Modular eigenvalues of the variable exponent p-Laplacian

Transcrição

Modular eigenvalues of the variable exponent
p-Laplacian
Joint Meeting AMS-EMS-SPMM
Porto, June 2015
Osvaldo Mendez
University of Texas-El Paso
Table of contents
Section 1
Modular Spaces
Section 2
Sobolev Spaces, Poincares inequality, Sobolev Embedding.
Section 3
Euler-Lagrange equations
Section 5
Existence of the rst eigenvalue
Section no. 6
Stability
Section 7
Remark
Section 7
Open problems
,
D. Edmunds, J. Lang, O.M.
Dierential Operators on Spaces of Variable Integrability
,
J. Lang, O. Mendez
,
J. Lang, O. Mendez
,
,
Nonlinear An., 2014
Journal of Di. Eq, 2015
Journal d'Analyse Mathematique, 2015
Preliminaries
Rn bounded, Lipschitz domain,
Borel-measurable.
p:
! (1; 1)
Preliminaries
Borel-measurable.
p:
! (1; 1)
Z
p();
(f ) = jf (x )jp(x ) dx
is a left-continuous convex modular on M(
); M(
) being the
set of all Lebesgue-measurable, scalar-valued functions on ;
f = g i f = g a.e..
Preliminaries
Borel-measurable.
p:
! (1; 1)
Z
p();
(f ) = jf (x )jp(x ) dx
is a left-continuous convex modular on M(
); M(
) being the
set of all Lebesgue-measurable, scalar-valued functions on ;
f = g i f = g a.e..
kf kp();
:= inf > 0 : p();
(f =) 1
is a norm on the space
Lp() (
) = f 2 M(
) : p();
(f ) < 1 for some > 0 :
Particular case of Musielak-Orlicz spaces.
n
W 1;p() (
) = u 2 Lp() (
) : jru j 2 Lp() (
)
o
n
W 1;p() (
) = u 2 Lp() (
) : jru j 2 Lp() (
)
kukW 1
( ) (
)
;p o
= kuk1;p() = kukp();
+ kjrujkp();
:
n
W 1;p() (
) = u 2 Lp() (
) : jru j 2 Lp() (
)
kukW 1
( ) (
)
;p o
+ kjrujkp();
:
W01;p() (
)
is the closure of C01(
) in W 1;p()(
).
n
W 1;p() (
) = u 2 Lp() (
) : jru j 2 Lp() (
)
kukW 1
( ) (
)
;p o
+ kjrujkp();
:
W01;p() (
)
) in W 1;p()(
). If p 2 C (
) the embedding
is compact
E : W01;p() (
) ,! Lp() (
)
n
W 1;p() (
) = u 2 Lp() (
) : jru j 2 Lp() (
)
kukW 1
( ) (
)
;p o
+ kjrujkp();
:
W01;p() (
)
) in W 1;p()(
). If p 2 C (
) the embedding
E : W01;p() (
) ,! Lp() (
)
is compact and Poincares inequality holds:
n
W 1;p() (
) = u 2 Lp() (
) : jru j 2 Lp() (
)
kukW 1
( ) (
)
;p o
+ kjrujkp();
:
W01;p() (
)
) in W 1;p()(
). If p 2 C (
) the embedding
E : W01;p() (
) ,! Lp() (
)
kukp() C (p; n; )kjrujkp():
n
W 1;p() (
) = u 2 Lp() (
) : jru j 2 Lp() (
)
kukW 1
( ) (
)
;p o
+ kjrujkp();
:
W01;p() (
)
) in W 1;p()(
). If p 2 C (
) the embedding
E : W01;p() (
) ,! Lp() (
)
n
W 1;p() (
) = u 2 Lp() (
) : jru j 2 Lp() (
)
kukW 1
( ) (
)
;p o
+ kjrujkp();
:
W01;p() (
)
) in W 1;p()(
). If p 2 C (
) the embedding
E : W01;p() (
) ,! Lp() (
)
kuk = kjrujkp()
n
W 1;p() (
) = u 2 Lp() (
) : jru j 2 Lp() (
)
kukW 1
( ) (
)
;p o
+ kjrujkp();
:
W01;p() (
)
) in W 1;p()(
). If p 2 C (
) the embedding
E : W01;p() (
) ,! Lp() (
)
kuk = kjrujkp()
is a norm on W01;p()(
) equivalent to the original one.
The THERMISTOR PROBLEM (Zhikov, 1996)
p()u := div jrujp(x ) 2ru = 0
Existence, uniqueness and regularity of the solution!!!
Fan-Zhang, 2003
Fan-Zhang, 2003
Modeling of Non-Newtonian uid hydrodynamics,
Electrorheological uids.
Fan-Zhang, 2003
Modeling of Non-Newtonian uid hydrodynamics,
Electrorheological uids.
Electrorheological uids: Modeling and Mathematical Theory.
R_ucika
Modular Eigenvalue problem
p()u = jujp()
2u
(1)
p()u = jujp()
(Fan-Zhang-Zhao, 2005)
2u
(1)
p()u = jujp()
Z
8h 2 C01(
):
Z
2u
jrujp() 2rurh = jujp() 2uh
(1)
(2)
p()u = jujp()
Z
8h 2 C01(
):
2u
Z
If p is constant ! rst positive eigenvalue:
1 :
=
kE kp
(1)
(2)
p()u = jujp()
Z
8h 2 C01(
):
2u
Z
1 :
=
kE kp
If p has either a local min or a local max in ..........
(1)
(2)
p()u = jujp()
Z
8h 2 C01(
):
2u
Z
1 :
=
kE kp
If p has either a local min or a local max in ..........
R
p (x )
inf1 ( ) R
jrjuu(x(x)j)pj(x ) = 0!!!!:
06=u2W0 (
) (Fhang-Zhang-Zhao (2005).)
;p (1)
(2)
The modular eigenvalue problem as Euler-Lagrange
Equations
Does the modular eigenvalue problem have a solution (; u)?
Equations
Abstract theory of Ljusternik-Schnirelman:
Equations
Abstract theory of Ljusternik-Schnirelman: k = 1; 2; 3:::
R
k
jrujp() :
R
= Cinf
sup
2C v 2C ju jp()
k
Equations
R
k
jrujp() :
R
= Cinf
sup
2C v 2C ju jp()
k
Ck =: fC W01;p()(
) ; C closed ; (C ) k g:
Equations
R
k
jrujp() :
R
= Cinf
sup
2C v 2C ju jp()
k
Ck =: fC W01;p()(
: Krasnoselskii genus:
Equations
R
k
jrujp() :
R
= Cinf
sup
2C v 2C ju jp()
k
Ck =: fC W01;p()(
Not a tangible description of the eigenfunctions.
Equations
R
k
jrujp() :
R
= Cinf
sup
2C v 2C ju jp()
k
Ck =: fC W01;p()(
Not a tangible description of the eigenfunctions.
There are eigenvalues "per se" that are NOT obtained via
Ljiusternik-Schnirelmann. (Bining-Rynne(2008))
Opt for an ad-hoc treatment (Sobolev embedding)
F : W01;p() (
) ! R
R
()
F (u ) = jrup((xx))j dx
p x
Opt for an ad-hoc treatment (Sobolev embedding)
F : W01;p() (
) ! R
R
()
F (u ) = jrup((xx))j dx
p x
G : Lp() (
) ! R
R
()
G (u ) = ju(px()xj ) dx
p x
Then
hF (u); hi =
0
and
Z
hG (u); hi =
0
jru(x )jp(x ) 2ru(x )rh(x ) dx
Z
ju(x )jp(x ) 2u(x )h(x ) dx
(3)
(4)
The modular eigenvalue problem is the Euler-Lagrange equation
for the constrained problem
max G (u) subject to F (u) = constant
(5)
r > 0 ) 9(u0 ; 0 )
solution of modular eigenvalue problem:
r > 0 ) 9(u0 ; 0 )
Z
ju0(x )jp(x ) dx =
p (x )
R
jru
sup( )
( )j
()
x
p x
p x
Z
dx r ju(x )jp(x ) dx :
p (x )
r > 0 ) 9(u0 ; 0 )
Z
ju0(x )jp(x ) dx =
p (x )
R
Z
jru
sup( )
( )j
()
x
p x
p x
Z
dx r jru0(x )jp(x ) dx = r
p (x )
ju(x )jp(x ) dx :
p (x )
Stability
( p ; u p )
(q ; uq )
eigenvalues corresponding to p; q;
Stability
( p ; u p )
(q ; uq )
)
pq
Stability
( p ; u p )
(q ; uq )
)
pq
p
q ??
Stability
( p ; u p )
(q ; uq )
)
pq
p
q ??
up uq ??
(Lindqvist (1990), p constant.)
Holder's inequality:
Z
jf (x )g (x )j dx (1 + 1=p
1=p+) kf kp kg kp :
0
Z
jf (x )g (x )j dx (1 + 1=p
Not suitable for "stability" analysis.
1=p+) kf kp kg kp :
0
Z
jf (x )g (x )j dx (1 + 1=p
1=p+) kf kp kg kp :
However: If
p (x ) q (x ) p (x ) + " a.e. in :
0
(6)
Z
jf (x )g (x )j dx (1 + 1=p
1=p+) kf kp kg kp :
However: If
p (x ) q (x ) p (x ) + " a.e. in :
q (f ) 1; then
p (f ) " j
j + " " :
0
(6)
Z
jf (x )g (x )j dx (1 + 1=p
1=p+) kf kp kg kp :
However: If
p (x ) q (x ) p (x ) + " a.e. in :
q (f ) 1; then
p (f ) " j
j + " " :
Edmunds-Lang-Nekvinda, 2009
0
(6)
Z
jf (x )g (x )j dx (1 + 1=p
1=p+) kf kp kg kp :
0
However: If
p (x ) q (x ) p (x ) + " a.e. in :
q (f ) 1; then
p (f ) " j
j + " " :
Edmunds-Lang-Nekvinda, 2009
This is the substitute of Holder's inequality for the stability
analysis.
(6)
(pi )i C (
)
pi increases uniformly to q
inf
q > n;
(pi )i C (
)
inf
q > n;
Then, there exists a solution to the q-eigenvalue problem,
such that
and
(q ; uq )
up * uq
i
in W01;q (
)
(pi )i C (
)
inf
q > n;
such that
and
Moreover:
(q ; uq )
up * uq
i
up
i
in W01;q (
)
! uq in Lq (
)
(pi )i C (
)
inf
q > n;
such that
and
Moreover:
and
(q ; uq )
up * uq
i
up
i
in W01;q (
)
! uq in Lq (
)
R
jrujq dx
q
L R
q
q+
ju j dx
L;
q 2
L q;1 L:
q+
(7)
(pi )i C (
)
inf
q > n;
such that
and
Moreover:
and
(q ; uq )
up * uq
i
up
i
in W01;q (
)
! uq in Lq (
)
R
jrujq dx
q
L R
q
q+
ju j dx
L;
q 2
L q;1 L:
q+
Similar result for (pi ) decreasing uniformly to p.
(7)
For
p; q 2
the convergence is actually strong in the corresponding Sobolev
Space.
Remarks
The norm on W01;p()(
),
u
! kjrujkp() = kuk1;p()
is Frechet dierentiable with derivative given by
p (x ) kjrf jkp(p)(x ) jjrf (x )jjp(x ) 1 sgn rf (x )
(grad kjf jk1;p())(x ) = R
:
p (x ) kjrf jkp(p)(x ) 1 jrf (x )jp(x ) dx
Remarks
),
u
:
Dinca-Mathei (2009)
Remarks
),
u
:
Dinca-Mathei (2009) Therefore:
Z
Z
grad ku0k1;p() (x )rh(x ) dx = p grad ku0kp() (x )h(x ) dx ;
with p = kE1 k :
Remarks
),
u
:
Z
Z
with p = kE1 k :
is the Euler-Lagrange equation corresponding to the maximal
functions of the Sobolev embedding!!!!
Remarks
),
u
:
Z
Z
with p = kE1 k :
is the Euler-Lagrange equation corresponding to the maximal
functions of the Sobolev embedding!!!! Bennewitz (2007),
Lindqvist-Francina (2013)
Open problems
Open problems
Strong convergence in general?
Open problems
Uniqueness of positive eigenfunction?
Open problems
Uniqueness of positive eigenfunction? Implied by the uniform
convexity of
W01;p (
)
with respect to the norm
u ! kjru jkp :
Open problems
Uniqueness of positive eigenfunction? Implied by the uniform
convexity of
W01;p (
)
with respect to the norm
u ! kjru jkp :
Other boundary conditions? (e.g. Neumann, Robin??)
MUITO OBRIGADO!
MUITO OBRIGADO!
THANKS!

Modular eigenvalues of the variable exponent p-Laplacian

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