A simple proof of the generalized Leibniz rule on bounded Euclidean domains

Quoc-Hung Nguyen; Yannick Sire; Juan-Luis Vázquez

doi:10.1515/forum-2020-0228

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A simple proof of the generalized Leibniz rule on bounded Euclidean domains

Quoc-Hung Nguyen , Yannick Sire und Juan-Luis Vázquez

Veröffentlicht/Copyright: 25. September 2021

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Aus der Zeitschrift Forum Mathematicum Band 33 Heft 6

Abstract

This paper is devoted to a simple proof of the generalized Leibniz rule in bounded domains. The operators under consideration are the so-called spectral Laplacian and the restricted Laplacian. Equations involving such operators have lately been considered by Constantin and Ignatova in the framework of the SQG equation [P. Constantin and M. Ignatova, Critical SQG in bounded domains, Ann. PDE 2 2016, 2, Article ID 8] in bounded domains, and by two of the authors [Q.-H. Nguyen and J. L. Vázquez, Porous medium equation with nonlocal pressure in a bounded domain, Comm. Partial Differential Equations 43 2018, 10, 1502–1539] in the framework of the porous medium with nonlocal pressure in bounded domains. We will use the estimates in this work in a forthcoming paper on the study of porous medium equations with pressure given by Riesz-type potentials.

Keywords: Fractional Laplacian operators on domains; commutator estimates; Leibniz rule

MSC 2010: 42B37; 35J25

Communicated by Christopher D. Sogge

Funding statement: The first author is supported by the Shanghai Tech University startup fund. The third author was partially funded by Project PGC2018-098440-B-I00 from MICINN, of the Spanish Government. Also, he partially performed as an honorary professor at University Complutense de Madrid.

A Appendix

In this appendix, we provide two results supporting our conjecture on the failure of the usual form of the Leibniz rule in the case of the restricted Laplacian. Our purpose is to relate a weighted (by a suitable power of the distance function) Lp-norm for p=1,2 of the function to the Lp-norm of its fractional Laplacian. We would like to make in particular three comments:

By the very definition of the restricted Laplacian, since the functions are supported on Ω, the Leibniz rule reduces to estimate the integrals
∫ℝd(f⁢(x)-f⁢(y))⁢(g⁢(x)-g⁢(y))|x-y|d+2⁢α⁢𝑑y.
Since we are interested in estimating L2-norms in Ω, one is led to consider quantities of the type
∫Ω×Ω(f⁢(x)-f⁢(y))⁢(g⁢(x)-g⁢(y))|x-y|d+2⁢α⁢𝑑x⁢𝑑y,∫Ω×Ωc(f⁢(x)-f⁢(y))⁢(g⁢(x)-g⁢(y))|x-y|d+2⁢α⁢𝑑x⁢𝑑y.
It is by now well-known that smooth functions that are compactly supported in Ω and have finite Hα semi-norm behave like dist(x,∂Ω)α close to the boundary of Ω.
Finally, notice that there are two different ways to define a semi-norm (even in ℝd) in W˙α,p⁢(ℝd), namely
∫ℝd×ℝd|f⁢(x)-f⁢(y)|p|x-y|d+p⁢α⁢𝑑x⁢𝑑y and ∫ℝd|(-Δ)α2⁢f⁢(x)|p⁢𝑑x.
In the case of the whole space ℝd and p=2, these latter norms are equivalent. Actually, according to [24], depending on p, these spaces are ordered for every α∈(0,1) in bounded domains and they are still equivalent for p=2.

According to the previous remarks, if one seeks for a counter-example, one would need to understand how the L2-norm in ℝd of the commutator behaves with respect to its L2-norm in Ω. The following computations show that the boundary behavior plays a crucial role.

Let α∈(0,12). Let uε∈Cc∞⁢(B1⁢(0)) be a cut-off function such that uε=1 in B1-2⁢ε, uε=0 in B1-εc, and |∇⁡uε|≤C⁢ε, 0≤uε≤1. We easily get first

∫B1uε⁢(x)2(1-|x|2)2⁢α⁢𝑑x∼1 for all ⁢ε∈(0,110).

and for any 0<α<α0<12,

∫B1∫B1|uε⁢(x)-uε⁢(y)|2|x-y|d+2⁢α⁢𝑑x⁢𝑑y≲s0ε1-2⁢α0.

Indeed,

∫B1∫B1|uε⁢(x)-uε⁢(y)|2|x-y|d+2⁢α⁢𝑑x⁢𝑑y=2⁢∫B1∖B1-3⁢ε∫B1∖B1-3⁢ε|uε⁢(x)-uε⁢(y)|2|x-y|d+2⁢α⁢𝑑x⁢𝑑y+2⁢∫B1∖B1-2⁢ε∫B1-3⁢ε|1-uε⁢(y)|2|x-y|d+2⁢α⁢𝑑x⁢𝑑y

≲ε-2⁢α0⁢∫B1∖B1-3⁢ε∫B1∖B1-3⁢ε1|x-y|d-2⁢(α0-α)⁢𝑑x⁢𝑑y+∫B1∖B1-2⁢ε∫B1-3⁢ε1|x-y|d+2⁢α⁢𝑑x⁢𝑑y

≲ε-2⁢α0⁢∫B1∖B1-3⁢ε𝑑y+∫B1∖B1-2⁢εε-2⁢α⁢𝑑y

≲ε1-2⁢α0+ε1-2⁢α≲ε1-2⁢α0.

On the other hand, for any x∈B1,

(A.1)∥(-Δ)α2⁢uε∥L2⁢(B1)∼1.

Indeed, for x∈B1,

|(-Δ)α2⁢uε⁢(x)-∫B1uε⁢(x)-uε⁢(y)|x-y|d+α⁢𝑑y|=|∫B1cuε⁢(x)|x-y|d+α⁢𝑑y|≲|uε⁢(x)|(1-|x|2)α.

So,

(A.2)∫B1|(-Δ)α2⁢uε⁢(x)-∫B1uε⁢(x)-uε⁢(y)|x-y|d+α⁢𝑑y|2⁢𝑑x=|∫B1cuε⁢(x)|x-y|d+α⁢𝑑y|≲∫B1|uε⁢(x)|2(1-|x|2)2⁢α⁢𝑑x∼1.

Moreover, for α<α1<α2<12,

∫B1|∫B1uε⁢(x)-uε⁢(y)|x-y|d+α⁢𝑑y|2⁢𝑑x≲α1∫B1∫B1|uε⁢(x)-uε⁢(y)|2|x-y|d+2⁢α1⁢𝑑y⁢𝑑x≲α1,α2ε1-2⁢α2,

where the second relation follows by (2.2). Combining this with (A.2) yields (A.1). As a consequence, we have

∫B1uε⁢(x)2(1-|x|2)2⁢α⁢𝑑x∼∥(-Δ)α2⁢uε∥L2⁢(B1)2∼1,

but

∫B1∫B1|uε⁢(x)-uε⁢(y)|2|x-y|d+2⁢α⁢𝑑x⁢𝑑y→0 as ⁢ε→0.

This is an explicit example that in a bounded domain for α<12 the Hardy inequality in L2 does not hold.

However, the analogous result in L1 does hold.

Theorem A.1.

Let u∈Cc∞⁢(Ω) be a solution to

∫ℝdu⁢(x)-u⁢(y)|x-y|d+2⁢α⁢𝑑y=f⁢(x)

in Ω, where Ω is a smooth bounded domain. Then

(A.3)∫Ω|u⁢(x)|d⁢i⁢s⁢t⁢(x,∂⁡Ω)2⁢α⁢𝑑x+∥u∥W2⁢α-δ,1⁢(ℝd)≲Ω,δ∥f∥L1⁢(Ω)

for any δ∈(0,α4)

Proof.

Using Tε⁢(u⁢(x))=sign⁡(u⁢(x))⁢min⁡{ε,|u⁢(x)|} as test function, we obtain

∫ℝd∫ℝd|u⁢(x)-u⁢(y)|⁢|Tε⁢(u⁢(x))-Tε⁢(u⁢(y))||x-y|d+2⁢α⁢𝑑x⁢𝑑y=2⁢∫Tε⁢(u⁢(x))⁢f⁢(x)⁢𝑑x.

This implies

∫Ωc∫Ω|u⁢(x)|⁢|Tε⁢(u⁢(x))||x-y|d+2⁢α⁢𝑑x⁢𝑑y=∫Ωc∫Ω|u⁢(x)-u⁢(y)|⁢|Tε⁢(u⁢(x))-Tε⁢(u⁢(y))||x-y|d+2⁢α⁢𝑑x⁢𝑑y≤ε⁢∥f∥L1⁢(Ω).

So,

∫Ωc∫Ω|u⁢(x)|⁢ε-1⁢Tε⁢(|u⁢(x)|)|x-y|d+2⁢α⁢𝑑x⁢𝑑y≤∥f∥L1⁢(Ω).

Letting ε→0 yields

∫Ωc∫Ω|u⁢(x)||x-y|d+2⁢α⁢𝑑x⁢𝑑y≤∥f∥L1⁢(Ω).

Since

∫Ωc1|x-y|d+2⁢αdy∼dist(x,∂Ω)-2⁢α,

we have

(A.4)∫Ω|u⁢(x)|d⁢i⁢s⁢t⁢(x,∂⁡Ω)2⁢α⁢𝑑x≲∥f∥L1⁢(Ω).

Moreover, we can write

∫ℝdu⁢(x)-u⁢(y)|x-y|d+2⁢α⁢𝑑y=𝟏x∈Ω⁢f⁢(x)+𝟏x∉Ω⁢∫Ω-u⁢(y)|x-y|d+2⁢α⁢𝑑y:=g⁢(x)

for any x∈ℝd. Thus, by the standard regularity theory, we have

∥u∥W2⁢α-δ,1⁢(ℝd)≲Ω,δ∥g∥L1⁢(ℝd),

for any δ∈(0,α2). Since

∥g∥L1⁢(ℝd)≤∥f∥L1⁢(Ω)+∫Ωc∫Ω|u⁢(y)||x-y|d+2⁢α⁢𝑑y⁢𝑑x≲∥f∥L1⁢(Ω)+∫Ω|u⁢(y)|dist(y,∂Ω)2⁢α⁢𝑑y≲∥f∥L1⁢(Ω),

where the second relation follows by (A.4), we get (A.3). ∎

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Received: 2020-08-17

Revised: 2021-08-23

Published Online: 2021-09-25

Published in Print: 2021-11-01

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Fractional Laplacian operators on domains; commutator estimates; Leibniz rule