Grillo

Transcript

Grillo

Functional inequalities
and asymptotics of solutions
to weighted porous media equations
Gabriele Grillo - Politecnico di Milano (Italy)
joint work with Matteo Muratori - Maria Michaela Porzio
Pisa, July 2, 2012
Weighted porous media equation
Functional inequalities and asymptotics
of linear semigroups-a short reminder
Consider a linear, self-adjoint, nonnegative operator L on L2 (X , m),
its associated semigroup Tt = e−tL , its quadratic form E(u) =
kL1/2 uk22 on D(E) := Dom L1/2 .
kL1/2 uk22 on D(E) := Dom L1/2 .
A well known topic in linear analysis is connecting functional
inequalities involving E to asymptotic properties involving the
semigroup Tt = e−tL . Among the type of inequalities considered one has, e.g.:
kL1/2 uk22 on D(E) := Dom L1/2 .
A well known topic in linear analysis is connecting functional
inequalities involving E to asymptotic properties involving the
semigroup Tt = e−tL . Among the type of inequalities considered one has, e.g.:
Poincaré-type, or gap, inequalities:
kf k2 ≤ C E(f ), kf − f k2 ≤ C E(f )
where f is the mean of f w.r.t. the measure m, provided it is
finite. This is a spectral information on L.
Logarithmic Sobolev inequalities. Suppose that m(X ) = 1.
The inequality
Z
f
dm ≤ C E(f ), ∀f ∈ D(E)
f 2 log
kf k2
X
is stronger than the Poincaré inequality kf − f k2 ≤ C E(f ).
2
A typical example is the case in which dm = ce−|x| /2 dx.
Logarithmic Sobolev inequalities. Suppose that m(X ) = 1.
The inequality
Z
f
dm ≤ C E(f ), ∀f ∈ D(E)
f 2 log
kf k2
X
is stronger than the Poincaré inequality kf − f k2 ≤ C E(f ).
2
A typical example is the case in which dm = ce−|x| /2 dx.
The Nash-type inequality
kf k2+ϑ
≤ C E(f )kf kϑ1 ,
2
∀f ∈ D(E) ∩ L1 .
It turns out that such Nash inequality is equivalent to a
Sobolev-type inequality if ϑ satisfy a suitable bound.
The Sobolev inequality (µ > 2)
kf k22µ/(µ−2) ≤ C E(f ),
∀f ∈ D(E).
It can be shown that the last two inequalities are equivalent
if one puts ϑ = 4/µ. They are also equivalent to families of
logarithmic Sobolev inequalities.
kf k22µ/(µ−2) ≤ C E(f ),
∀f ∈ D(E).
How are this functional inequalities related to suitable properties
of the semigroup Tt ?
kf k22µ/(µ−2) ≤ C E(f ),
∀f ∈ D(E).
Poincaré inequalities in general only imply exponential
decay in time of the L2 norm of u(t) or of u(t) − u (suppose
that L1 = 0).
kf k22µ/(µ−2) ≤ C E(f ),
∀f ∈ D(E).
that L1 = 0).
A single logarithmic Sobolev inequality implies
hypercontractivity : for all q > p ≥ 1there exists T (p, q) > 0
such that kTt kp,q (operator norm) is finite exactly for
t ≥ T (p, q): there is a waiting time to get regularization into
Lq . Solutions need not be bounded at any time.
kf k22µ/(µ−2) ≤ C E(f ),
∀f ∈ D(E).
that L1 = 0).
A single logarithmic Sobolev inequality implies
hypercontractivity : for all q > p ≥ 1there exists T (p, q) > 0
such that kTt kp,q (operator norm) is finite exactly for
t ≥ T (p, q): there is a waiting time to get regularization into
Lq . Solutions need not be bounded at any time.
A Nash, or a Sobolev, inequality, implies ultracontractivity ,
namely to a bound of the form
kTt (u)k∞ ≤
C
t µ/2
kuk1 ,
∀u ∈ L1 :
(1)
kTt (u)k∞ ≤
C
t µ/2
kuk1 ,
∀u ∈ L1 :
(1)
solutions become instantaneously bounded at any time,
with quantitative bounds on the rate of explosion for short t
and on the rate of decay for t large. It is well-known that (1)
is equivalent to, say, the previous Sobolev-type inequality.
Versions for Neumann b.c. exist.
kTt (u)k∞ ≤
C
t µ/2
kuk1 ,
∀u ∈ L1 :
(1)
solutions become instantaneously bounded at any time,
with quantitative bounds on the rate of explosion for short t
and on the rate of decay for t large. It is well-known that (1)
is equivalent to, say, the previous Sobolev-type inequality.
Versions for Neumann b.c. exist.
Similar problems have been studied for a while also in the nonlinear setting. To make an example of the kind of new phenomena
which can arise, consider a p-energy functional:
Z
1
|∇u|Px eV dm.
Ep (u) =
p N
Here N ⊂ M, M is a manifold, m is the Riemannian measure,
V a function, and Dirichlet boundary conditions are assumed in
a suitable sense. Consider the nonlinear semigroup associated
to the subgradient of Ep , provided the functional is convex and
l.s.c..
l.s.c.. Then one has:
THEOREM (G., JDE 2010). Consider the p-Laplacian type evolution (p > 2) associated to Ep and let u(t) be a solution to such
a flow. Then the functional inequality (in an appropriate space
taking into account the Dirichlet b.c.)
kf kpp ≤ C Ep (u)
kf kpp ≤ C Ep (u)
is equivalent to the bound (1 ≤ q0 ≤ % < ∞):
ku(t)k% ≤
C
t (%−q0 )/[%(p−2)]
ku(0)kq0 , ∀t > 0.
kf kpp ≤ C Ep (u)
is equivalent to the bound (1 ≤ q0 ≤ % < ∞):
ku(t)k% ≤
C
t (%−q0 )/[%(p−2)]
ku(0)kq0 , ∀t > 0.
The last bound need not hold if % = ∞.
Weighted porous media equations-The Poincaré case
Motivated by such result, we started the study of classes of twoweights porous media equations, whose models are formally
(Ω ⊆ Rn , m > 1):
(Ω ⊆ Rn , m > 1):

m

ρν ut = div (ρµ ∇(u )) in Ω × (0, ∞)
(2)
u=0
on ∂Ω × (0, ∞)


u(·, 0) = u0 (·)
in Ω.
(Ω ⊆ Rn , m > 1):

m

ρν ut = div (ρµ ∇(u )) in Ω × (0, ∞)
(2)
u=0
on ∂Ω × (0, ∞)


u(·, 0) = u0 (·)
in Ω.


div (ρµ ∇(u m )) in Ω × (0, ∞)
ρν ut =
m
)
ρµ ∂(u
on ∂Ω × (0, ∞) .
∂n = 0


u(·, 0) = u0 (·)
in Ω
(3)
(Ω ⊆ Rn , m > 1):

m

ρν ut = div (ρµ ∇(u )) in Ω × (0, ∞)
(2)
u=0
on ∂Ω × (0, ∞)


u(·, 0) = u0 (·)
in Ω.


div (ρµ ∇(u m )) in Ω × (0, ∞)
ρν ut =
m
)
ρµ ∂(u
on ∂Ω × (0, ∞) .
∂n = 0


u(·, 0) = u0 (·)
in Ω
(3)
−1 −1
Here ρν , ρµ are measurable weight in L∞
loc and s.t. ρν , ρµ are in
∞
Lloc (singularities or degeneracies are only “at infinity"). We set
dν = %ν dx, dµ = %µ dx.
The one weight case (ρµ = 1) is thoroughly studied for special classes of weights, see Eidus, Kamin, Rosenau, Reyes,
Vazquez....: see e.g. Kamin, Reyes, Vazquez DCDS 2010 and
references quoted.
references quoted.
I start with the definition of solution (Dirichlet case), see also Dolbeault, Gentil, Guillin, Wang, Pot. Anal. 2008. Here v ∈ V0 (Ω; µ)
1,1
if v ∈ Wloc
(Ω), ∇v ∈ L2 (Ω; µ) and there exists a sequence
∞
{ϕn } ⊂ Cc (Ω) such that k∇v − ∇ϕn k2;µ → 0.
references quoted.
I start with the definition of solution (Dirichlet case), see also Dolbeault, Gentil, Guillin, Wang, Pot. Anal. 2008. Here v ∈ V0 (Ω; µ)
1,1
if v ∈ Wloc
(Ω), ∇v ∈ L2 (Ω; µ) and there exists a sequence
∞
{ϕn } ⊂ Cc (Ω) such that k∇v − ∇ϕn k2;µ → 0.
Definition. A function u ∈ L1 ((0, T ); L1loc (Ω; ν)) with u m (t) ∈
V0 (Ω; µ) for a.a. t, ∇(u m ) ∈ L1 ((0, T ); [L2 (Ω; µ)]N ) is a weak
solution of (2) with initial datum u0 ∈ L1loc (Ω; ν) if it satisfies:
Z TZ
Z
u(x, t)ηt (x, t) dν dt = − u0 (x)η(x, 0) dν
0
Ω
Ω
Z TZ
+
0
∇(u m )(x, t) · ∇η(x, t) dµdt
Ω
∀η ∈ C 1 (Ω × [0, T ]) : supp η(·, t) b Ω, η(x, T ) = 0
Ω , ∀t ∈ [0, T ].
∀x ∈
Uniqueness is a delicate issue. In fact, it does not necessarily
hold even in the linear case. But (the forthcoming results are
from G., Muratori, Porzio, preprint 2012):
Proposition. There exists at most one weak solution satisfying:
m+1
m+1
u m ∈ L m ((0, T ), V0 m (Ω; ν, µ)), ∇(u m ) ∈ L2 ((0, T ); [L2 (Ω; µ)]N )
∀T > 0. Here V0p (Ω; ν, µ) is the closure of Cc∞ (Ω) with respect
to the norm kϕkp,2;ν,µ = kϕkp;ν + k∇ϕk2;µ .
m+1
m+1
This uses a known method due to Oleǐnik test function. Such
solutions, if any, are called weak energy solutions. This solution
will be referred to as “the" solution to the problem at hand.
m+1
m+1
This uses a known method due to Oleǐnik test function. Such
solutions, if any, are called weak energy solutions. This solution
will be referred to as “the" solution to the problem at hand.
Proposition. Let Ω ⊆ RN be a domain, and let ρν , ρµ be two
−1
∞
sufficiently regular weights such that ρν , ρµ , ρ−1
ν , ρµ ∈ Lloc (Ω).
1
r
If u0 ∈ L (Ω; ν) ∩ L (Ω; ν), with r ≥ m + 1, then there exists
a unique weak energy solution u. Moreover, given two energy
solutions u, v one has
Z
Z
(u(x, T ) − v (x, T ))+ dν ≤ (u0 (x) − v0 (x))+ dν.
Ω
Ω
Remark. 1) The proof uses appropriate approximating problems.
2) Extension to data in L1 (ν) is possible through the concept of
limit solutions.
limit solutions.
3) Similar results also hold for the Neumann problem.
limit solutions.
We now consider, as a sample of results concerning the relations between functional inequalities and short time regularization of the evolution, the Neumann problem which is much less
studied in the literature (ν(Ω) < ∞).
limit solutions.
Theorem. If the inequality
kv k2;ν ≤ WP k∇v k2;µ + kv k1;ν ∀v ∈ W 1,2 (Ω; ν, µ)
limit solutions.
Theorem. If the inequality
kv k2;ν ≤ WP k∇v k2;µ + kv k1;ν ∀v ∈ W 1,2 (Ω; ν, µ)
holds, then the solution u(t) corresponding to u0 ∈ Lq0 (Ω; ν)
satisfies the bound
%−q
q0
0
− %(m−1)
%
ku(t)k%;ν ≤ K2 t
ku0 kq0 ;ν + ku0 kq0 ;ν
for a.e. t > 0
for all q0 ∈ (1, ∞) ∩ [m − 1, ∞), % > q0 ..
Remark. 1) The proof uses a Moser iteration technique, together
with a (Gross-like) differential method which involves suitable
logarithmic Sobolev inequalities deduced from the Poincaré-type
inequality.
inequality.
2) The assumed functional inequality is weaker than the usual
Poincaré inequality. An explicit counterexample shows that no
L∞ regularization holds in general.
inequality.
A converse implication also holds:
inequality.
Theorem. If there exist a constant K > 0 and a given q0 ∈
[m, m + 1) such that, for all u0 ∈ Lq0 (Ω; ν), the solution u corresponding to the initial datum u0 satisfies the estimate
q0
m+1−q0
−
ku(t)km+1;ν ≤ K t (m+1)(m−1) ku0 kqm+1
+
ku
k
for a.e. t > 0,
0 q0 ;ν
0 ;ν
inequality.
Theorem. If there exist a constant K > 0 and a given q0 ∈
[m, m + 1) such that, for all u0 ∈ Lq0 (Ω; ν), the solution u corresponding to the initial datum u0 satisfies the estimate
q0
m+1−q0
−
ku(t)km+1;ν ≤ K t (m+1)(m−1) ku0 kqm+1
+
ku
k
for a.e. t > 0,
0 q0 ;ν
0 ;ν
then there exists a constant B > 0 such that the functional inequality
kv k2;ν ≤ B k∇v k2;µ + kv k q0 ;ν
∀v ∈ W 1,2 (Ω; ν, µ)
m
holds as well.
Corollary. With the above notations, the bound
m
− %−m
%
ku(t)k%;ν ≤ K t %(m−1) ku0 km;ν
+ ku0 km;ν
for a.e. t > 0
for solutions to the weighted PME, for a given % ≥ m + 1, is
equivalent to the validity of the functional inequality
kv k2;ν ≤ WP k∇v k2;µ + kv k1;ν
∀v ∈ W 1,2 (Ω; ν, µ).
Corollary. With the above notations, the bound
m
− %−m
%
ku(t)k%;ν ≤ K t %(m−1) ku0 km;ν
+ ku0 km;ν
for a.e. t > 0
for solutions to the weighted PME, for a given % ≥ m + 1, is
equivalent to the validity of the functional inequality
kv k2;ν ≤ WP k∇v k2;µ + kv k1;ν
∀v ∈ W 1,2 (Ω; ν, µ).
The use of the Poincaré inequality also allow to establish asymptotic results for t large. Decay rates (polinomial) for suitable entropies are given in Dolbeault, Nazaret, Savaré (Comm. Math.
Sci. 2008). Following an old idea of Alikakos and Rostamian,
Indiana Univ. Math. J. 1981 the key Lemma consists in showing that, for Φ approximately a power and ξ with ξ = 0 and
Φ(ξ) ∈ W 1,2 (Ω; ν, µ) the inequality
kΦ(ξ)k2;ν ≤ CΦ k∇Φ(ξ)k2;µ
holds. The existing proof uses compactness, which we have not,
but one can proceed otherwise.
Theorem. Let q0 ∈ [1, ∞), u0 ∈ Lq0 (Ω; ν) and u0 = 0.
Theorem. Let q0 ∈ [1, ∞), u0 ∈ Lq0 (Ω; ν) and u0 = 0. If the
inequality
kv − v k2;ν ≤ MP k∇v k2;µ
∀v ∈ W 1,2 (Ω; ν, µ)
holds, then the solution u with initial datum u0 satisfies the following absolute bound:
1
ku(t)k%;ν ≤ Q2 t − m−1 for a.e. t > 0 ,
for any % ∈ [1, ∞).
inequality
∀v ∈ W 1,2 (Ω; ν, µ)
1
ku(t)k%;ν ≤ Q2 t − m−1 for a.e. t > 0 ,
for any % ∈ [1, ∞). Moreover
%−q
ku(t)k%;ν ≤ Q1 t
0
− %(m−1)
q0
ku0 kq%0 ;ν for a.e. t > 0.
inequality
∀v ∈ W 1,2 (Ω; ν, µ)
1
ku(t)k%;ν ≤ Q2 t − m−1 for a.e. t > 0 ,
q0
%−q
ku(t)k%;ν ≤ Q1 t
0
− %(m−1)
If instead u0 6= 0, we have, for an unknown constant C:
ku(t) − uk%;ν ≤ e−C|u|
m−1 t
ku0 − uk%;ν
∀ t > 0.
inequality
∀v ∈ W 1,2 (Ω; ν, µ)
1
ku(t)k%;ν ≤ Q2 t − m−1 for a.e. t > 0 ,
q0
%−q
ku(t)k%;ν ≤ Q1 t
0
− %(m−1)
If instead u0 6= 0, we have, for an unknown constant C:
ku(t) − uk%;ν ≤ e−C|u|
m−1 t
ku0 − uk%;ν
∀ t > 0.
Convergence to the mean value in general is not uniform even if
the datum is bounded, but in that case it is locally uniform.
Examples (weighted Poincaré inequalities)
For the following weights, zero-mean Poincaré inequalities hold in
W 1,2 but no Sobolev-type inequality holds in the same space (see e.g.
Kufner-Opic):
Kufner-Opic):
• Intervals:
◦ (x β−2 , x β ) for β > 1 on (0, b) or for β < 1 on (a, +∞);
◦ x1 | log x|β−2 , x| log x|β for β 6= 1 on (0, c), with c ∈ (0, 1);
◦ (eα|x| , eα|x| ) for α < 0 on R;
Kufner-Opic):
• Intervals:
◦ (eα|x| , eα|x| ) for α < 0 on R;
• Bounded convex domains:
◦ (δ β−2 , δ β ) for β ≥ 2;
Kufner-Opic):
• Intervals:
◦ (eα|x| , eα|x| ) for α < 0 on R;
• Bounded convex domains:
◦ (δ β−2 , δ β ) for β ≥ 2;
• The Euclidean space RN :
◦ ((1 + |x|2 )α−1 , (1 + |x|2 )α ) for α < 1 −
2
N
2;
2
◦ (e−a|x| , e−a|x| ) for a > 0.
The Sobolev case
The Neumann problem has been considered first in Alikakos-Rostamian,
Indiana Univ. Math. J. 1982 (no weights), for L∞ data.
The Sobolev case
Indiana Univ. Math. J. 1982 (no weights), for L∞ data. In Bonforte,
G.(JFA 2005), it is shown that solutions corresponding to L1 data are
instantaneously smoothed out into L∞ .
The Sobolev case
Short time asymptotics, and long time one as well for zero mean data,
are polynomial. Long-time asymptotics for non-zero mean data is exponential (see below). The existing bounds are not sharp.
The Sobolev case
Consider our weighted problem with weights s.t., for some σ > 1,
∀v ∈ W 1,2 (Ω; ν, µ).
(4)
kv k2σ;ν ≤ B k∇v k2;µ + kv k1;ν
The Sobolev case
∀v ∈ W 1,2 (Ω; ν, µ).
(4)
The following results are proved in Muratori, G., preprint 2012
The Sobolev case
∀v ∈ W 1,2 (Ω; ν, µ).
(4)
The following results are proved in Muratori, G., preprint 2012
THEOREM . Let ν(Ω) < ∞ and let inequality (4) hold true for
some σ > 1. Then for the solution u corresponding to an initial
datum u0 ∈ Lq0 (Ω; ν) with q0 ∈ [1, ∞) the following estimate
holds ∀t > 0:
!
(σ−1)q
ku(t)k∞ ≤ K
t
− (σ−1)q
σ
0 +σ(m−1)
0
(σ−1)q0 +σ(m−1)
ku0 kq0 ;ν
+ ku0 kq0 ;ν
,
holds ∀t > 0:
!
(σ−1)q
ku(t)k∞ ≤ K
t
− (σ−1)q
σ
0 +σ(m−1)
0
(σ−1)q0 +σ(m−1)
ku0 kq0 ;ν
+ ku0 kq0 ;ν
,
The bound is sharp in the non-weighted case (Barenblatt solutions...). In the weighted case, there are explicit weights for
which (4) holds for a best possible σ and such that the power of
time is attained for special solutions (whose initial trace is singular “at infinity").
holds ∀t > 0:
!
(σ−1)q
ku(t)k∞ ≤ K
t
− (σ−1)q
σ
0 +σ(m−1)
0
(σ−1)q0 +σ(m−1)
ku0 kq0 ;ν
+ ku0 kq0 ;ν
,
The bound is sharp in the non-weighted case (Barenblatt solutions...). In the weighted case, there are explicit weights for
which (4) holds for a best possible σ and such that the power of
time is attained for special solutions (whose initial trace is singular “at infinity").
Also a converse result holds true
THEOREM . Let ν(Ω) < ∞ and suppose that there exist a constant K > 0 and σ > 1, q0 ∈ [m, m + 1) such that, for all
u0 ∈ Lq0 (Ω; ν), the solution u corresponding to the initial datum
u0 satisfies ∀t > 0:
!
(σ−1)q
ku(t)k∞ ≤ K
t
− (σ−1)q
σ
0 +σ(m−1)
0
(σ−1)q0 +σ(m−1)
ku0 kq0 ;ν
+ ku0 kq0 ;ν
.
(5)
!
(σ−1)q
ku(t)k∞ ≤ K
t
− (σ−1)q
σ
0 +σ(m−1)
0
(σ−1)q0 +σ(m−1)
ku0 kq0 ;ν
+ ku0 kq0 ;ν
.
(5)
Then there exists a constant B > 0 such that the functional inequality
kv k2σ;ν ≤ B k∇v k2;µ + kv k q0 ;ν
∀v ∈ W 1,2 (Ω; ν, µ)
(6)
m
holds as well.
!
(σ−1)q
ku(t)k∞ ≤ K
t
− (σ−1)q
σ
0 +σ(m−1)
0
(σ−1)q0 +σ(m−1)
ku0 kq0 ;ν
+ ku0 kq0 ;ν
.
(5)
Then there exists a constant B > 0 such that the functional inequality
kv k2σ;ν ≤ B k∇v k2;µ + kv k q0 ;ν
∀v ∈ W 1,2 (Ω; ν, µ)
(6)
m
holds as well. In particular, the validity of (5) for some, hence
all, q0 ≥ m (and, a posteriori, hence for all q0 ≥ 1) and of (6) for
q0 = m, are equivalent.
As for the long-time behaviour, the bounds for zero mean data
are qualitatively similar to one proved in the Poincaré case, but
bounds hold in L∞ : e.g. one can prove that, for such data and
any t ≥ 0 (u0 = 0):
ku(t)k∞ ≤ Q1 t
σ
0 +σ(m−1)
− (σ−1)q
Q2 t +
ku0 k1−m
q0 ;ν
1
(σ−1)q0
(m−1)[(σ−1)q0 +σ(m−1)]
from which an L∞ absolute bound in terms of t −(1−m) follows.
,
As for the long-time behaviour, the bounds for zero mean data
are qualitatively similar to one proved in the Poincaré case, but
bounds hold in L∞ : e.g. one can prove that, for such data and
any t ≥ 0 (u0 = 0):
ku(t)k∞ ≤ Q1 t
σ
0 +σ(m−1)
− (σ−1)q
Q2 t +
ku0 k1−m
q0 ;ν
1
(σ−1)q0
(m−1)[(σ−1)q0 +σ(m−1)]
,
from which an L∞ absolute bound in terms of t −(1−m) follows.
The situation is more interesting for non-zero mean solutions.
In fact, the only existing result in the weighted, Neumann-type
case is due to Kamin and Rosenau (CPAM 1982, 1-dim). They
prove, for some explicit weights, local uniform convergence to
the mean value. As we saw in the Poincaré case, data with
compact support may generate solutions which are compactly
supported for all times.
THEOREM. Let ν(Ω) < ∞ and let the Sobolev-type inequality
kv − v k2σ;ν ≤ CS k∇v k2;µ
∀v ∈ W 1,2 (Ω; ν, µ)
hold true for some σ > 1.
∀v ∈ W 1,2 (Ω; ν, µ)
hold true for some σ > 1. For any solution u corresponding to
an initial datum u0 ∈ L1 (Ω; ν) with u0 = u 6= 0 there exists a
constant G such that the following estimate holds:
ku(t) − uk∞ ≤ G e
−
m
C2
P
|u|m−1 t
∀t ≥ 1 ,
∀v ∈ W 1,2 (Ω; ν, µ)
−
m
C2
P
|u|m−1 t
∀t ≥ 1 ,
CP being the smallest constant such that the inequality
kv − v k2;ν ≤ CP k∇v k2;µ
∀v ∈ W 1,2 (Ω; ν, µ)
holds.
∀v ∈ W 1,2 (Ω; ν, µ)
−
m
C2
P
|u|m−1 t
∀t ≥ 1 ,
∀v ∈ W 1,2 (Ω; ν, µ)
holds. In particular, the support of a solution corresponding to a
compactly supported datum blows up in finite time.
∀v ∈ W 1,2 (Ω; ν, µ)
−
m
C2
P
|u|m−1 t
∀t ≥ 1 ,
∀v ∈ W 1,2 (Ω; ν, µ)
The bounds are sharp in the non-weighted case (a matching
lower bound holds for some solutions) and the rate is the one
predicted by linearization in all cases.
∀v ∈ W 1,2 (Ω; ν, µ)
−
m
C2
P
|u|m−1 t
∀t ≥ 1 ,
∀v ∈ W 1,2 (Ω; ν, µ)
The bounds are sharp in the non-weighted case (a matching
lower bound holds for some solutions) and the rate is the one
predicted by linearization in all cases.
Examples (weighted Sobolev inequalities)
• Intervals:
i
α+1
;
◦ (x α , x β ) on (0, b) (b > 0): β > 1, α > β − 2 and σ ∈ 1, β−1
◦ (x α ,x β ) on (a,
i +∞) (a > 0): β < 1, α < β − 2 and
α+1
σ ∈ 1, β−1 ;
◦ x1 | log x|α , x| logi x|β on (0, c) (c ∈ (0, 1)): β < 1, α < β − 2
α+1
and σ ∈ 1, β−1
;
i
◦ (eα|x| , eβ|x| ) on R: β < 0, α < β and σ ∈ 1, α
β .
Examples (weighted Sobolev inequalities)
• Intervals:
i
α+1
;
◦ (x α , x β ) on (0, b) (b > 0): β > 1, α > β − 2 and σ ∈ 1, β−1
◦ (x α ,x β ) on (a,
i +∞) (a > 0): β < 1, α < β − 2 and
α+1
σ ∈ 1, β−1 ;
◦ x1 | log x|α , x| logi x|β on (0, c) (c ∈ (0, 1)): β < 1, α < β − 2
α+1
and σ ∈ 1, β−1
;
i
◦ (eα|x| , eβ|x| ) on R: β < 0, α < β and σ ∈ 1, α
β .
• Bounded Lipschitz domains (N > 1):
i
N
◦ (δ α , δ β ): β ≤ 1, α > −1 and σ ∈ 1, min N−2
, α+N
OR
N−1
i
N
α+N
β > 1, α > β − 2 and σ ∈ 1, min N−2 , β+N−2 (δ denotes
the distance function from ∂Ω).
• The Euclidean space RN (N > 1):
i
N
◦ ((1 + |x|)α , (1 + |x|)β ): β ≥ 2 − N, α < −N and σ ∈ 1, N−2
i
N
α+N
OR β < 2 − N, α < β − 2 and σ ∈ 1, min N−2
, β+N−2
;
i
N
α|x|
β|x|
◦ (e , e ): β ≥ 0, α < 0 and σ ∈ 1, N−2 OR β < 0,
i
N
α < β and σ ∈ 1, min N−2
.
,α
β
THANK YOU FOR YOUR ATTENTION!

Grillo

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