[∫Ω

|log(1

+|u

n |)| 2∗]

22∗

≤1

S2α

2

∫Ω

|E| 2

+2

S2α

∫Ω

|f|

⇒

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Appro

x.

pro

ble

ms

[∫Ω

|log(1

+|u

n |)| 2∗]

22∗

≤1

S2α

2

∫Ω

|E| 2

+2

S2α

∫Ω

|f|

⇒

measu

rex∈

Ω:|u

n (x)|

>k|≤

C( ‖

E‖

2 ,‖f‖

1 )

[log(1

+k)] 2

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Appro

x.

pro

ble

ms

Tools

-

6

−k

k

Gk(s)

-

6

−k

k

k

−k

Tk(s)

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Appro

x.

pro

ble

ms

(Note

:we

need

(only

!)|E|∈

L2(Ω

))

un∈

W1,2

0(Ω

):−div

(M(x

)∇u

n )=−

div

(u

n

1+

1n |un |

E

1+

1n |E|

)

+f

Use

Tk (u

n ):α

∫Ω

|∇T

k (un )| 2

≤

∫Ω

|un ||E

n ||∇T

k (un )|+

∫Ω

fTk (u

n )

≤k

∫Ω

|E||∇

Tk (u

n )|+k

∫Ω

|f|

≤

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Appro

x.

pro

ble

ms

(Note

:we

need

(only

!)|E|∈

L2(Ω

))

un∈

W1,2

0(Ω

):−div

(M(x

)∇u

n )=−

div

(u

n

1+

1n |un |

E

1+

1n |E|

)

+f

Use

Tk (u

n ):α

∫Ω

|∇T

k (un )| 2

≤

∫Ω

|un ||E

n ||∇T

k (un )|+

∫Ω

fTk (u

n )

≤k

∫Ω

|E||∇

Tk (u

n )|+k

∫Ω

|f|

≤α2

∫Ω

|∇T

k (un )| 2

+2α

k2

∫Ω

|E| 2

+k

∫Ω

|f|⇒

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Appro

x.

pro

ble

ms

(Note

:we

need

(only

!)|E|∈

L2(Ω

))

un∈

W1,2

0(Ω

):−div

(M(x

)∇u

n )=−

div

(u

n

1+

1n |un |

E

1+

1n |E|

)

+f

Use

Tk (u

n ):α

∫Ω

|∇T

k (un )| 2

≤

∫Ω

|un ||E

n ||∇T

k (un )|+

∫Ω

fTk (u

n )

≤k

∫Ω

|E||∇

Tk (u

n )|+k

∫Ω

|f|

≤α2

∫Ω

|∇T

k (un )| 2

+2α

k2

∫Ω

|E| 2

+k

∫Ω

|f|⇒

α2

∫Ω

|∇T

k (un )| 2

≤2α

k2

∫Ω

|E| 2

+k

∫Ω

|f|

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

2startin

gblo

cks

−div

(M(x

)∇u

n )=−

div

(u

n

1+

1n |un |

E

1+

1n |E|

)

+f

measu

rex∈

Ω:|u

n (x)|

>k|≤

C(‖

E‖

2 ,‖f‖

1 )

[log(1

+k)] 2

α2

∫Ω

|∇T

k (un )| 2

≤2α

k2

∫Ω

|E| 2

+k

∫Ω

|f|

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Basic

estim

ate

topro

veth

eW

1,2

0(Ω

)co

ercivity

−div

(M(x

)∇u

n )=−

div

(u

n

1+

1n |un |

E

1+

1n |E|

)

+f

use

Gk (u

n )⇒

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Basic

estim

ate

topro

veth

eW

1,2

0(Ω

)co

ercivity

−div

(M(x

)∇u

n )=−

div

(u

n

1+

1n |un |

E

1+

1n |E|

)

+f

use

Gk (u

n )⇒

α

∫Ω

|∇G

k (un )| 2

≤1S

(

∫

k≤|u

n |

|E| N

)1N

∫Ω

|∇G

k (un )| 2

+ǫ

∫Ω

|∇G

k (un )| 2

+k

2

4ǫ

∫

k≤|u

n |

|E| 2+

ǫ

∫Ω

|∇G

k (un )| 2

+S

2

4ǫ

[∫

k≤|u

n |

|f|

2N

N+

2

]N

+2

N

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Existe

nce

inW

1,2

0(Ω

)

[B-2

008]”dedica

toalm

ioM

aestro

”

Tu

seilo

mio

maestro

eil

mio

auto

re;

tuse

iso

loco

luida

cuiio

tolsi

lobello

stilo...

(Dante

:In

fern

oI)

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Existe

nce

inW

1,2

0(Ω

)

-**$

22(,-

% #"!

#!"

&#

"

#

$$"

#"

31$(*-

+(-

+ $120

-$(*+(-

32-0$

231$(1-*-

"-*3(# "3((-

2-*1(

*-!$**-

12(*-

#!

"#!

## "

$$

$

")

%$

$$"$$

&"#

")#$#$

"$#"

#$%

"#"

$ "

#"$)

)"#

#"

"&

#"#$

#) )$##

#

$'#'"$$

$#$)

$$#

)'%

&%#

&

$$$

$#

"

#$")

'#$)%$

%$

$!%$

#'$

#

$

%%#

$#

"##$

##

#

!#

#$"#

'

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Existe

nce

inW

1,2

0(Ω

)

[B-2

008]:|E|∈

LN(Ω

),f∈

Lm(Ω

),m≥

2N

/(N

+2)

−div

(M(x

)∇u

n )=−

div

(u

n

1+

1n |un |

E

1+

1n |E|

)

+f

un

=T

k (un )

+G

k (un )

Then

∫Ω

|∇G

k (un )| 2

≤C

k∗

and

∫Ω

|∇T

k (un )| 2

≤2α

k2

∫Ω

|E| 2

+k

∫Ω

|f|

⇒

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Existe

nce

inW

1,2

0(Ω

)

[B-2

008]:|E|∈

LN(Ω

),f∈

Lm(Ω

),m≥

2N

/(N

+2)

−div

(M(x

)∇u

n )=−

div

(u

n

1+

1n |un |

E

1+

1n |E|

)

+f

un

=T

k (un )

+G

k (un )

Then

∫Ω

|∇G

k (un )| 2

≤C

k∗

and

∫Ω

|∇T

k (un )| 2

≤2α

k2

∫Ω

|E| 2

+k

∫Ω

|f|

⇒u

n bounded

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Existe

nce

inW

1,2

0(Ω

)

[B-2

008]:|E|∈

LN(Ω

),f∈

Lm(Ω

),m≥

2N

/(N

+2)

−div

(M(x

)∇u

n )=−

div

(u

n

1+

1n |un |

E

1+

1n |E|

)

+f

un

=T

k (un )

+G

k (un )

Then

∫Ω

|∇G

k (un )| 2

≤C

k∗

and

∫Ω

|∇T

k (un )| 2

≤2α

k2

∫Ω

|E| 2

+k

∫Ω

|f|

⇒u

n bounded⇒∃

u∈

W1,2

0(Ω

)weak

solu

tion

of

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Weak

solu

tions

(|E|∈

LN

(Ω))

Existe

nce

inW

1,2

0(Ω

)

[B-2

008]:|E|∈

LN(Ω

),f∈

Lm(Ω

),m≥

2N

/(N

+2)

−div

(M(x

)∇u

n )=−

div

(u

n

1+

1n |un |

E

1+

1n |E|

)

+f

un

=T

k (un )

+G

k (un )

Then

∫Ω

|∇G

k (un )| 2

≤C

k∗

and

∫Ω

|∇T

k (un )| 2

≤2α

k2

∫Ω

|E| 2

+k

∫Ω

|f|

⇒u

n bounded⇒∃

u∈

W1,2

0(Ω

)weak

solu

tion

of

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Distrib

utio

nalso

lutio

ns

(|E|∈

LN

(Ω))

[B-2

008]:|E|∈

LN(Ω

);f∈

Lm(Ω

),1≤

m≤

N2

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

∃u

sam

epro

pertie

sof

wso

lutio

nof

−

div

(M(x

)∇w

))=

f(x

):

Ω,

w=

0:

∂Ω

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Distrib

utio

nalso

lutio

ns

(|E|∈

LN

(Ω))

(Nonlin

ear

Cald

ero

n-Z

ygm

und)

−

div

(a(x,u

,∇u))

=f(x

)∈

Lm(Ω

):

Ω,

u=

0:

∂Ω

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Distrib

utio

nalso

lutio

ns

(|E|∈

LN

(Ω))

|E|∈

LN

(Ω)

like

E=

0

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

)∈

Lm(Ω

):

Ω,

u=

0:

∂Ω

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Distrib

utio

nalso

lutio

ns

(|E|∈

LN

(Ω))

|E|∈

LN

(Ω)

like

E=

0

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

)∈

Lm(Ω

):

Ω,

u=

0:

∂Ω

∃u

weak,unbounded

2N

N+

2≤

m<

N2⇒

u∈

W1,2

0(Ω

)∩

Lm

∗∗(Ω

);

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Distrib

utio

nalso

lutio

ns

(|E|∈

LN

(Ω))

|E|∈

LN

(Ω)

like

E=

0

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

)∈

Lm(Ω

):

Ω,

u=

0:

∂Ω

∃u

weak,unbounded

2N

N+

2≤

m<

N2⇒

u∈

W1,2

0(Ω

)∩

Lm

∗∗(Ω

);

∃distrib

utio

nal

1<

m<

2N

N+

2⇒

u∈

W1,m

∗

0(Ω

);

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Distrib

utio

nalso

lutio

ns

(|E|∈

LN

(Ω))

|E|∈

LN

(Ω)

like

E=

0

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

)∈

Lm(Ω

):

Ω,

u=

0:

∂Ω

∃u

weak,unbounded

2N

N+

2≤

m<

N2⇒

u∈

W1,2

0(Ω

)∩

Lm

∗∗(Ω

);

∃distrib

utio

nal

1<

m<

2N

N+

2⇒

u∈

W1,m

∗

0(Ω

);

∃distrib

utio

nal|E|∈

LN(Ω

),m

=1⇒

u∈

W1,q

0(Ω

),q

<N

N−

1 .

u∈

W1,2

0(Ω

)∩

L∞

(Ω),∀φ∈D

:∫Ω

M(x

)∇u∇

φ=

∫Ω

uE∇

φ+

∫Ω

fφ.

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Distrib

utio

nalso

lutio

ns

(|E|∈

LN

(Ω))

|E|∈

LN

(Ω)

like

E=

0

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

∃u∈

W1,2

0(Ω

)bounded

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Distrib

utio

nalso

lutio

ns

(|E|∈

LN

(Ω))

|E|∈

LN

(Ω)

like

E=

0

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

∃u∈

W1,2

0(Ω

)bounded

|E|∈

Lr(Ω

),r

>N

,f∈

Lm(Ω

),m

>N2⇒

u∈

W1,2

0(Ω

)∩

L∞

(Ω),∀v∈

W1,2

0(Ω

):

∫Ω

M(x

)∇u∇

v=

∫Ω

uE∇

v+

∫Ω

fv

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Distrib

utio

nalso

lutio

ns

(|E|∈

LN

(Ω))

Open

pro

ble

m

|E|∈

LN(Ω

),f∈

Mm(Ω

),m≥

1

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

)∈

Mm(Ω

):

Ω,

u=

0:

∂Ω

?u

sam

epro

pertie

sofw

?

1[B-2

008,ded

icatedto

Juan

Luis]

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Distrib

utio

nalso

lutio

ns

(|E|∈

LN

(Ω))

Open

pro

ble

m

|E|∈

LN(Ω

),f∈

Mm(Ω

),m≥

1

−

div

(M(x

)∇u))

=−

div

(uE

(x))

+f(x

)∈

Mm(Ω

):

Ω,

u=

0:

∂Ω

?u

sam

epro

pertie

sofw

?

−

div

(M(x

)∇w

))=

f(x

)∈

Mm(Ω

):

Ω,

w=

0:

∂Ω

1

1[B-2

008,ded

icatedto

Juan

Luis]

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Bord

erlin

e:

0∈

Ω,|E

|≤

A|x|

Ex.

1:|E|=

A|x| ;

f∈

Lm(Ω

),∀

m<

N2.

−∆

u=

+Adiv

(

ux|x| 2

)

+A

N−

2|x| 2

:B

(0,1),

u=

0:

∂B

(0,1)

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Bord

erlin

e:

0∈

Ω,|E

|≤

A|x|

Ex.

1:|E|=

A|x| ;

f∈

Lm(Ω

),∀

m<

N2.

−∆

u=

+Adiv

(

ux|x| 2

)

+A

N−

2|x| 2

:B

(0,1),

u=

0:

∂B

(0,1)

uA(x

)=

1|x| A−

1

weak

solu

tion

ifA

<1

+N

/2

distrib

utio

nalso

lutio

nif

1+

N/2≤

A<

N−

2.

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Bord

erlin

e:

0∈

Ω,|E

|≤

A|x|

Ex.

2uA

=rA−

r2

sol.

of

−∆

u=

+Adiv

(

ux|x| 2

)

+(2−

A)N

:B

(0,1),

u=

0:

∂B

(0,1)

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Bord

erlin

e:

0∈

Ω,|E

|≤

A|x|

Ex.

2uA

=rA−

r2

sol.

of

−∆

u=

+Adiv

(

ux|x| 2

)

+(2−

A)N

:B

(0,1),

u=

0:

∂B

(0,1)

A<

0,u

Aunbounded,

but

f(x

)=

(2−

A)N

bounded!

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Bord

erlin

e:

0∈

Ω,|E

|≤

A|x|

0∈

ΩH

(∫Ω

|v| 2

|x| 2

)12

≤

(∫Ω

|∇v| 2

)12,

∀v∈

W1,2

0(Ω

)2

2H

=H

ardy-S

obolev-B

rezis-Vazq

uez-P

eralcon

stant

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Bord

erlin

e:

0∈

Ω,|E

|≤

A|x|

0∈

Ω,

|E|≤

x|x| 2

−

div

(M(x

)∇u))

=−

Adiv

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

f∈

Lm(Ω

)

2N

N+

2≤

m<

N2,|A|<

2∗αH

m∗∗⇒

u∈

W1,2

0(Ω

)∩

Lm

∗∗(Ω

);

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Bord

erlin

e:

0∈

Ω,|E

|≤

A|x|

0∈

Ω,

|E|≤

x|x| 2

−

div

(M(x

)∇u))

=−

Adiv

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

f∈

Lm(Ω

)

2N

N+

2≤

m<

N2,|A|<

2∗αH

m∗∗⇒

u∈

W1,2

0(Ω

)∩

Lm

∗∗(Ω

);

1<

m<

2N

N+

2 ,|A|<

2∗αH

m∗∗⇒

u∈

W1,m

∗

0(Ω

);

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Bord

erlin

e:

0∈

Ω,|E

|≤

A|x|

0∈

Ω,

|E|≤

x|x| 2

−

div

(M(x

)∇u))

=−

Adiv

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

f∈

Lm(Ω

)

2N

N+

2≤

m<

N2,|A|<

2∗αH

m∗∗⇒

u∈

W1,2

0(Ω

)∩

Lm

∗∗(Ω

);

1<

m<

2N

N+

2 ,|A|<

2∗αH

m∗∗⇒

u∈

W1,m

∗

0(Ω

);

f∈

L1(Ω

),|A|<

2αH⇒

u∈

W1,q

0(Ω

).

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Bord

erlin

e:

0∈

Ω,|E

|≤

A|x|

0∈

Ω,

|E|≤

x|x| 2

−

div

(M(x

)∇u))

=−

Adiv

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

f∈

Lm(Ω

)

2N

N+

2≤

m<

N2,|A|<

2∗αH

m∗∗⇒

u∈

W1,2

0(Ω

)∩

Lm

∗∗(Ω

);

1<

m<

2N

N+

2 ,|A|<

2∗αH

m∗∗⇒

u∈

W1,m

∗

0(Ω

);

f∈

L1(Ω

),|A|<

2αH⇒

u∈

W1,q

0(Ω

).

Note

that

limm→

1

2∗αH

m∗∗

=2αH

,

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Bord

erlin

e:

0∈

Ω,|E

|≤

A|x|

0∈

Ω,

|E|≤

x|x| 2

−

div

(M(x

)∇u))

=−

Adiv

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

f∈

Lm(Ω

)

2N

N+

2≤

m<

N2,|A|<

2∗αH

m∗∗⇒

u∈

W1,2

0(Ω

)∩

Lm

∗∗(Ω

);

1<

m<

2N

N+

2 ,|A|<

2∗αH

m∗∗⇒

u∈

W1,m

∗

0(Ω

);

f∈

L1(Ω

),|A|<

2αH⇒

u∈

W1,q

0(Ω

).

Note

that

limm→

1

2∗αH

m∗∗

=2αH

,lim

m→

N2

2∗αH

m∗∗

=0.

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Entro

py

solu

tions

|E|∈

L2(Ω

):existe

nce

ofentro

py

solu

tions,

B-2

010

∃u

:

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Entro

py

solu

tions

|E|∈

L2(Ω

):existe

nce

ofentro

py

solu

tions,

B-2

010

∃u

:

Tj (u

)∈

W1,2

0(Ω

),∀j>

0,

log(1

+|u|)∈

W1,2

0(Ω

),∫Ω

M(x

)∇u∇

Tk [u−

ϕ]≤

∫Ω

u(x

)E(x

)∇T

k [u−

ϕ]+

∫Ω

fT

k [u−

ϕ]

∀ϕ∈

W1,2

0(Ω

)∩

L∞

(Ω)

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Pb.

with

lower

ord

er

term

s

λ>

0−

div

(M(x

)∇u))

+λu

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

.

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Pb.

with

lower

ord

er

term

s

λ>

0−

div

(M(x

)∇u))

+λu

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

.

λ

∫Ω

|u|≤

∫Ω

|f|.

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Pb.

with

lower

ord

er

term

s

λ>

0−

div

(M(x

)∇u))

+λu

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

.

λ

∫Ω

|u|≤

∫Ω

|f|.

IfN≥

5,in

B(0,1),

u(x

)=|x| −

2−

1is

adistrib

utio

nal,

unbounded

solu

tion

ofth

eboundary

valu

epro

ble

m

−∆

u+

(N−

2)u=−

div

(

u

[

−2

x

|x| 2−

3x

])

+2(N

+1)

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Pb.

with

lower

ord

er

term

s

Dualpro

ble

m

Lin

ear

dualpro

ble

m

λ>

0,E∈

L2(Ω

):

−

div

(M(x

)∇u))

+λu

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

.

∗

−

div

(M(x

)∇w

))+

λw

=E

(x)·∇

w+

g(x

):

Ω,

w=

0:

∂Ω

.

∣∣∣∣

∫Ω

fw

∣∣∣∣=

∣∣∣∣

∫Ω

gu

∣∣∣∣≤‖g‖∞

∫Ω

|u|≤‖g‖∞

1λ

∫Ω

|f|

3.

g∈

L∞

(Ω)⇒

∃w∈

W1,2

0(Ω

)∩

L∞

(Ω).

3dep

ends

on

λ;in

dep

enden

tof

α.

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Pb.

with

lower

ord

er

term

s

Dualpro

ble

m

Nonlin

ear

dualpro

ble

m

−

div

(a(x,∇

w))

+λw

=E

(x)·∇

w+

g(x

):

Ω,

w=

0:

∂Ω

.

pap

erby

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Pb.

with

lower

ord

er

term

s

Dualpro

ble

m

Nonlin

ear

dualpro

ble

m

−

div

(a(x,∇

w))

+λw

=E

(x)·∇

w+

g(x

):

Ω,

w=

0:

∂Ω

.

pap

erby

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Pb.

with

lower

ord

er

term

s

Dualpro

ble

m

Nonlin

ear

dualpro

ble

m

−

div

(a(x,∇

w))

+λw

=E

(x)·∇

w+

g(x

):

Ω,

w=

0:

∂Ω

.

paper

by

Tom

maso

Leonori

and

Fra

nce

scoPetitta

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Pb.

with

lower

ord

er

term

s

Dualpro

ble

m

Ex.

3

−∆

w=

(N−

2)x·∇

w

|x| 2

+0

inB

1 (0)

solution

s

w0

=0,

wL

=log|x|.

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Anonlin

ear

pro

ble

m

|E|∈

L2(Ω

),f∈

L1(Ω

),γ

>1

−

div

([a(x)+|u| γ]∇

u))

=−

div

(uE

(x))

+f(x

):

Ω,

u=

0:

∂Ω

.

Use

T1 [u−

Tk (u

)]

kγ

∫

k≤|u|<

k+

1

|∇u| 2≤

∫Ω

|f|

⇒

∫Ω

|∇u| 2≤

Cα [ ∫Ω

|E| 2+

∫Ω

|f|]+

∫Ω

|f|∞

∑k=

1

k−

γ⇒∃

u∈

W1,2

0(Ω

).

(see

also

an

old

paper

by

Ale

ssio)

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

G-co

nverg

ence

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

G-co

nverg

ence

join

tpaper

dedica

ted

to

Fra

nco

isfo

rhis

60th

birth

day

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Para

bolic

pro

ble

m

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Para

bolic

pro

ble

m

Join

tpaper

with

and

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Syste

m1

Join

tpaper

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Syste

m1

Join

tpaper

−div

(M(x

)∇u)

=−

adiv

(u|z| s)

+f(x

)∈

Lp(Ω

):

Ω,

−div

(N(x

)∇z)

=−

bdiv

(z|u| r)

+g(x

)∈

Lq(Ω

):

Ω,

u=

z=

0:

∂Ω

.

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Syste

m2

(entro

py,

weak)

solu

tions

“N

onlin

ear

signalkin

etics

model(th

esa

tura

ting

signalpro

ductio

nm

odel)”

f≥

0,A

sym.

0<

λ

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Syste

m2

(entro

py,

weak)

solu

tions

“N

onlin

ear

signalkin

etics

model(th

esa

tura

ting

signalpro

ductio

nm

odel)”

f≥

0,A

sym.

0<

λ

−div

(M(x

)∇u)+

u=−

div

(uA(x

)∇z)+

f(x

):

Ω,

−div

(A(x

)∇z)

=u

1+

λu

:Ω

,

u=

z=

0:

∂Ω

.

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Syste

m3

(entro

py,

weak)

solu

tions

−div

(M(x

)∇u)+

u=−

div

(

uA(x

)∇z

1

(1+

z)γ

)

+f(x

):Ω

,

−div

(A(x

)∇z)

=u

θ:Ω

,u

=z

=0

:∂Ω

.

Let

0<

θ<

1an

dγ≥

0su

chth

at

(i)if

0<

θ≤

N+

22N

,γ≥

0;

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Syste

m3

(entro

py,

weak)

solu

tions

−div

(M(x

)∇u)+

u=−

div

(

uA(x

)∇z

1

(1+

z)γ

)

+f(x

):Ω

,

−div

(A(x

)∇z)

=u

θ:Ω

,u

=z

=0

:∂Ω

.

Let

0<

θ<

1an

dγ≥

0su

chth

at

(i)if

0<

θ≤

N+

22N

,γ≥

0;

(ii)if

N+

22N

<θ

<1,

γ>

2θN−

(N+

2)

4(θ

N−

2)

.

Then

there

existu≥

0en

tropysolu

tionan

dz≥

0weak

solution

ofth

esystem

.

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Cia

oD

onot

forg

et

our

”get-to

geth

er”

durin

gour

young

age

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

Cia

oD

onot

forg

et

our

”get-to

geth

er”

durin

gour

young

age

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

”Ce

que

jevo

us

souhaite

pour

les

pro

chain

s60

ans”

Cia

o!

Dirich

let

pro

ble

ms

with

singular

conve

ction

term

sand

applica

tions

SH

ALO

M