Addendum: Local Elliptic Regularity for the Dirichlet Fractional Laplacian

Umberto Biccari; Mahamadi Warma; Enrique Zuazua

doi:10.1515/ans-2017-6020

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Addendum: Local Elliptic Regularity for the Dirichlet Fractional Laplacian

Ein Erratum zu diesem Artikel finden Sie hier: https://doi.org/10.1515/ans-2017-0014

Umberto Biccari , Mahamadi Warma und Enrique Zuazua

Veröffentlicht/Copyright: 3. August 2017

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Aus der Zeitschrift Advanced Nonlinear Studies Band 17 Heft 4

Abstract

In [1], for 1<p<∞, we proved the Wloc2⁢s,p local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian (-Δ)s on an arbitrary bounded open set of ℝN. Here we make a more precise and rigorous statement. In fact, for 1<p<2 and s≠12, local regularity does not hold in the Sobolev space Wloc2⁢s,p, but rather in the larger Besov space (Bp,22⁢s)loc.

Keywords: Fractional Laplacian; Dirichlet Boundary Condition; Weak Solutions; Local Regularity

MSC 2010: 35B65; 35R11; 35S05

In [1], we have analyzed the local regularity for the solutions to the Dirichlet problem

(1) { ( - Δ ) s ⁢ u = f in ⁢ Ω , u = 0 on ⁢ ℝ N ∖ Ω ,

where Ω⊂ℝN is an arbitrary bounded open set and the fractional Laplace operator (-Δ)s is defined for s∈(0,1) as the singular integral

( - Δ ) s ⁢ u ⁢ ( x ) := C N , s ⁢ P . V . ∫ ℝ N u ⁢ ( x ) - u ⁢ ( y ) | x - y | N + 2 ⁢ s ⁢ 𝑑 y ,

where CN,s is a normalization constant.

In [1, Theorem 1.4], we stated and proved the following maximal local regularity result for the weak solutions to (1): if f∈Lp⁢(Ω), 1<p<∞, the corresponding weak solution to system (1) satisfies u∈Wloc2⁢s,p⁢(Ω).

However, although this is true for p≥2, when 1<p<2 the result is correct only for s=12. When 1<p<2 and s≠12, instead, u belongs to the Besov space (Bp,22⁢s)loc⁢(Ω), which is strictly larger than Wloc2⁢s,p⁢(Ω).

This is due to the fact that in the proof [1, Theorem 1.4], the Wloc2⁢s,p combines a cut-off argument and global results for the fractional Poisson-type equation

(2) ( - Δ ) s ⁢ u = F in ⁢ ℝ N .

When F∈Lp⁢(ℝN), the solution u belongs to W2⁢s,p⁢(ℝN) for p≥2 and for 1<p<2, s=12. But not when 1<p<2 and s≠12. However [1, Theorem 2.7] does not make this distinction stating W2⁢s,p⁢(ℝN) regularity for all 1<p<∞ and all 0<s<1. When 1<p<2 and s≠12, we have regularity in the Besov space Bp,22⁢s⁢(ℝN), which is strictly larger than W2⁢s,p⁢(ℝN). This has been also mentioned in [4, Remark 7.1].

The correct statement of [1, Theorem 2.7] should be the following.

Theorem 1.

Let 1<p<∞. Given F∈Lp⁢(RN), let u be the unique weak solution to the fractional Poisson-type equation

( - Δ ) s ⁢ u = F in ⁢ ℝ N .

Then u∈L2⁢sp⁢(RN), where

(3) ℒ 2 ⁢ s p ⁢ ( ℝ N ) := { u ∈ L p ⁢ ( ℝ N ) : ( - Δ ) s ⁢ u ∈ L p ⁢ ( ℝ N ) }

has been introduced for example in [5, Chapter V, Section 3.3, formula (38)].

As a consequence, we have the following:

If 1<p<2 and s≠12, then u∈Bp,22⁢s⁢(ℝN).
If 1<p<2 and s=12, then u∈W2⁢s,p⁢(ℝN)=W1,p⁢(ℝN).
If 2≤p<∞, then u∈W2⁢s,p⁢(ℝN).

Remark 2.

The space ℒ2⁢sp⁢(ℝN), which in the literature is called potential space, is sometimes denoted as Hps⁢(ℝN) (see, e.g., [7, Section 1.3.2]).

Accordingly, [1, Theorem 1.4] should be rephrased as follows.

Theorem 3.

Let 1<p<∞. Given f∈Lp⁢(Ω), let u be the unique weak solution to the Dirichlet problem (1). Then u∈(L2⁢sp)loc⁢(Ω). As a consequence, we have the following result:

If 1 < p < 2 and s ≠ 1 2 , then

u ∈ ( B p , 2 2 ⁢ s ) loc ⁢ ( Ω ) .
If 1 < p < 2 and s = 1 2 , then

u ∈ W loc 2 ⁢ s , p ⁢ ( Ω ) = W loc 1 , p ⁢ ( Ω ) .
If 2 ≤ p < ∞ , then

u ∈ W loc 2 ⁢ s , p ⁢ ( Ω ) .

We provide below the explanation of these facts:

According to [5, Chapter V, Section 5.3, Theorem 5 (B)], when 1<p<2, the space ℒ2⁢sp⁢(ℝN) introduced in (3) is included in the Besov space B2⁢sp,2⁢(ℝN). Moreover, an explicit counterexample showing that sharper inclusions are not possible has been given in [5, Chapter V, Section 6.8]. But if 2⁢s is an integer, that is, if s=12, then

ℒ 2 ⁢ s p ⁢ ( ℝ N ) = ℒ 1 p ⁢ ( ℝ N ) = W 1 , p ⁢ ( ℝ N )

(see [5, Chapter V, Section 3.3, Theorem 3]). In view of these observations, we have that if F∈Lp⁢(ℝN), 1<p<2, then the solution u to (2) belongs to the Besov space B2⁢sp,2⁢(ℝN) if s≠12, and belongs to W1,p⁢(ℝN) if s=12.
Recall also that the Sobolev space W2⁢s,p⁢(ℝN) is, by definition, the space B2⁢sp,p⁢(ℝN) (see, e.g., [5, Chapter V, Section 5.1, formula (60)]) and that the latter is strictly included in

B 2 ⁢ s p , 2 ⁢ ( ℝ N ) for ⁢ 1 < p < 2 .

Hence, the counterexample mentioned above implies that the space of functions whose fractional Laplacian is in Lp⁢(ℝN) is strictly larger than B2⁢sp,p⁢(ℝN) (which is by definition W2⁢s,p⁢(ℝN)) when 1<p<2 and s≠12. In other words, when 1<p<2 and s≠12, there are functions whose fractional Laplacian belongs to Lp⁢(ℝN), but they do not belong to W2⁢s,p⁢(ℝN).

These arguments provide a proof of Theorem 1. Instead, the proof of Theorem 3 presented in [1] is correct. Indeed, it is based on a cut-off argument which is not affected by the discussion above.

During the revision process of our original manuscript, we became aware that similar results were obtained using pseudo-differential calculus (see, e.g., [3, Section 7] or [6, Chapter XI, Proposition 2.4, Theorem 2.5 and Exercise 2.1]). We already pointed out this fact in [1], saying that, in [3, Section 7], Grubb proved that, under the restriction s>Np, the assumption f∈Wτ,p⁢(Ω) for some real number τ≥0 implies that the corresponding solution u of (1) belongs to

W loc τ + 2 ⁢ s , p ⁢ ( Ω ) .

A more careful reading of Grubb’s work and a discussion with the author made us realize that, for 1<p<2, this result does not hold in the classical Sobolev setting, but rather in (ℒτ+2⁢sp)loc⁢(Ω). Also, the restriction s>Np mentioned above is not necessary and this regularity is true for all 1<p<∞.

Our approach complements the pseudo-differential one, using merely classical PDE techniques in the context of linear and nonlinear elliptic and parabolic equations.

In fact, our techniques and results extend to the following parabolic problem:

(4) { u t + ( - Δ ) s ⁢ u = f in ⁢ Ω × ( 0 , T ) , u = 0 on ⁢ ( ℝ N ∖ Ω ) × ( 0 , T ) , u ⁢ ( ⋅ , 0 ) = 0 in ⁢ Ω .

In particular, we have the following theorem.

Theorem 4.

Let 1<p<∞. Given f∈Lp⁢(Ω×(0,T)), let u be the unique weak solution to the parabolic problem (4). Then

u ∈ L p ⁢ ( ( 0 , T ) ; ( ℒ 2 ⁢ s p ) loc ⁢ ( Ω ) ) .

As a consequence, we have the following result:

If 1 < p < 2 and s ≠ 1 2 , then

u ∈ L p ⁢ ( ( 0 , T ) ; ( B p , 2 2 ⁢ s ) loc ⁢ ( Ω ) ) .
If 1 < p < 2 and s = 1 2 , then

u ∈ L p ⁢ ( ( 0 , T ) ; W loc 2 ⁢ s , p ⁢ ( Ω ) ) = L p ⁢ ( ( 0 , T ) ; W loc 1 , p ⁢ ( Ω ) ) .
If 2 ≤ p < ∞ , then

u ∈ L p ⁢ ( ( 0 , T ) ; W loc 2 ⁢ s , p ⁢ ( Ω ) ) .

We refer to [2] for more details on this topic.

Acknowledgements

The authors wish to thank Gerd Grubb (University of Copenhagen) and Xavier Ros-Oton (University of Texas at Austin), for the interesting discussions that led to the clarification reported in this addendum.

References

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Received: 2017-05-10

Accepted: 2017-05-10

Published Online: 2017-08-03

Published in Print: 2017-10-01

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Fractional Laplacian; Dirichlet Boundary Condition; Weak Solutions; Local Regularity

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