Dirichlet’s principle and wellposedness of solutions for a nonlocal p-Laplacian system

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Abstract

We prove Dirichlet’s principle for a nonlocal p-Laplacian system which arises in the nonlocal setting of peridynamics when p=2. This nonlinear model includes boundary conditions imposed on a nonzero volume collar surrounding the domain. Our analysis uses nonlocal versions of integration by parts techniques that resemble the classical Green and Gauss identities. The nonlocal energy functional associated with this “elliptic” type system exhibits a general kernel which could be weakly singular. The coercivity of the system is shown by employing a nonlocal Poincaré’s inequality. We use the direct method in calculus of variations to show existence and uniqueness of minimizers for the nonlocal energy, from which we obtain the wellposedness of this steady state diffusion system.

Introduction

In classical continuum mechanics materials are assumed to be continuously distributed entirely throughout the space they occupy. However, in some materials cracks and other discontinuities arise, hindering the use of the classical model, as the governing equations collapse at singularities. Similarly, other phenomena exhibit discontinuities of solutions. Thus nonlocal theories have been introduced in the mechanics of solids, image processing, and biology models.

Motivated by these nonlocal models we consider here the following nonlocal “elliptic” boundary value problemLpu(x)=b(x),xΩ,u(x)=g(x),xΓ,whereLpu(x)2ΩΓ|u(x)-u(x)|p-2(u(x)-u(x))μ(x,x)dx.Here Ω denotes an open bounded subset of Rn,ΓRnΩ denotes a “collar” domain surrounding Ω which has nonzero volume, and 1<p<. The nonhomogeneities b and g are L2 integrable functions over Ω, respectively Γ. The kernel μ(x,x) denotes a positive, symmetric function of its arguments, with a singularity around x=x which records the interaction of xΩ with its neighboring points x. The operator Lp is a nonlocal extension of the classical p-Laplacian known as div(|u|p-2u), or the operator from the porous medium equation Δ(|u|m-1u). An in-depth discussion of the evolution equation associated with this nonlocal operator may be found in Chapter 6 of the monograph [10].

The use of nonlocal operators such as Lp has been proven valuable in applications from several areas, including image processing [18], [19], [20], sandpile formation [8], swarm [23] and other population density models [14], [13]. Current literature from the last decade contains an in depth investigation of nonlocal systems such as (1.1), but usually this is performed under smoothness assumptions for the kernel (see for example [6], [7], [9], [10] and the references therein).

The main contribution of this paper is to prove existence and uniqueness of solutions of the nonlinear nonlocal system 1.1. These results have been obtained from (and are complemented by) the establishment of (i) a nonlocal version of Dirichlet’s principle and (ii) ellyptic-type properties for the nonlocal operator. These theoretical contributions motivate the validity of different algorithms developed for computing nonlocal solutions that have been designed [3], [11], [17], [24]. The mathematical theory of establishing wellposedness and providing rigorous proofs for qualitative and quantitative properties of these systems remains in its early development stages. Our results are also suitable for weak solutions for which the well posedness constitutes the groundwork for conforming finite element method. Thus in [17] linear elements are used for a one-dimensional nonlocal equation, whereas in [3] the issue of domain decomposition preconditioning is investigated.

For p=2 the operator Lp is the operator considered in linear peridynamic models of elasticity and heat diffusion [5], [12], [26], [25]. The theory of peridynamics was introduced by Silling in [26] and prescribes that spatial derivatives may be avoided in modeling the interactions between particles. In this nonlocal version of continuum theory, internal forces acting on material particles are assumed to form a network of pairwise forces, known as bonds. Each material point x interacts with all the neighboring points within a given region surrounding the point; this region is called the horizon and is denoted here by Hx. In peridynamics a fracture is seen as a breaking of the aforementioned pairwise bonds, which can be modeled by a vanishing pairwise force function. Thus the strength of the peridynamic formulation is that the same equations can be applied to all points in the domain, eliminating the previous need for special techniques from fracture mechanics. Note that the nonlocality of the model forces the equations to be considered on the larger domain ΩΓ. Indeed, to account for the interaction of x with points outside the domain Ω, when x belongs to the boundary Ω, we allow x to belong to the collar domain Γ which contains all horizons Hx as x moves along Ω.

In peridynamics a prototype kernel μ(x,x) is of the formμ(x,x)=1|x-x|βfor|x-x|<δ,0for|x-x|δ,where β>0. In the linear case (when p=2), with the above choice for μ, for β>n the natural framework to study regularity properties of the operator Lp is that provided by fractional derivative spaces (see [1]). For β<n the kernel is weakly singular and the derivation of classical regularity results cannot be done following standard techniques. Some of the difficulties arising in this situation stem from the fact that the operator is non-smoothing; also, there is no higher integrability for the function based on the L2 bounds of the nonlocal gradient as given by the nonlocal version of Poincaré’s inequality (see Lemma 3.5 and Remark 3.7).

An important aspect in the development of the peridynamic theory is the consideration of boundary conditions. Due to the nonlocal property of the operator Lp, classical boundary conditions (imposed on boundaries of zero volume) will not yield well posed systems. Gunzburger and Lehoucq resolved this issue in [21] where they define weak formulations for the nonlocal boundary value problem (1.1) in the scalar case for p=2 and show wellposedness of these problems after developing a calculus for nonlocal operators. A rigorous framework for nonlocal calculus of vectors and tensors was recently developed. Note also that in [21], the integral operator on the domain Ω alone. We chose to employ the integral operator over ΩΓ so the nonlocal equations and identities satisfied by this operator appear as natural generalizations of the classical PDEs. Indeed, in Lemma 2.1 and Proposition 2.5 we verify properties of Lp that are nonlocal analogs of properties of the classical Laplacian. Gunzburger and Lehoucq have shown in [21] that for p=2 the problem (1.1) provides a nonlocal equivalent of the classical elliptic boundary value problemΔu(x)=b(x),xΩ,u(x)=g(x),xΓ.More specifically, they show that the solution of Lu=δx=x becomes the fundamental solution of the Laplacian when one chooses μ as a combination of Dirac masses and its derivatives. In addition, [21] provides wellposedness results of the system (1.1), when p=2 and μ is weakly singular, by using a variational approach based on the Lax–Milgram lemma.

The nonlinearity of our system renders the Lax–Milgram variational approach unapplicable, so we use the direct method in calculus of variations to show existence and uniqueness of solutions. Our first contribution is the proof of Dirichlet’s principle in the given nonlocal setting. More precisely, we show that the minimizers of the energy functionalF[u]=1pΩΓΩΓ|u(x)-u(x)|pμ(x,x)dxdx+Ωb(x)u(x)dxsatisfy (1.1) and conversely, any solution of (1.1) is a minimizer for F. This result then enables us to prove the wellposedness of the system (1.1) by showing existence and uniqueness of minimizers. As in the classical setting (see [15]) the existence of minimizers relies on convexity and coercivity properties of the integrand. For our functional, the convexity is immediate due to the structure of our integrand, and this ensures the necessary weakly lower semicontinuity. The coercivity property requires the use of a nonlocal Poincaré type inequality which is proven in [3]. Since our proof follows the direct method in calculus of variations it will not need the completeness assumptions on the space of functions, therefore, our kernels could be chosen in great generality. Thus, we do not rely on the existence of a uniform radius δ for the horizon in the peridynamic representation (see Remark 2.3). Also, the kernel function μ=μ(x,x) does not need to be solely a function of |x-x|, as it only needs to satisfy the assumption (A1) with the bound (2.8). The existence of minimizers was proven in [16] for the problem on Rn, but under growth restrictions on the growth of the kernel. Our results hold for any β when the domain Ω is bounded.

The paper is organized as follows: Section 2 contains a discussion of notation, definitions, and elliptic-type properties involving the operator Lp. Also, in Section 2 we collect and prove some nonlocal calculus results developed in [9], [21] which provide nonlocal analogues of both Gauss’s Theorem and Green’s identities. In Section 3, we define the nonlocal energy functional associated with (1.1) and prove the Dirichlet’s principle in the nonlocal setting. SubSection 3.2 contains the proof for existence and uniqueness of minimizers based on the nonlocal Poincaré’s inequality given in Lemma 3.5. These results yield the well-posedness of the system (1.1).

Section snippets

Preliminaries

For α:(ΩΓ)×(ΩΓ)Rn,u:ΩΓR, and f:(ΩΓ)×(ΩΓ)Rn we define as in [21] the following generalized nonlocal operators:

  • (i) Generalized gradientG(u)(x,x)(u(x)-u(x))α(x,x),x,xΩΓ,

  • (ii) Generalized nonlocal divergenceD(f)(x)ΩΓ(f(x,x)·α(x,x)-f(x,x)·α(x,x))dx,xΩ,

  • (iii) Generalized normal componentN(f)(x)-ΩΓ(f(x,x)·α(x,x)-f(x,x)·α(x,x))dx,xΓ.

    With the above notation in place, the authors in [21] demonstrate that for v:ΩΓR and s:(ΩΓ)×(ΩΓ)Rn the following identity holdsΩvD(s)dx+

Well-posedness of the nonlocal boundary-value problem

In this section we will first prove a nonlocal version of Dirichlet’s principle (Theorem 3.9). Next, we will establish the existence and uniqueness of minimizers of the functional F defined below. Combining these results, we obtain the well-posedness of the problem (1.1). First we will introduce the spaces and the framework in which we will present the results.

Let μ:(ΩΓ)2R a nonnegative measurable function and consider its support given by the closure of the set of all the points where μ does

Acknowledgments

The authors thank the referees for their valuable comments that helped to improve this paper.

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    The authors were supported by NSF Grant DMS – 0908435.

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