A Note on the Sobolev and Gagliardo--Nirenberg Inequality when 𝑝 > 𝑁

Alessio Porretta

doi:10.1515/ans-2020-2086

Open Access Published by De Gruyter April 15, 2020

A Note on the Sobolev and Gagliardo--Nirenberg Inequality when 𝑝 > 𝑁

Alessio Porretta

From the journal Advanced Nonlinear Studies

https://doi.org/10.1515/ans-2020-2086

Abstract

It is known that the Sobolev space W1,p⁢(ℝN) is embedded into LN⁢p/(N-p)⁢(ℝN) if p<N and into L∞⁢(ℝN) if p>N. There is usually a discontinuity in the proof of those two different embeddings since, for p>N, the estimate ∥u∥∞≤C⁢∥D⁢u∥pN/p⁢∥u∥p1-N/p is commonly obtained together with an estimate of the Hölder norm. In this note, we give a proof of the L∞-embedding which only follows by an iteration of the Sobolev–Gagliardo–Nirenberg estimate ∥u∥N/(N-1)≤C⁢∥D⁢u∥1. This kind of proof has the advantage to be easily extended to anisotropic cases and immediately exported to the case of discrete Lebesgue and Sobolev spaces; we give sample results in case of finite differences and finite volumes schemes.

Keywords: Sobolev spaces; Gagliardo–Nirenberg Inequality; Discrete Sobolev Inequalities

MSC 2010: 35A23

1 Introduction

Let ℝN denote the euclidean N-dimensional space, and assume that N≥2. In its basic form, the celebrated Sobolev inequality [13] asserts that, for every 1≤p<N,

(1.1) ∥ u ∥ L p * ⁢ ( ℝ N ) ≤ C S ⁢ ∥ D ⁢ u ∥ L p ⁢ ( ℝ N ) , p * = N ⁢ p N - p ,

for every C1 function u with compact support in ℝN. It is common knowledge that the inequality for p>1 can be easily deduced from the case p=1, sometimes called the Gagliardo inequality. This latter one, in its general form, reads as

(1.2) ∥ u ∥ L N N - 1 ⁢ ( ℝ N ) N ≤ ∏ i = 1 N ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L 1 ⁢ ( ℝ N )

and was proved independently by Gagliardo [9] and Nirenberg [11].

It is well known that (1.1) fails to be true if p=N and p*=∞. However, for p>N, it holds that

(1.3) ∥ u ∥ L ∞ ⁢ ( ℝ N ) ≤ K p ⁢ [ ∥ u ∥ L p ⁢ ( ℝ N ) + ∥ D ⁢ u ∥ L p ⁢ ( ℝ N ) ]

for every C1 function u with compact support in ℝN.

The usual proof of (1.3) goes together with the Morrey estimate (see e.g. [1, 7]), which states the embedding of W1,p⁢(ℝN), for p>N, into the space of Hölder continuous functions [10]. Even if the Morrey embedding gives, of course, a fundamental piece of information, it seems a natural question whether (1.3) could not be obtained itself from the scale of Sobolev inequalities, rather than as a byproduct of an estimate on the oscillation of u. This question is especially motivated by the application to discrete numerical schemes for PDEs since discrete-type Sobolev inequalities are more efficiently proved by using only scaling arguments.

The purpose of this note is to give a proof of (1.3) which relies only on the (recursive application of) Gagliardo inequality (1.2). In particular, we wish to give evidence to the following two remarks:

Inequality (1.3) can be directly obtained from (1.2) through an iteration scheme.
This approach preserves the natural form of (1.2) and easily extends, for instance, to anisotropic cases and discrete versions.

To be precise, the natural generalized form of inequality (1.2) that we prove in this paper for the case p>N is the following one.

Theorem 1.1.

Let pi, i=1⁢…⁢N, be such that ∑i=1N1pi<1. For every r≥1, there exists a constant C, only depending on pi, N and r, such that

(1.4) ∥ u ∥ L ∞ ⁢ ( ℝ N ) ≤ C ( ∏ i = 1 N ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p i ⁢ ( ℝ N ) ) θ N ∥ u ∥ L r ⁢ ( ℝ N ) 1 - θ , θ : = N N + r ⁢ ( 1 - ∑ i = 1 N 1 p i ) ,

for every C1 function u with compact support in RN.

As we said before, the interest here lies in the proof of (1.4), which is obtained with an elementary iteration from the case p=1, using only algebraic steps and Hölder inequalities. In this approach, inequality (1.3) for the case p>N follows in the same spirit as inequality (1.1) for the case p<N, up to replacing a finite with an infinite iteration. In fact, in order to get at the sup-norm, one needs a limit as q→∞ of the embedding in Lq⁢(ℝN), obtained by applying (1.2) to increasing powers of u, like in Moser-type iterations. The convergence of the iteration and scaling arguments, which fix the precise form of the embedding estimate, are the only ingredients required. In particular, compared to the usual proof (see [1, 7]), we do not make any use of the geometry of the underlying euclidean space since we only rely on the starting inequality (1.2).

Apart from the pedagogical interest of this proof, we think that it may have an interest in cases where the structure of the state space is more complex than the euclidean flat case. As a motivation, and an application of our approach, we consider the case of discrete-type inequalities which are needed in numerical schemes for partial differential equations.

An extensive literature now exists about discrete-type Sobolev inequalities; we mention in particular [3, 4, 6, 8] (and many other references cited therein) for the case of finite volumes. However, the case p>N, p*=∞ is often outside the range of those results, despite the fact that, in numerical analysis, the case of low dimension (N=2, N=3) is very relevant. At the end of this note, we provide a discrete-type version of (1.3) for finite volume schemes, see Theorem 4.1, which is obtained as in the continuous case with an iteration method. Notice that, in the finite volumes setting, it is very convenient to start with inequality (1.2), applied to BV functions, a very natural frame for piecewise constant functions. We hope that the kind of discrete Gagliardo–Nirenberg inequality proved in Theorem 4.1 may have an interest for people working in numerical analysis: indeed, a special case for N=2 was needed in our recent paper [12], for a finite difference scheme used to show numerical hypocoercivity of the Kolmogorov equation.

2 The Iteration Scheme

We start by showing that the iteration of the Gagliardo–Nirenberg–Sobolev inequality (1.2) is convergent and leads to the estimate of the sup-norm. This is the main technical step in our approach.

Lemma 2.1.

Let p>N. For every r≥1, there exist constants α,β,C, only depending on p, N and r, such that

(2.1) ∥ u ∥ L ∞ ⁢ ( ℝ N ) ≤ C ⁢ ( ∏ i = 1 N ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p ⁢ ( ℝ N ) ) α ⁢ ∥ u ∥ L r ⁢ ( ℝ N ) β

for every C1 function u with compact support in RN.

Proof.

Given γ>p, we apply (1.2) to uγ obtaining

(2.2) ∥ u γ ∥ L N N - 1 ⁢ ( ℝ N ) N ≤ γ N ⁢ ∏ i = 1 N ∥ u γ - 1 ⁢ ∂ ⁡ u ∂ ⁡ x i ∥ L 1 ⁢ ( ℝ N ) ≤ γ N ⁢ ( ∏ i = 1 N ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p ⁢ ( ℝ N ) ) ⁢ ( ∫ u ( γ - 1 ) ⁢ p ′ ⁢ d ⁢ x ) N p ′ .

Since γ>p and p>N, we have γ<(γ-1)⁢p′<γ⁢NN-1, so we interpolate

( ∫ u ( γ - 1 ) ⁢ p ′ ⁢ d ⁢ x ) N p ′ ≤ ( ∫ u γ ⁢ N N - 1 ⁢ d ⁢ x ) θ ⁢ N p ′ ⁢ ( ∫ u γ ) N ⁢ ( 1 - θ ) p ′ ,

where (γ-1)⁢p′=θ⁢γ⁢NN-1+(1-θ)⁢γ, which means

θ = ( γ - 1 ) ⁢ p ′ - γ γ ⁢ 1 N - 1 .

We deduce from (2.2)

( ∫ u γ ⁢ N N - 1 ⁢ d ⁢ x ) N - 1 - θ ⁢ N p ′ ≤ γ N ⁢ ( ∏ i = 1 N ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p ⁢ ( ℝ N ) ) ⁢ ( ∫ u γ ) N ⁢ ( 1 - θ ) p ′ ,

which yields

(2.3) ( ∫ u γ ⁢ N N - 1 ⁢ d ⁢ x ) N - 1 γ ⁢ N ≤ { γ N ⁢ ( ∏ i = 1 N ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p ⁢ ( ℝ N ) ) } ( N - 1 ) ⁢ p ′ γ ⁢ N ⁢ ( p ′ ⁢ ( N - 1 ) - θ ⁢ N ) ⁢ ( ∫ u γ ) ( N - 1 ) ⁢ ( 1 - θ ) γ ⁢ ( p ′ ⁢ ( N - 1 ) - θ ⁢ N ) .

We use the value of θ in terms of γ,N,p and the two exponents in the right term become respectively

( N - 1 ) ⁢ p ′ γ ⁢ N ⁢ ( p ′ ⁢ ( N - 1 ) - θ ⁢ N ) = p ′ N ⁢ [ 1 γ ⁢ ( N - ( N - 1 ) ⁢ p ′ ) + N ⁢ p ′ ] = 1 N ⁢ [ 1 γ ⁢ ( 1 - N p ) + N ] ,

( N - 1 ) ⁢ ( 1 - θ ) γ ⁢ ( p ′ ⁢ ( N - 1 ) - θ ⁢ N ) = 1 γ ⁢ [ γ ⁢ ( N - ( N - 1 ) ⁢ p ′ ) + ( N - 1 ) ⁢ p ′ γ ⁢ ( N - ( N - 1 ) ⁢ p ′ ) + N ⁢ p ′ ] = 1 γ ⁢ [ 1 - 1 γ ⁢ ( 1 - N p ) + N ] .

Henceforth, we set the iteration scheme. For r>p, we define the recursive sequence

{ γ n = ( N N - 1 ) ⁢ γ n - 1 , γ 0 = r ↝ γ n = r ( N N - 1 ) n ,

and we define

σ n : = 1 γ n ⁢ ( 1 - N p ) + N .

With the above notations, using (2.3) with γ=γn-1, we have

(2.4) ∥ u ∥ L γ n ⁢ ( ℝ N ) ≤ { γ n - 1 N ⁢ ( ∏ i = 1 N ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p ⁢ ( ℝ N ) ) } σ n - 1 N ⁢ ∥ u ∥ L γ n - 1 ⁢ ( ℝ N ) 1 - σ n - 1 ,

which holds for every n≥1. To shorten notations, we define

C n - 1 : = { γ n - 1 N ( ∏ i = 1 N ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p ⁢ ( ℝ N ) ) } σ n - 1 N

so that (2.4) takes the form

∥ u ∥ L γ n ⁢ ( ℝ N ) ≤ C n - 1 ⁢ ∥ u ∥ L γ n - 1 ⁢ ( ℝ N ) 1 - σ n - 1 for all ⁢ n ≥ 1 .

We can now iterate this estimate, and we get

∥ u ∥ L γ n ⁢ ( ℝ N ) ≤ C n - 1 ⁢ ∥ u ∥ L γ n - 1 ⁢ ( ℝ N ) 1 - σ n - 1 ≤ C n - 1 ⁢ ( C n - 2 ⁢ ∥ u ∥ L γ n - 2 ⁢ ( ℝ N ) 1 - σ n - 2 ) 1 - σ n - 1 ≤ C n - 1 ⁢ C n - 2 1 - σ n - 1 ⁢ ( C n - 3 ⁢ ∥ u ∥ L γ n - 3 ⁢ ( ℝ N ) 1 - σ n - 3 ) ( 1 - σ n - 1 ) ⁢ ( 1 - σ n - 2 ) ≤ C n - 1 ⁢ C n - 2 1 - σ n - 1 ⁢ C n - 3 ( 1 - σ n - 1 ) ⁢ ( 1 - σ n - 2 ) ⁢ … ⁢ ∥ u ∥ r ∏ j = 0 n - 1 ( 1 - σ j ) .

Finally, we deduce the estimate

(2.5) ∥ u ∥ L γ n ⁢ ( ℝ N ) ≤ C n - 1 ⁢ ∏ k = 0 n - 2 C k ∏ j = k + 1 n - 1 ( 1 - σ j ) ⁢ ∥ u ∥ r ∏ j = 0 n - 1 ( 1 - σ j ) for all ⁢ n ≥ 1 .

We observe that

∏ j = k n - 1 ( 1 - σ j ) = exp ⁡ ( ∑ j = k n - 1 log ⁡ ( 1 - σ j ) ) ,

and since, by definition of σn and γn,

log ⁡ ( 1 - σ j ) ∼ - σ j ∼ - 1 r ⁢ ( 1 - N p ) ⁢ ( N N - 1 ) - j ,

the above sum is convergent and there exists some c0>0 such that

(2.6) 0 < c 0 ≤ ∏ j = k n - 1 ( 1 - σ j ) ≤ 1 for all ⁢ k ≤ n - 1 ⁢ and all ⁢ n ≥ 1 .

We recall that, by definition of Ck and γk, we have

C k = { γ k N ( ∏ i = 1 N ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p ⁢ ( ℝ N ) ) } σ k N = A σ k ( N N - 1 ) k ⁢ σ k , where A : = r ( ∏ i = 1 N ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p ⁢ ( ℝ N ) ) 1 N .

Hence

∏ k = 0 n - 2 C k ∏ j = k + 1 n - 1 ( 1 - σ j ) = ( A ) ∑ k = 0 n - 2 σ k ⁢ ∏ j = k + 1 n - 1 ( 1 - σ j ) ⁢ ( N N - 1 ) ∑ k = 0 n - 2 k ⁢ σ k ⁢ ∏ j = k + 1 n - 1 ( 1 - σ j ) .

Using (2.6) and σk=O⁢((NN-1)-k), we notice that the above quantity is bounded uniformly with respect to n. Therefore, from (2.5), we deduce that

lim sup n → ∞ ⁡ ∥ u ∥ L γ n ⁢ ( ℝ N ) ≤ C ⁢ ( ∏ i = 1 N ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p ⁢ ( ℝ N ) ) α ⁢ ∥ u ∥ L r ⁢ ( ℝ N ) β

for some positive constants C,α,β only depending on r,p,N. Since u has compact support, the left-hand side converges to ∥u∥L∞⁢(ℝN) and (2.1) is proved, at least for r>p. In addition, the above iteration clearly shows that the exponent β is smaller than one (see (2.6)). Since, for any s<r, we have

∥ u ∥ r ≤ ∥ u ∥ s s r ⁢ ∥ u ∥ ∞ 1 - s r ,

the estimate is immediately extended to all Lebesgue spaces Ls⁢(ℝN), with values s≤p, with an estimate

∥ u ∥ L ∞ ⁢ ( ℝ N ) ≤ C ⁢ ( ∏ i = 1 N ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p ⁢ ( ℝ N ) ) α ⁢ ∥ u ∥ L s ⁢ ( ℝ N ) β

for possibly different C,α,β. ∎

As we now show, there is no need of a difficult inspection of the above iteration argument in order to detect the values of α and β. Once inequality (2.1) is obtained, the precise value of α and β can be easily found through scaling arguments.

Lemma 2.2.

Assume that (2.1) holds true. Then we have

(2.7) α = p r ⁢ ( p - N ) + N ⁢ p 𝑎𝑛𝑑 β = r ⁢ ( p - N ) r ⁢ ( p - N ) + N ⁢ p .

Proof.

First of all, replacing u with λ⁢u, λ>0, we deduce from (2.1) that

(2.8) N ⁢ α + β = 1 .

Secondly, we take u=uR:=ζ(xR), where ζ∈Cc1⁢(ℝN) is a cut-off function such that 0≤ζ≤1, ζ⁢(x)≡0 if |x|>2 and ζ⁢(x)≡1 if |x|≤1. Applying (2.1) to uR, we get

1 ≤ C ⁢ ( ∏ i = 1 N ∥ ∂ ⁡ ζ ∂ ⁡ x i ∥ L p ⁢ ( ℝ N ) ) α ⁢ ∥ ζ ∥ L r ⁢ ( ℝ N ) β ⁢ R N ⁢ β r ⁢ R N ⁢ α ⁢ ( N p - 1 ) ,

and since R is arbitrary, this implies

(2.9) β r + α ⁢ ( N p - 1 ) = 0 .

Putting together (2.8) and (2.9) gives (2.7). ∎

Collecting the above lemmas, we deduce the embedding of W1,p⁢(ℝN)∩Lr⁢(ℝN) (r≥1) into L∞⁢(ℝN).

Corollary 2.1.

Let p>N. For every r≥1, there exists a constant C, only depending on p, N and r, such that

(2.10) ∥ u ∥ L ∞ ⁢ ( ℝ N ) ≤ C ⁢ ( ∏ i = 1 N ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p ⁢ ( ℝ N ) ) p r ⁢ ( p - N ) + N ⁢ p ⁢ ∥ u ∥ L r ⁢ ( ℝ N ) r ⁢ ( p - N ) r ⁢ ( p - N ) + N ⁢ p

for every C1 function u with compact support in RN.

Remark 2.1.

If we simply use ∥∂⁡u∂⁡xi∥Lp⁢(ℝN)≤∥D⁢u∥Lp⁢(ℝN), then (2.10) implies

(2.11) ∥ u ∥ L ∞ ⁢ ( ℝ N ) ≤ C ⁢ ∥ D ⁢ u ∥ L p ⁢ ( ℝ N ) N ⁢ p r ⁢ ( p - N ) + N ⁢ p ⁢ ∥ u ∥ L r ⁢ ( ℝ N ) r ⁢ ( p - N ) r ⁢ ( p - N ) + N ⁢ p ,

which is one standard form of the so-called Gagliardo–Nirenberg inequality.

Remark 2.2.

If r=p, then (2.10) reads as

∥ u ∥ L ∞ ⁢ ( ℝ N ) ≤ C ⁢ ( ∏ i = 1 N ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p ⁢ ( ℝ N ) ) 1 p ⁢ ∥ u ∥ L p ⁢ ( ℝ N ) p - N p ,

which implies (1.3) by Young’s inequality.

3 The Anisotropic Case

The same strategy used before can be applied to the more general anisotropic case.

Lemma 3.1.

Let pi, i=1⁢…⁢N, be such that ∑i=1N1pi<1. For every r≥1, there exist constants α,β,C, only depending on pi, N and r, such that

(3.1) ∥ u ∥ L ∞ ⁢ ( ℝ N ) ≤ C ⁢ ( ∏ i = 1 N ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p i ⁢ ( ℝ N ) ) α ⁢ ∥ u ∥ L r ⁢ ( ℝ N ) β

for every C1 function u with compact support in RN.

Proof.

Let {si}1≤i≤N be real numbers and s:=∑i=1Nsi. We start from the inequality

(3.2) ∥ u ∥ L s N - 1 ⁢ ( ℝ N ) s ≤ ∏ i = 1 N s i ⁢ ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p i ⁢ ( ℝ N ) ⁢ ∥ u ∥ L ( s i - 1 ) ⁢ p i ′ ⁢ ( ℝ N ) s i - 1

Inequality (3.2) is the usual starting point for the anisotropic Sobolev inequality; see e.g. [2, 14]. For the sake of completeness, let us recall how this is obtained, using once more the Gagliardo argument. Since

we have

| u ⁢ ( x ) | s N - 1 ≤ ∏ i = 1 N f i ⁢ ( x - i ) ,

where x-i:=(x1,…,xi-1,xi+1,…) and

f i : = s i 1 N - 1 ( ∫ ℝ | ∂ ⁡ u ∂ ⁡ x i ( x ) | p i d x i ) 1 p i ⁢ ( N - 1 ) ( ∫ ℝ | u ( x ) | ( s i - 1 ) ⁢ p i ′ d x i ) 1 p i ′ ⁢ ( N - 1 ) .

Applying the Gagliardo inequality (see e.g. [5])

∥ ∏ i = 1 N f i ∥ L 1 ⁢ ( ℝ N ) ≤ ∏ i = 1 N ∥ f i ∥ L 1 ⁢ ( ℝ N - 1 ) ,

we get

∫ | u ⁢ ( x ) | s N - 1 ⁢ d ⁢ x ≤ ∏ i = 1 N s i 1 N - 1 ⁢ ∫ ℝ N - 1 ( ∫ ℝ | ∂ ⁡ u ∂ ⁡ x i ⁢ ( x ) | p i ⁢ d ⁢ x i ) 1 p i ⁢ ( N - 1 ) ⁢ ( ∫ ℝ | u ⁢ ( x ) | ( s i - 1 ) ⁢ p i ′ ⁢ d ⁢ x i ) 1 p i ′ ⁢ ( N - 1 ) ⁢ d ⁢ x - i

which implies, by the Hölder inequality,

∫ | u ⁢ ( x ) | s N - 1 ⁢ d ⁢ x ≤ ∏ i = 1 N s i 1 N - 1 ⁢ ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p i ⁢ ( ℝ N ) 1 N - 1 ⁢ ∥ | u | s i - 1 ∥ L p i ′ ⁢ ( ℝ N ) 1 N - 1 .

This proves (3.2). Now we choose {si} so that the N-1 conditions

(3.3) ( s i - 1 ) ⁢ p i ′ = ( s j - 1 ) ⁢ p j ′ for all ⁢ i ≠ j

are satisfied. Thanks to (3.3), and since ∑i1pi<1, we claim that

(3.4) s N < ( s i - 1 ) ⁢ p i ′ < s N - 1 for all ⁢ i = 1 , … , N

provided

(3.5) s i > 1 + N p i ′ ⁢ ∑ j = 1 N 1 p j .

Indeed, using (3.3), and the standard conjugate relation for pj′, we have

(3.6) s = s i + ∑ j ≠ i ( s j - 1 ) + N - 1 = s i + ( s i - 1 ) ⁢ p i ′ ⁢ ∑ j ≠ i 1 p j ′ + N - 1 = s i + ( s i - 1 ) ⁢ p i ′ ⁢ ∑ j ≠ i ( 1 - 1 p j ) + N - 1 = s i + ( N - 1 ) ⁢ ( 1 + ( s i - 1 ) ⁢ p i ′ ) - ( s i - 1 ) ⁢ p i ′ ⁢ ∑ j ≠ i 1 p j = ( N - 1 ) ⁢ ( s i - 1 ) ⁢ p i ′ + s i ⁢ ( 1 - p i ′ ⁢ ∑ j ≠ i 1 p j ) + N - 1 + p i ′ ⁢ ∑ j ≠ i 1 p j .

Since the condition ∑j1pj<1 implies pi′⁢∑j≠i1pj<1, we immediately deduce that

s > ( N - 1 ) ⁢ ( s i - 1 ) ⁢ p i ′ ,

which gives the right-hand inequality in (3.4). In addition, we compute

s - N ⁢ ( s i - 1 ) ⁢ p i ′ = - s i ⁢ ( p i ′ - 1 + p i ′ ⁢ ∑ j ≠ i 1 p j ) + p i ′ + N - 1 + p i ′ ⁢ ∑ j ≠ i 1 p j = - s i ⁢ p i ′ ⁢ ∑ j 1 p j + N + p i ′ ⁢ ∑ j 1 p j ,

and therefore s<N⁢(si-1)⁢pi′ provided (3.5) holds true.

Let us suppose, by now, that (3.5) is satisfied. Then we have shown that (3.4) holds true, and we can use the interpolation inequality to deduce from (3.2)

(3.7) ∥ u ∥ L s N - 1 ⁢ ( ℝ N ) s ≤ ∏ i = 1 N s i ⁢ ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p i ⁢ ( ℝ N ) ⁢ ( ∫ | u | s N ) 1 - θ i p i ′ ⁢ ( ∫ | u | s N - 1 ) θ i p i ′ ,

where (si-1)⁢pi′=θi⁢sN-1+(1-θi)⁢sN, which means

θ i = N - 1 s ⁢ [ N ⁢ ( s i - 1 ) ⁢ p i ′ - s ] .

Simplifying (3.7), we obtain

( ∫ | u | s N - 1 ) N - 1 - ∑ i θ i p i ′ ≤ { ∏ i = 1 N s i ⁢ ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p i ⁢ ( ℝ N ) } ⁢ ( ∫ | u | s N ) ∑ i 1 - θ i p i ′ .

Since

∑ θ i p i ′ = N - 1 s ⁢ ( s ⁢ ∑ j 1 p j - N 2 ) , ∑ 1 - θ i p i ′ = N s ⁢ [ s ⁢ ( 1 - ∑ j 1 p j ) + N ⁢ ( N - 1 ) ] ,

we get

(3.8) ∥ u ∥ L s N - 1 ⁢ ( ℝ N ) ≤ { ∏ i = 1 N s i ⁢ ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p i ⁢ ( ℝ N ) } 1 s ⁢ ( 1 - ∑ j 1 p j ) + N 2 ⁢ ∥ u ∥ L s N ⁢ ( ℝ N ) s ⁢ ( 1 - ∑ j 1 p j ) + N ⁢ ( N - 1 ) s ⁢ ( 1 - ∑ j 1 p j ) + N 2 .

We set henceforth

γ n : = r ( N N - 1 ) n ,

and we apply (3.8) with s=N⁢γn-1. Notice that the N-1 conditions (3.3) and the choice s=N⁢γn-1 yield a unique choice of the {si}i=1,…,N used above^[1]. In addition, we have that si depends on n and goes to infinity as n→∞; this makes sure that condition (3.5) is satisfied for n large. Indeed, by taking r sufficiently large, we can suppose that this condition holds for all n≥1.

With the above choice, (3.8) reads as the recursive estimate

∥ u ∥ L γ n ⁢ ( ℝ N ) ≤ { ∏ i = 1 N s i ⁢ ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p i ⁢ ( ℝ N ) } σ n - 1 N ⁢ ∥ u ∥ L γ n - 1 ⁢ ( ℝ N ) 1 - σ n - 1 for all ⁢ n ≥ 1 ,

where we have set

σ n : = 1 γ n ⁢ ( 1 - ∑ j 1 p j ) + N .

Defining as well

C n - 1 : = { ∏ i = 1 N s i ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p i ⁢ ( ℝ N ) } σ n - 1 N ,

the above estimate takes the form

∥ u ∥ L γ n ⁢ ( ℝ N ) ≤ C n - 1 ⁢ ∥ u ∥ L γ n - 1 ⁢ ( ℝ N ) 1 - σ n - 1 for all ⁢ n ≥ 1 ,

exactly as in Lemma 2.1. Therefore, we obtain again the estimate

(3.9) ∥ u ∥ L γ n ⁢ ( ℝ N ) ≤ C n - 1 ⁢ ∏ k = 0 n - 2 C k ∏ j = k + 1 n - 1 ( 1 - σ j ) ⁢ ∥ u ∥ r ∏ j = 0 n - 1 ( 1 - σ j ) for all ⁢ n ≥ 1 ,

and we conclude with similar arguments: as in the isotropic case, there exists c0>0 such that

0 < c 0 ≤ ∏ j = k n - 1 ( 1 - σ j ) ≤ 1 for all ⁢ k ≤ n - 1 ⁢ and all ⁢ n ≥ 1 ,

and since we have si≤θ⁢γn-1 for some constant θ only depending on pi,N, we estimate

C n - 1 ≤ { ( θ ⁢ γ n - 1 ) N ⁢ ∏ i = 1 N ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p i ⁢ ( ℝ N ) } σ n - 1 N .

Therefore, we conclude, exactly as in Lemma 2.1, that there exist constants α,C>0 such that

C n - 1 ⁢ ∏ k = 0 n - 2 C k ∏ j = k + 1 n - 1 ( 1 - σ j ) ≤ C ⁢ { ∏ i = 1 N ∥ ∂ ⁡ u ∂ ⁡ x i ∥ L p i ⁢ ( ℝ N ) } α ,

and passing to the limit in (3.9), we finally obtain (3.1), at least for r sufficiently large.

The conclusion is then extended to any value r≥1 as in Lemma 2.1, with a straightforward interpolation argument. ∎

Finally, the same proof as in Lemma 2.2 gives the unique values of α,β. Thus the proof of Theorem 1.1 is concluded.

4 Discrete Inequalities

Discrete inequalities similar to those proved in the previous sections can be obtained with the same approach. In order to keep things simpler, we only consider the isotropic case as in Section 2. We start with the easier case of finite difference schemes.

4.1 Finite Difference Schemes

A discrete inequality as (2.10) was obtained in [12] for a finite difference scheme approximating the Kolmogorov equation in ℝ2. In that situation, we proved that, for any p>2 and r≥1, there exists a constant C, only depending on p,r, such that

∥ f ∥ ∞ ≤ C ⁢ ( ∥ D x ⁢ f ∥ p ⁢ ∥ D y ⁢ f ∥ p ) p r ⁢ ( p - 2 ) + 2 ⁢ p ⁢ ∥ f ∥ r r ⁢ ( p - 2 ) r ⁢ ( p - 2 ) + 2 ⁢ p

for every f∈ℓhs⁢(ℝ2) with s=min⁡(2,r), where x,y are coordinates in ℝ2, f=(fi⁢j) is a function defined on the scheme and Dx, Dy are finite difference versions of the derivatives.

Here we give an extension of this kind of estimate to N-dimensional finite difference schemes. To this purpose, let us consider a uniform grid on ℝN with mesh step h, and let P=(i1⁢h,…,iN⁢h) denote a generic point in ℝhN, with i1,…,iN∈ℤ. The values of a function f at P are denoted by fi1,…,iN, and the natural Lebesgue space ℓhp is defined as

f ∈ ℓ h p ⇔ ∥ f ∥ ℓ h p : = ( h N ∑ i 1 , … , i N | f i 1 , … , i N | p ) 1 p < ∞ .

For a function f defined on the scheme, the discrete derivatives can be defined, for example, as follows:

D x k f ( P ) : = f i 1 , … , i k - 1 , i k , i k + 1 , … , i N - f i 1 , … , i k - 1 , i k - 1 , i k + 1 , … , i N h .

Of course, right-sided derivatives, or centered derivatives, could be used alternatively.

A discrete Gagliardo inequality can be the starting point here. Assume that f has compact support. Since, in any direction k, one has

| f i 1 , … , i N | ≤ ∑ j = - ∞ i k | f i 1 , … , i k - 1 , j , i k + 1 , … , i N - f i 1 , … , i k - 1 , j - 1 , i k + 1 , … , i N | ,

by taking power 1N-1 and N copies of this inequality in the N directions, one obtains

| f i 1 , … , i N | N N - 1 ≤ ∏ k = 1 N ( ∑ j = - ∞ ∞ | f i 1 , … , i k - 1 , j , i k + 1 , … , i N - f i 1 , … , i k - 1 , j - 1 , i k + 1 , … , i N | ) 1 N - 1 .

Then, exactly as in the continuous case, integrating and using the Hölder inequality (and scaling the powers of h), one gets

(4.1) ∥ f ∥ ℓ h N N - 1 N ≤ ∏ k = 1 N ∥ D x k ⁢ f ∥ ℓ h 1

for any f which has compact support. Inequality (4.1) is the discrete equivalent of (1.2). Now, applying (4.1) to |f|γ and using that

| D x k | f | γ | ≤ γ ( | f | γ - 1 + | f i 1 , … , i k - 1 , i k - 1 , i k + 1 , … , i N | γ - 1 ) | D x k f | ,

one obtains with the Hölder inequality

∥ f ∥ ℓ h γ ⁢ N N - 1 γ ≤ c ⁢ γ ⁢ ( ∏ k = 1 N ∥ D x k ⁢ f ∥ ℓ h p ) 1 N ⁢ ∥ f ∥ ℓ h ( γ - 1 ) ⁢ p ′ γ - 1 .

The iteration scheme then follows exactly as in Lemma 2.1 and provides with the inequality

∥ f ∥ ∞ ≤ c ⁢ ( ∏ k = 1 N ∥ D x k ⁢ f ∥ ℓ h p ) α ⁢ ∥ f ∥ ℓ h r β

for some α,β>0 and for every f with compact support. The consistency of the scheme implies that the same inequality be true for C1 functions with compact support, so the values α,β are fixed once again by Lemma 2.2. Finally, we end up with the inequality

(4.2) ∥ f ∥ ∞ ≤ c ( ∏ k = 1 N ∥ D x k f ∥ ℓ h p ) θ N ∥ f ∥ ℓ h r 1 - θ , θ : = N ⁢ p N ⁢ p + r ⁢ ( p - N ) ,

which is proved to hold (with a standard density argument) for every f∈ℓhs, s=min⁡(p,r).

Remark 4.1.

We point out that different choices could as well be done for the discrete derivatives Dxk in different directions. The proof adapts easily, for example, to left, right or centered choices. For instance, in [12], we obtained (4.2) using the choice Dxf:=fi,j-fi-1,jh, Dyf:=fi,j+1-fi,j-12⁢h in order to match the anisotropic behavior of the Kolmogorov equation ∂t⁡f-∂x⁢x⁡f-x⁢∂y⁡f=0 in ℝ2.

4.2 Finite Volume Schemes

Let us now show similar discrete inequalities in the more general context of finite volume schemes. We follow here [8] for the reference functional setting. An admissible mesh of ℝN is given by:

A family ℳ of control volumes, denoted by K, which are bounded convex polyhedral subsets of ℝN with positive measure, and such that ℳ realizes a locally finite partition of ℝN.
A family ℰ of relatively open subsets of hyperplanes of ℝN, denoted by σ, with positive N-1 measure, which represent the faces of each volume K. Indeed, for each K, there exists ℰK⊂ℰ such that ∂⁡K=⋃σ∈ℰKσ¯, and ℰ=⋃K∈ℳℰK. We also assume that the cardinality of ℰK is uniformly bounded for all K∈ℳ (i.e. there is a uniform bound on the number of faces of the control volumes).
A family of points {xK}K∈ℳ such that xK belongs to the interior of K, and for any two neighboring cells K,L, the line through xK,xL, denoted by [xK,xL], intersects and is orthogonal to the face σK⁢L:=∂K∩∂L.

The size of the mesh is defined as h:=supK∈ℳdiam(K) and supposed to be finite. For any xK,xL, their distance is denoted as

d σ : = | x K - x L | for σ = σ K ⁢ L .

Notice that the above condition on the line [xK,xL] implies dσ≥d⁢(xK,σK⁢L). The mesh is called regular if those distances are equivalent (uniformly in the mesh), namely,

(4.3) there exists ⁢ c 0 > 0 ⁢ such that ⁢ d σ ≤ c 0 ⁢ d ⁢ ( x K , σ ) ⁢ for all ⁢ σ ∈ ℰ K ⁢ and all ⁢ K ∈ ℳ .

Functions u defined on the scheme are nothing but a collection of real numbers (uK)K∈ℳ, and clearly identified with the space of measurable functions in ℝN which are piecewise constant on ℳ. In particular, we set

ℓ p ( ℳ ) : = { u = ∑ K ∈ ℳ u K 𝟏 K , u K ∈ ℝ : ∑ K ∈ ℳ | u K | p | K | < ∞ }

with its natural norm

∥ u ∥ ℓ p : = ( ∑ K ∈ ℳ | u K | p | K | ) 1 p if 1 ≤ p < ∞ , ∥ u ∥ ℓ ∞ : = sup K ∈ ℳ | u K | .

The discrete version of the seminorm of W1,p is given by the following:

(4.4) | u | 1 , p : = ( ∑ σ ∈ ℰ , σ = K | L | σ | d σ ( | u K - u L | d σ ) p ) 1 p ,

where, here and later, we denote by |⋅| the Lebesgue measure, used for both N-1 and N-dimensional sets, which will be clear in the context. In (4.4), we assume, without loss of generality, that dσ>0 for all σ=K|L (see also [8]).

We now establish a discrete version of (2.11). The following result may be seen as a complement of similar embedding estimates proved in [3, 4] for p<N.

Theorem 4.1.

Let M be a discrete regular mesh as defined above. Let p>N. For any r≥1, there exists a constant C, only depending on p,r,N, and c0 given by (4.3), such that

∥ u ∥ ℓ ∞ ≤ C | u | 1 , p θ ∥ u ∥ ℓ r 1 - θ , θ : = N ⁢ p r ⁢ ( p - N ) + N ⁢ p ,

for every u∈ℓq, q=min⁡(p,r).

Proof.

Without loss of generality, we assume that u has compact support, i.e. uK≠0 for only a finite number of K∈ℳ (the general case is recovered by density). We recall the Sobolev inequality for functions of bounded variation

∥ v ∥ L N N - 1 ⁢ ( ℝ N ) ≤ C N ⁢ ∥ v ∥ B ⁢ V ⁢ ( ℝ N ) ,

where CN only depends on N. If applied to u∈ℓ1⁢(ℳ) (piecewise constant functions, compactly supported, belong to B⁢V⁢(ℝN)), the previous inequality reads as

∥ u ∥ ℓ N N - 1 ≤ C N ⁢ 1 2 ⁢ ∑ K ∈ ℳ ∑ σ ∈ ℰ K , σ = K | L | σ K ⁢ L | ⁢ | u K - u L | .

We apply this inequality to |u|γ and use that ||uK|γ-|uL|γ|≤γ⁢(|uK|γ-1+|uL|γ-1)⁢|uK-uL|. We get

∥ u ∥ ℓ γ ⁢ N N - 1 γ ≤ C N ⁢ γ 2 ⁢ ∑ K ∈ ℳ ∑ σ ∈ ℰ K , σ = K | L | σ K ⁢ L | ⁢ ( | u K | γ - 1 + | u L | γ - 1 ) ⁢ | u K - u L | .

Assuming γ>p and using the Hölder inequality, we obtain (we shorten the notations in the summation index, where σ=K|L is omitted)

∥ u ∥ ℓ γ ⁢ N N - 1 γ ≤ C N ⁢ γ 2 ⁢ ( ∑ K ∈ ℳ ∑ σ ∈ ℰ K | σ | ⁢ d σ ⁢ ( | u K | γ - 1 + | u L | γ - 1 ) p ′ ) 1 p ′ ⁢ ( ∑ K ∈ ℳ ∑ σ ∈ ℰ K | σ | ⁢ d σ 1 - p ⁢ | u K - u L | p ) 1 p .

Using condition (4.3) on the mesh, and the definition of discrete Sobolev seminorm, we deduce that

∥ u ∥ ℓ γ ⁢ N N - 1 γ ≤ C ⁢ c 0 1 p ′ ⁢ γ ⁢ ( ∑ K ∈ ℳ ∑ σ ∈ ℰ K | σ | ⁢ d ⁢ ( x K , σ ) ⁢ | u K | ( γ - 1 ) ⁢ p ′ ) 1 p ′ ⁢ | u | 1 , p ≤ C ⁢ c 0 1 p ′ ⁢ γ ⁢ ( ∑ K ∈ ℳ | K | ⁢ | u K | ( γ - 1 ) ⁢ p ′ ) 1 p ′ ⁢ | u | 1 , p ,

where C denotes possibly different constants only depending on N,p. Thus we obtain

∥ u ∥ ℓ γ ⁢ N N - 1 γ ≤ C ⁢ c 0 1 p ′ ⁢ γ ⁢ ∥ u ∥ ℓ ( γ - 1 ) ⁢ p ′ γ - 1 ⁢ | u | 1 , p ,

which is the discrete equivalent of (2.2) in Lemma 2.1. Starting from this inequality, the iteration scheme can be applied without changes; eventually, this leads to the estimate

∥ u ∥ l ∞ ≤ C ^ ⁢ | u | 1 , p α ⁢ ∥ u ∥ ℓ r β

for some α,β>0, and some C^ depending on N,p,r,c0. The consistency of the scheme implies that the same inequality should hold for smooth functions with compact support; hence the values of α,β>0 are fixed by Lemma 2.2. ∎

Dedicated to Laurent Véron, a mathematical gentleman, with esteem and friendship

Communicated by Julián López-Gómez and Patrizia Pucci

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Received: 2019-12-29

Accepted: 2020-02-28

Published Online: 2020-04-15

Published in Print: 2020-05-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

A Note on the Sobolev and Gagliardo--Nirenberg Inequality when 𝑝 > 𝑁

Abstract

1 Introduction

Theorem 1.1.

2 The Iteration Scheme

Lemma 2.1.

Proof.

Lemma 2.2.

Proof.

Corollary 2.1.

Remark 2.1.

Remark 2.2.

3 The Anisotropic Case

Lemma 3.1.

Proof.

4 Discrete Inequalities

4.1 Finite Difference Schemes

Remark 4.1.

4.2 Finite Volume Schemes

Theorem 4.1.

Proof.

References

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