Calderón–Zygmund theory for strongly coupled linear system of nonlocal equations with Hölder-regular coefficient

1.1 Motivation and notations

The goal of this work is to obtain Sobolev regularity estimates for solutions of the strongly coupled system of linear nonlocal equations ℒ A s ⁢ u = f , where the operator ℒ A s is formally given by

Here we take n ≥ 1 , 0 < s < 1 , and A : ℝ n × ℝ n → ℝ is taken to be symmetric and serves as a variable coefficient for the operator ℒ A s . For vectors a = ( a 1 , … , a n ) and b = ( b 1 , … , b n ) in ℝ n , the tensor product a ⊗ b is the rank one matrix with its ( i ⁢ j ) th entry being a i ⁢ b j .

Coupled systems of linear nonlocal equations of the above type appear in applications. In fact, the operator ℒ A s is related to the bond-based linearized peridynamic equation [31, 32]. To briefly describe where the operator comes from, consider a heterogeneous elastic solid occupying the domain Ω in ℝ n , n = 1 , 2 , or 3, that is linearly deforming when subjected to an external force field f. In the framework of the peridynamic model, a bounded domain hosting an elastic material is conceptualized as a sophisticated mass-spring system. Here, any pair of points x and y within the material is considered to interact through the bond vector x - y . When external load f is applied, the material undergoes a deformation, mapping a point x in the domain to the point x + u ⁢ ( x ) ∈ ℝ n , where the vector field u represents the displacement field. Adhering to the principles of uniform small strain theory [32], the strain of the bond x - y is given by the nonlocal linearized strain

The linearized bond-based peridynamic static model relates the displacement field u and the external load f by the equation (see [8, 18])

where the vector-valued pairwise force density function 𝒞 is given by

In the above A ⁢ ( x , y ) serves as a “spring constant” for the bond joining x and y and the function ρ is the interaction kernel that is radial and describes the force strength between material points. After noting that

ℒ A s is precisely the linearized bond-based peridynamic operator corresponding to the kernel of interaction

To describe the problem we study, let us first introduce some standard notations and define relevant function spaces. For t ∈ ( 0 , 2 ) the fractional Laplacian ( - Δ ) t 2 is, defined via the Fourier transform,

where the Fourier transform is defined as ℱ ⁢ ( u ) ⁢ ( ξ ) = u ^ ⁢ ( ξ ) = ∫ ℝ n e - 2 ⁢ π ⁢ ı ⁢ x ⋅ ξ ⁢ u ⁢ ( x ) ⁢ 𝑑 x . It also has a useful integral representation and for any vector field u in the Schwartz class

where p.v. stands for the principal value, whose mentioning we will suppress. The inverse operator of the fractional Laplacian is the Riesz potential whose integral representation is

for a vector field v in the Schwartz class. We introduce types of fractional Sobolev spaces that we need to state the main result: Bessel potential spaces H s , p and Besov spaces W s , p . For 1 < p < ∞ and s ∈ ( 0 , 1 ) , the Bessel potential spaces H s , p ⁢ ( ℝ n ) are defined as follows: f ∈ H s , p ⁢ ( ℝ n ) if f ∈ L p ⁢ ( ℝ n ) and ( - Δ ) s 2 ⁢ f ∈ L p ⁢ ( ℝ n ) . The associated norm is

For any open subset Ω ⊂ ℝ n , the Besov spaces W s , p ⁢ ( Ω ) , for s ∈ ( 0 , 1 ) , are induced by the semi-norm (called Sobolev–Slobodeckij or Gagliardo norm)

and ∥ ⋅ ∥ W s , p ⁢ ( Ω ) = ∥ ⋅ ∥ L p ⁢ ( Ω ) + [ ⋅ ] W s , p ⁢ ( Ω ) serves as a norm. For p = 2 , W s , 2 ⁢ ( ℝ n ) = H s , 2 ⁢ ( ℝ n ) , which we denote by H s ⁢ ( ℝ n ) . For p < 2 , we have W s , p ⁢ ( ℝ n ) ⊊ H s , p ⁢ ( ℝ n ) and for p > 2 , H s , p ⁢ ( ℝ n ) ⊊ W s , p ⁢ ( ℝ n ) . These spaces are particular examples of the more general Triebel–Lizorkin spaces and F p ⁢ p s ⁢ ( ℝ n ) = W s , p ⁢ ( ℝ n ) and F p , 2 s ⁢ ( ℝ n ) = H s , p ⁢ ( ℝ n ) , see [24].

Given s ∈ ( 0 , 1 ) , A ∈ L ∞ ⁢ ( ℝ n × ℝ n ) and u ∈ L loc 1 ⁢ ( ℝ n , ℝ n ) , we understand ℒ A s ⁢ u as a distribution defined as

for all φ ∈ C c ∞ ⁢ ( ℝ n , ℝ n ) . Moreover, if u ∈ H s ⁢ ( ℝ n , ℝ n ) , then from the above definition, ℒ A s ⁢ u ∈ H - s ⁢ ( ℝ n , ℝ n ) with the estimate that

where H - s ⁢ ( ℝ n , ℝ n ) represents the dual of H s ⁢ ( ℝ n , ℝ n ) .

Our interest is to address the question of regularity of solutions u to ℒ A s ⁢ u = f relative to the data f. To that end, we require the coefficient A to satisfy some continuity and boundedness assumptions. First, we say A satisfies a uniform Hölder continuity assumption if for some α ∈ ( 0 , 1 ) and Λ > 0 ,

Given λ , Λ > 0 and α ∈ ( 0 , 1 ) , we define the coefficient class

We note that members of the coefficient class 𝒜 ⁢ ( α , λ , Λ ) can be negative off diagonal. Indeed, as indicated in [19], the coefficient A ⁢ ( x , y ) = 2 + | x | α + | y | α 1 + | x | α + | y | α + 10 ⁢ ( sin ⁡ x + sin ⁡ y ) ⁢ | x - y | α 1 + | x - y | α belongs to 𝒜 ⁢ ( α , λ , Λ ) for some λ , Λ and yet can be negative off diagonal.

Now for an open set Ω ⊂ ℝ n and f ∈ H - s ⁢ ( ℝ n , ℝ n ) , a vector field u ∈ H s ⁢ ( ℝ n , ℝ n ) is a solution to ℒ A s ⁢ u = f in Ω if

In the event, A = 1 , then operator agrees with the integral operator defined as

where the integral converges in the sense of principal value for smooth vector fields. For vector fields u in the Schwarz space 𝒮 ⁢ ( ℝ n , ℝ n ) , we have

for some positive constants ℓ 1 and ℓ 2 depending only on n and s. As a consequence, as shown in [20] for any τ > 0 and f ∈ L p ⁢ ( ℝ n , ℝ n ) with 1 < p < ∞ , then the solution u to

lives in H 2 ⁢ s , p ⁢ ( ℝ n , ℝ n ) . For the nonlocal equation of variable coefficient (1.2), we would like to obtain a Sobolev regularity of the above type for solutions in the event that the right hand side f has additional regularity. We begin by noting that for some λ and Λ, A ∈ 𝒜 ⁢ ( α , λ , Λ ) , and f ∈ H - s ⁢ ( ℝ n , ℝ n ) , a solution to (1.2) exists under some volumetric condition on u and on the domain Ω. Indeed, if Ω is a Lipschitz domain, a minimizer of the energy

over the space H ~ s ⁢ ( Ω ) = { u ∈ H s ⁢ ( ℝ n , ℝ n ) : u = 0 ⁢  on  ℝ n ∖ Ω } will satisfy equation (1.2). For a Lipschitz domain Ω, H ~ s ⁢ ( Ω ) is precisely the closure of C c ⁢ ( Ω , ℝ n ) in H s ⁢ ( ℝ n , ℝ n ) , and therefore H ~ s ⁢ ( Ω ) is a Hilbert space with obvious inner product, [17]. The existence of a minimizer for the quadratic functional E over H ~ s ⁢ ( Ω ) , with a possible sign changing A ∈ 𝒜 ⁢ ( α , λ , Λ ) will be shown later using Lax–Milgram theorem. As has been demonstrated in [18], with a proper multiplicative constant c ⁢ ( s , n ) , in terms of the nonlocality parameter s, the operator c ⁢ ( s , n ) ⁢ ℒ A s ⁢ u that corresponds to A ⁢ ( x , y ) = 1 / 2 ⁢ ( a ⁢ ( x ) + a ⁢ ( y ) ) will converge in an appropriate sense to the Lamé differential operator

This operator is strongly elliptic in the sense of Legendre-Hadamard but not uniformly elliptic. One can then view (1.2) as a fractional analogue of the classical Navier-Lamé system of linearized elasticity equation.

1.2 Statement of the main result

The main result of the paper is the following interior regularity estimate which is the version of the regularity result proved in [19] for the coupled system of nonlocal equations under discussion.

The proof of Theorem 1.1 parallels the approach used in [19]. Namely, we compare the operator ℒ A s with the simpler operator 𝔏 ¯ A D s 1 , s 2 , where s 1 + s 2 = 2 ⁢ s , and is defined as, for u ∈ H s ⁢ ( ℝ n , ℝ n ) and φ ∈ C c ∞ ⁢ ( ℝ n , ℝ n ) ,

for constants c 1 and c 2 that will be determined as a function of s and n. In the above definition, the operator ℛ = ( ℛ 1 , ℛ 2 , … , ℛ n ) is the vector of Riesz transforms, and A D ⁢ ( z ) = A ⁢ ( z , z ) , the restriction of the coefficient A on the diagonal. Notice that for constant coefficients the two operators ℒ A s and 𝔏 ¯ A D s , s coincide. Indeed, if A ⁢ ( x , y ) = A , constant, then by using (1.3), for vector fields in the Schwarz space

with c 1 = ℓ 1 and c 2 = ℓ 2 . We will prove an optimal regularity result for solutions of the strongly coupled equation

and use those solutions as approximations of the solution to the original system of equations. The mechanism we accomplish this is via perturbation argument where we show that the difference operator

can be understood as a lower order term in the event that A is Hölder continuous.

While our work studies solutions to strongly coupled linear nonlocal PDEs, there has been a number of results in the literature that studied the regularity of solutions to scalar nonlocal PDEs. To name a few, optimal local regularity results are obtained in [2] for weak solutions to the Dirichlet problem associated with the fractional Laplacian. Similar results are obtained for the fractional heat equation in [12, 13]. Almost optimal regularity results are obtained in [5] for weak solutions to nonlocal equations with Hölder regular coefficients. Optimal Sobolev regularity are proved in [7] for strong solutions to nonlocal equations with translation invariant coefficients ( A ⁢ ( x , y ) = A ⁢ ( x - y ) ). For equations with less regular coefficients, higher integrability and higher differentiability results are obtained in [23, 21] for nonlocal equations with variable coefficients that have small mean oscillations. See also [9, 10] for related results. For elliptic, measurable, and bounded coefficients, solutions to nonlocal PDEs are proved in [15] to have a self-improvement property where higher integrability and higher differentiability are obtained without any smoothness assumption on the coefficients, see also [27]. Similar results are also verified in [30, 4] for solutions to nonlocal double phase problems. For a concise description of the results of the above mentioned manuscripts, we refer to [19]. See also [3, 22, 1].

The paper is organized as follows. In the next section, we will discuss some preliminary results we need in the sequel. In Section 3, we estimate 〈 𝒟 s , t ⁢ u , φ 〉 in terms of the Riesz potential I s = ( - Δ ) - s 2 . In Section 4, we will develop the optimal regularity result for a solution of equation (1.5). In Section 5, we prove the main result of the paper by using an iterative argument making use of the commutator estimate we prove in Section 3 and the optimal regularity result obtained in Section 4.

Throughout the paper, we work under the convention that domains of integrals are always open sets and we use the symbol ⊂ ⊂ to say compactly contained, e.g., Ω 1 ⊂ ⊂ Ω 2 if Ω 1 ¯ is compact and Ω 1 ¯ ⊂ Ω 2 . Constants change from line to line, and unless it is important we may not detail their dependence on various parameters. We will make frequent use of ≲ , ≳ and ≍ , which denotes inequalities with multiplicative constants (depending on non-essential data). For example we say A ≲ B if for some constant C > 0 we have A ≤ C ⁢ B . We will use the angle bracket 〈 ⋅ , ⋅ 〉 to represent the standard inner product or the duality pairing depending on the context.