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Smoothing and Decay Estimates for Nonlinear Diffusion EquationsEquations of Porous Medium Type$

Juan Luis Vázquez

Print publication date: 2006

Print ISBN-13: 9780199202973

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780199202973.001.0001

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(p.203) Appendix I. Some analysis topics

(p.203) Appendix I. Some analysis topics

Source:
Smoothing and Decay Estimates for Nonlinear Diffusion Equations
Publisher:
Oxford University Press

AI.1 Some integrals and constants

We list some of the integrals that enter the calculation of the best constants in the smoothing effect.

  1. (i) EULER'S GAMMA FUNCTION is defined as

    Γ ( p ) = 0 t p 1 e 1 d t , p > 0.
    We have Γ(p) = (p − 1) Γ (p − 1), and Γ ( 1 / 2 ) = π . As p → ∞ we have
    Γ ( p ) ( p / e ) p ( 2 π p ) 1 / 2 .

  2. (ii) EULER'S BETA FUNCTION is defined for p, q > 0 as

    B ( p , q ) = 0 1 t p 1 ( 1 t ) q 1 d t = 2 0 1 s 2 p 1 ( 1 s 2 ) q 1 d s .
    We have B(p, q) = B(q, p) and the basic relation
    B ( p , q ) = Γ ( p ) Γ ( q ) Γ ( p + q ) ,
    as well as the equivalent expressions with parameter r > 0
    B ( p , q ) = r 0 1 s r p 1 ( 1 s r ) q 1 d s = r 0 x r q 1 ( 1 + x r ) p + q d x .
    These expressions are usually found for the value r = 2.

  3. (iii) VOLUME OF THE UNIT SPHERE: It is well known that

    ω n = 2 π n / 2 n Γ ( n / 2 ) = π n / 2 Γ ( n / 2 + 1 ) .
    This formula easily follows from the properties of Euler's integrals by induction.

(p.204) AI.2 More on Marcinkiewicz spaces. Lorentz spaces

From the definition of Marcinkiewicz norm in Section 1.1, formula (1.13), we immediately obtain an estimate for the distribution function of f of the form:

μ f ( k ) ( f M p k ) p .
A norm in M p (R n) is obtained by taking the infimum of the constants in this estimate:
(AI.1)
| f | M p ( R n ) = inf { C : μ f ( k ) ( C / k ) p k > 0 } .
It is easily proved that the new norm is equivalent to the old one for all 1 < p < ∞, cf. [BBC75], page 548.

In this norm the explicit function U p(x) = A |x|n/p of Sections 3.3 and 5.4 is not only the most concentrated one in the class of rearranged functions of equal norm, it is even the greatest element up to rearrangement. The existence of greatest element is an additional simplification for comparison arguments.

The same definition and norms apply when the base space is a domain of R n, or even a measurable subset K. One of the important functional features of M p(R n) or M p(Ω) is the fact that they are not separable. According to [BS88], every separable Banach function space is the closure of the set of bounded functions with support in a set of finite measure. This is not the case here, and in fact, when Ω is bounded, L p(Ω) is a non-dense subset of M p(Ω). When Ω is unbounded or R n, then L loc p ( Ω ) M p ( Ω ) is a non-dense subset.

In order to clarify this fact, we introduce the following functional in M p (Ω):

(AI.2)
N p ( f ) = lim k ( | f | k ) + M p .
Note that the limit exists since the family (|f| − k)+ is non-increasing as k → ∞. Actually, the definition only needs f to be such that (|f| − k)+M p(R n) for some k > 0.

Lemma AI.1 Np is a continuous seminorm in M p(Ω). Given fM p(Ω) we have N p ( f ) = 0 i f f M 0 p ( Ω ) , which is defined as the closure of C c 1 ( Ω ) in M p (Ω). If Ω is bounded, the converse is true.

Proof The fact that Np is a seminorm and is continuous are both clear. Let us prove next that Np(f) = 0 for every f M 0 p ( Ω ) . In the case where f is C c 1 it is bounded, and then Np (f) = 0. The result follows for M 0 p ( Ω ) by density.

On the contrary, if N p(f) = 0, for every ε > 0 there is a value k such that ∥f kMp ≤ ε with f k = (|f| − k)+ sign (f). Now, ff kL (Ω) ⊂ M p (Ω) ⊂ L p(Ω), since Ω has bounded measure. The result follows.

Note that the limit Np(f) is not zero in the case of example (1.14); actually, the limit is still Np(U p) = Ak p.

(p.205) Lorentz spaces L p, q (Ω), 1 ≤ p, q < ∞, are defined by means of the norms

f p , q = ( 0 | Ω | s q p 1 f * q ( s ) d s ) 1 / q = n 1 q ω n 1 p ( 0 R r n q p 1 ( f * ) q ( r ) d r ) 1 / q < ,
where R is defined by |B R| = |Ω|. Here, f * denotes the one-dimensional symmetric representation of a function f ∈ ℒ0 defined by means of
(AI.3)
f * ( s ) = f * ( r ) , s = ω n r n .
Then, f * is defined in the interval [0, |Ω|], with |Ω| = meas (Ω). Notice that
(AI.4)
f * ( s ) = inf { t 0 : μ ( t ) < s } ,
which makes f * a generalized inverse of μf.

It is easy to see that L p, p(Ω) coincides with L p(Ω). The L p,q(Ω) form an increasing family of separable spaces as q increases. In the limit q → ∞ the norm becomes

f p , : = ess sup { s 1 / p f * ( s ) } < .
Comparing this with (AI.1), we see that L p,∞ (Ω) = M p (Ω).

AI.3 Morrey spaces

We have seen these in the study of extinction. Here is the definition.

Definition AI.1 Let p ∈ (1, ∞) and λ ∈ (0, n). The function f L loc 1 ( Ω ) is said to belong to the Morrey space p, λ (Ω) if

(AI.5)
f L , p , λ ( Ω ) = sup ρ > 0 , x Ω { ρ λ B ρ ( x ) Ω | f | p d x } < + s ,
where B ρ(x) denotes the n-dimensional ball of radius ρ centred at the point x.

AI.4 Maximal monotone graphs

The nonlinearity in the filtration equation is in principle a C 1 monotone function, so that the equation may be written as

u t = ( ϕ ( u ) D u )
and the formal parabolic character is easy to recognize. If ϕ′(u) > 0 for every u, than the problem is actually parabolic. But the applications point to the interest in dealing with extensions of the nonlinearity to include non-strictly increasing or multi-valued functions. In order to encompass all these applications the concept of maximal monotone (p.206) graph is used, m.m.g. for short. We think of ϕ as a graph consisting of pairs (u, υ) ∈ R 2, with u in a domain D(ϕ), which is an interval of the real line. The graph is monotone if for every two graph pairs (u 1, υ1) and (u 2, υ2) we have
( u 1 u 2 ) ( υ 1 υ 2 ) 0.
It is maximal if it cannot be extended further. A continuous monotone function defined in R is a good example of m.m.g. For a discontinuous monotone function the key point is that we have to add all the vertical segments corresponding to the multi-valued options, and it has to be defined in a maximal interval D(ϕ). This interval is not necessarily R. Typical maximal monotone graphs appearing in the proof of estimates for the types of PDEs we deal with in this book are the sign function
sign ( s ) { = 1 for s > 0 , = 1 for s < 0 , = [ 1 , 1 ] for s = 0 ;
and also its positive part, denoted by sign+ (s), where we modify sign (s) so that sign+ (s) = 0 for s < 0 and sign+ (0) = [0, 1].

One of the main advantages of this generality is the fact that the inverse of a m.m.g. is again a m.m.g.; actually, both graphs are symmetric with respect to the main bisectrix in R 2. The standard and somewhat awkward notation when using multivalued operators is set inclusion, so that when (a, b) is a point in the graph ϕ we write b ∈ ϕ(a) instead of b = ϕ(a), since generally ϕ(a) is not a singleton.

Maximal monotone operators in Hilbert spaces have been studied in full detail by Brezis in [Br73]. A review of the application of maximal monotone operators to the theory of the PME is contained in Chapter 10 of [V06].