DIRICHLET S PRINCIPLE AND WELLPOSEDNESS OF SOLUTIONS FOR A NONLOCAL p LAPLACIAN SYSTEM WITH APPLICATIONS IN PERIDYNAMICS

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1 DIRICHLET S PRINCIPLE AND WELLPOSEDNESS OF SOLUTIONS FOR A NONLOCAL LAPLACIAN SYSTEM WITH APPLICATIONS IN PERIDYNAMICS BRITTNEY HINDS, PETRONELA RADU Abstract. We rove Dirichlet s rincile for a nonlocal -Lalacian system which arises in the nonlocal setting of eridynamics when = 2. This nonlinear model includes boundary conditions imosed on a nonzero volume collar surrounding the domain. Our analysis uses nonlocal versions of integration by arts techniques that resemble the classical Green and Gauss identities. The nonlocal energy functional associated with this ellitic tye system exhibits a general kernel which could be weakly singular. The coercivity of the system is shown by emloying 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 wellosedness of this steady state diffusion system. 1. Introduction In classical continuum mechanics materials are assumed to be continuously distributed entirely throughout the sace they occuy. However, in some materials cracks and other discontinuities arise, hindering the use of the classical model, as the governing equations collase at singularities. As a result, a nonlocal theory called eridynamics was introduced in 2000 by Silling [24] who rescribes that satial derivatives may be avoided in modeling the interactions between articles. In this nonlocal version of continuum theory, internal forces acting on material articles are assumed to form a network of airwise forces, known as bonds. Each material oint interacts with all the neighboring oints within a given region surrounding the oint; this region is called the horizon. In eridynamics a fracture is seen as a breaking of the aforementioned airwise bonds, which can be modeled by a vanishing airwise force function. Thus, the strength of the eridynamic formulation is that the same equations can be alied to all oints in the domain, eliminating the revious need for secial techniques from fracture mechanics. The eridynamic theory has successfully dealt with with materials that are homogeneous [10, 24, 23] and also with materials that are heterogeneous [4] by relacing the artial differential oerators found in classical continuum theory with integral or integro-differential oerators. The theory of eridynamics has received considerable attention from the engineering community, however, the mathematical theory roviding rigorous roofs for qualitative and quantitative roerties of these systems is still in its early develoment stages. Date: July 24, Key words and hrases. Peridynamics, Dirichlet s rincile, nonlocal Lalacian, singular kernel. The authors were suorted by NSF grant DMS

2 2 BRITTNEY HINDS, PETRONELA RADU In this aer we consider the following nonlocal ellitic boundary value roblem { L u(x) = b(x), x (1.1) u(x) = g(x), x, where (1.2) L u(x) := 2 u(x ) u(x) 2 (u(x ) u(x))µ(x, x ) dx. Here denotes an oen bounded subset of R n, R n \ denotes a collar domain surrounding which has nonzero volume, and 1 < <. The nonhomogeneities b and g are L 2 integrable functions over, resectively. The kernel µ(x, x ) denotes a ositive, symmetric function of its arguments, with a singularity around x = x which records the interaction of x with its neighboring oints x from the horizon of x. To account for the interaction of x with oints outside the domain, when x, we allow x to belong to the collar domain which contains the horizons H x as x moves along. For = 2 the oerator L is the oerator considered in eridynamic models of elasticity and heat diffusion [10, 24]. This oerator is also a nonlocal extension of the classical -Lalacian div( u 2 u), or the oerator from the orous medium equation ( u m 1 u). An in-deth discussion of the evolution equation associated with this nonlocal oerator may be found in Chater 6 of the monograh [9]. The use of nonlocal oerators such as L has been roven valuable in alications from several areas, including image rocessing [17, 18, 19], sandile formation [7], swarm [22] and other oulation density models [13, 12]. Current literature from the last decade contains an in deth investigation of nonlocal systems such as (1.1), but usually this is erformed under smoothness assumtions for the kernel (see for examle [5, 6, 8, 9] and the references therein). The system (1.1) is encountered in eridynamic models when = 2 with a rototye kernel µ(x, x ) of the form (1.3) µ(x, x ) = { 1 x x β for x x < δ 0 for x x δ, where β > 0. In the linear case (when = 2), with the above choice for µ, for β > n the natural framework to study regularity roerties of the oerator L is that rovided by fractional derivative saces (see [1]). For β < n the kernel is weakly singular and the derivation of classical regularity results can not be done following standard techniques. Some of the difficulties arising in this situation stem from the fact that the oerator is non-smoothing; also, there is no higher integrability for the function based on the L 2 bounds of the nonlocal gradient as given by the nonlocal version of Poincaré s inequality (see Lemma 3.5 and Remark 3.7). An imortant asect in the develoment of the eridynamic theory is the consideration of boundary conditions. Due to the nonlocal roerty of the oerator L, classical boundary conditions (imosed on boundaries of zero volume) will not yield well osed systems. Gunzburger and Lehoucq resolved this issue in [20] where they define weak formulations for the nonlocal boundary value roblem (1.1) in the scalar case for = 2 and show wellosedness of these roblems after develoing a calculus for nonlocal oerators. A rigorous framework for nonlocal calculus of vectors and tensors was recently develoed in [15].

3 DIRICHLET S PRINCIPLE IN PERIDYNAMICS 3 The nonlocal Lalacian oerator defined here is an extension of the oerator L from [20], that corresonds to the case = 2. In [20], however, the integral is considered over the domain alone. We chose to emloy the integral oerator over because the nonlocal equations associated with this oerator have the form of natural generalizations of the classical PDEs, and the identities resented throughout the aer are simlified. Indeed, in Lemma 2.1 and Proosition 2.5 we verify roerties of L that are nonlocal analogues of roerties of the classical Lalacian. Gunzburger and Lehoucq have shown in [20] that for = 2 the roblem (1.1) rovides a nonlocal equivalent of the classical ellitic boundary value roblem { u(x) = b(x), x u(x) = g(x), x. More secifically, they show that the solution of Lu = δ(x x) becomes the fundamental solution of the Lalacian when one chooses µ = δ(x x)+ 2 x δ(x x). 2 In addition, [20] rovides wellosedness results of the system (1.1), when = 2 and µ is weakly singular, by using a variational aroach based on the Lax-Milgram lemma. The Lax-Milgram variational aroach is not alicable for our nonlinear system, so we use the direct method in calculus of variations to show existence and uniqueness of solutions. Our first contribution is the roof of Dirichlet s rincile in the given nonlocal setting. More recisely, we show that the minimizers of the energy functional (1.4) F[u] = 1 u(x ) u(x) µ(x, x )dx dx + b(x)u(x)dx, 2 satisfy (1.1) and conversely, any solution of (1.1) is a minimizer for F. This result then enables us to rove the wellosedness of the system (1.1) by showing existence and uniqueness of minimizers. As in the classical setting (see [14]) the existence of minimizers relies on convexity and coercivity roerties 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 roerty requires the use of a nonlocal Poincaré tye inequality which is roven in [2]. Since our roof follows the direct method in calculus of variations it will not need the comleteness assumtions on the sace 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 eridynamic reresentation (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 assumtion (A1) with the bound (2.8). The existence of minimizers was roven in [16] for the roblem on R n, but under growth restrictions on the growth of the kernel. Our results hold for any β when the domain is bounded. The aer is organized as follows: Section 2 contains a discussion of notation, definitions, and ellitic-tye roerties involving the oerator L. Also, in Section 2 we collect and rove some nonlocal calculus results develoed in [8] and [20] which rovide 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 rove the Dirichlet s Princile in the nonlocal setting. Subsection 3.2 contains the roof for existence and uniqueness of minimizers based on the nonlocal Poincaré s

4 4 BRITTNEY HINDS, PETRONELA RADU inequality given in Lemma 3.5. These results yield the well-osedness of the system (1.1). 2. Preliminaries For α : ( ) ( ) R n, u : R, and f : ( ) ( ) R n we define as in [20] the following generalized nonlocal oerators: (i) Generalized gradient (2.1) G(u)(x, x ) := (u(x ) u(x))α(x, x ), x, x, (ii) Generalized nonlocal divergence (2.2) D(f)(x) := (f(x, x ) α(x, x ) f(x, x) α(x, x))dx, x, (iii) Generalized normal comonent (2.3) N (f)(x) := (f(x, x ) α(x, x ) f(x, x) α(x, x))dx, x. With the above notation in lace, the authors in [20] demonstrate that for v : R and s : ( ) ( ) R n the following identity holds (2.4) vd(s)dx + s G(v)dx dx = vn (s)dx Note that by choosing v(x) to be a constant, we arrive at the following nonlocal Gauss s theorem: D(f)dx = N (f)dx. Its connection to the classical Gauss s theorem is shown in [20]. For the remainder of the aer we let µ : ( ) ( ) R be given by µ(x, x ) := α(x, x ). We will assume throughout the aer that α is symmetric (i.e. we have α(x, x ) = α(x, x), for every x, x ), hence µ is symmetric as well. With this choice for µ the following identity : (2.5) D( G(u) 2 G(u)) = 2 (u(x ) u(x) 2 (u(x ) u(x))µ(x, x )dx, was shown in [20], eq.(5.3) for = 2. We state and rove the general case below. Lemma 2.1. For every measurable function u : R we have the following identity (2.6) L u = D( G(u) 2 G(u)), where L, G, and D are defined in (1.1), (2.1), and (2.2). The equality holds whenever both sides are finite.

5 DIRICHLET S PRINCIPLE IN PERIDYNAMICS 5 Proof. We have the following identities D( G(u) 2 G(u))(x) = G(u)(x) 2 G(u)(x)α(x, x ) G(u)(x ) 2 G(u)(x )α(x, x)dx = (u(x ) u(x))α 2 (x, x )( G(u)(x) 2 + G(u)(x ) 2 )dx = 2 (u(x ) u(x))α 2 (x, x ) u(x ) u(x) 2 α(x, x ) 2 dx = 2 u(x ) u(x) 2 α(x, x ) (u(x ) u(x))dx = 2 u(x ) u(x) 2 (u(x ) u(x))µ(x, x )dx = L u(x). Hence, L u = D( G(u) 2 G(u)), the desired result, a generalization of the classical identity div( u 2 u) = u. The following lemma rovides an integration by arts tye identity; it can also be found in [9] (Lemma 6.5 on age 125); we rovide a roof for comleteness below. Note that in contrast with the local theory, this nonlocal version of the integration by arts formula does not contain any boundary terms, hence, the values of u on the domain could be nonzero without affecting the equality. Lemma 2.2. Let u, v : R be measurable. Then one has the following integration by arts identity: (2.7) (L u)v dx = G(u) 2 G(u) G(v)dx dx. Proof. Alying (2.4) with s(x, x ) = G(u) 2 G(u)(x, x ) and using the result in Lemma 2.1 we obtain (L u)vdx = (L u)vdx G(u) 2 G(u) G(v)dx dx. Thus (L u)vdx = G(u) 2 G(u) G(v)dx dx, the desired result Assumtions and roerties of the nonlocal oerator L. Throughout the aer we will work under the following assumtion: (A1). Let µ : ( ) ( ) R be a nonnegative measurable function. For every x H x, there exists a constant C 0 > 0 such that µ(x, x ) C 0. In other words, for all x we have: (2.8) C 0 χ δ (x, x ) µ(x, x ),

6 6 BRITTNEY HINDS, PETRONELA RADU { where χ δ (x, x 1, x x δ ) = 0, x x > δ. (A2). The domain is an oen bounded domain in R n and is a collar surrounding, i.e. =. Also, from a hysical oint of view we imose that contains all the horizons H x ; mathematically, in light of assumtion (A1) above, we only need that x B δ (x). Remark 2.3. Note that the rototye function µ given in (1.3), satisfies this assumtion. Also, our assumtion allows for horizons which are not sherical; all we need is to find a ball centered at x (inside the horizon), where the kernel is bounded uniformly from below. This resents the following interesting asect: hysically, the interactions that the material exeriences may take lace on a large irregular domain, but mathematically all that is necessary is the existence of a small region where the interactions are non-zero. Remark 2.4. In [20] the kernel µ is assumed to satisfy (as given by inequalities (6.1)): µ(x, x )dx C 1 and µ(x, x )dx C 2. These assumtions essentially imose that µ has the growth of a weakly singular oerator; in other words, the exonent β in (1.3) must satisfy β < n for all x. Here we only imose lower bounds through (2.8) which give that β 0 for a kernel of a form found in (1.3). We will rove next some roerties of the nonlocal oerator L. These roerties aear in [18] under a connectivity assumtion which is satisfied, but we will not use it here (instead, we have (2.8) which ensures that µ 0 inside the horizon). Proosition 2.5. The oerator L admits the following roerties: (a) If u constant then L u = 0. (b) Let x. For any maximal oint x 0 that satisfies u(x 0 ) u(x), we have L u(x 0 ) 0. Similarly, if x 1 is a minimal oint such that u(x 1 ) u(x), then L u(x 1 ) 0. (c) L u is a ositive semidefinite oerator, i.e. L u, u 0, where, denotes the L 2 ( ) inner roduct. (d) L u(x)dx = 0. Remark 2.6. The converse of art (a), which states that the solution u of the system { L u = 0, x u = 0, x must be identically zero follows from the wellosedness of the system, which is a result of Theorems 3.8 and Proof. Proerty (a) is trivial because if u constant, then for all x and x we have u(x) u(x ) = 0, hence L u = 0. For roerty (b), let x 0 be a maximal oint. Then for all x we have u(x 0 ) u(x). Then L u(x 0 ) = 2 (u(x 0 ) u(x )) u(x 0 ) u(x ) 2 µ(x 0, x )dx 0

7 DIRICHLET S PRINCIPLE IN PERIDYNAMICS 7 since µ(x 0, x ) 0 by assumtion (2.8) and u(x 0 ) u(x ) 0. A similar argument can be used for a minimal oint. To obtain (c), note that we have by Lemma 2.2 that L u, u = G(u) G(u) 2, G(u) 0 which shows that L u is a ositive semidefinite oerator. For (d), we use again an integration by arts argument L u(x)dx = L u, 1 = G(u) G(u) 2, G(1) = 0 since G(1) = Well-Posedness of the Nonlocal Boundary-Value Problem In this section we will first rove a nonlocal version of Dirichlet s Princile (Theorem 3.8). Next, we will establish the existence and uniqueness of minimizers of the functional F defined below. Combining these results, we obtain the well-osedness of the roblem (1.1). First we will introduce the saces and the framework in which we will resent the results. Let µ : () 2 R a nonnegative measurable function and consider its suort given by the closure of the set of all the oints where µ does not vanish: N µ := {(x, x ) ( ) 2 µ(x, x ) 0} = su(µ). Remark 3.1. Note that the assumtion (A1) with the bound (2.8) imlies that N µ contains the band {(x, x ) ( ) 2 x x δ}. As a consequence of this fact, we have that N µ, since contains all the balls B δ (x) for x. For all 1 < and E a Lebesgue measurable set, E ( ) 2 consider the sace of weighted L µ functions: L µ(e) := {f : ( ) 2 R f measurable, f(x, x ) µ(x, x ) dx dx < }. Notice that these functions are defined over the entire domain, but they are L µ integrable only on the set E. Due to the definition of N µ we have that L µ(n µ ) = L µ[( ) 2 ]. We have that L µ(e) is a vector sace (in fact, Banach sace see Remark 3.3) with resect to the norm ( 1/ f L µ (E) := f(x, x ) µ(x, x ) dx dx). E Remark 3.2. Note that the equivalence classes of functions on L µ(e) distinguish only among values for functions inside N µ, since outside N µ the weight µ = 0. Thus f L µ (E) = 0 iff f = 0 a.e. on N µ E. Remark 3.3. Note that since µ is a nonnegative and measurable, the set function λ(e) := µ(x, x )dxdx E is a measure on the Lebesgue measurable sets E subsets of ( ) ( ) (according to Theorem 1.47 g. 17 in [1]). (Observe that the measure λ may not E

8 8 BRITTNEY HINDS, PETRONELA RADU be absolutely continuous with resect to the Lebesgue measure, since the kernel µ could be non-integrable). Thus L µ(e) is the sace of integrable functions with resect to the new measure µ(x, x )dxdx, hence it is a Banach sace. For u : R denote (3.1) û(x, x ) := u(x) u(x ). Using this notation define the subsace of the functions with a zero trace : (3.2) W := {w : R ŵ L µ[( ) 2 ], and w = 0 on } endowed with the norm ( (3.3) w W = ŵ L µ [() 2 ] = w(x ) w(x) µ(x, x ) dx dx) 1/. Remark 3.4. From Remark 3.2 we have that u W = 0 imlies that u = constant on N µ, and due to the boundary conditions we have that u = 0 on N µ. Observe that u may not be constant outside N µ ; however, the values of u outside N µ are irrelevant due to the fact that u is integrated against µ, and µ = 0 on the comlement of N µ. A version of the following nonlocal Poincaré-tye inequality was roved in [2]; this inequality will aid in roving coercivity for the functional F[u]. Other nonlocal versions of Poincaré s inequality may be found in [3] and [21]. Lemma 3.5. (Nonlocal Poincaré s Inequality.) Let g L () with > 1, n 1. If u g +W, and G is as defined in (2.1), then there exist λ P ncr = λ P ncr (,, δ, n) > 0 and C g > 0 (unless g 0) such that the following inequality holds: (3.4) λ P ncr u L () G(u) L [() 2 ] + C g, which can also be written as (3.5) λ P ncr u L () û L µ [() 2 ] + C g. Moreover, if g 0, then C g = 0. Remark 3.6. The above Lemma has as a consequence the fact that for every > 1 one has u W u L (). Remark 3.7. In the classical case Poincaré s inequality yields higher integrability than L for every u H 1 0 (): u q C u, 1 q n n. For the nonlocal theory, however, we can only obtain L integrability for u as a consequence of the L bound for G(u). This is the case also with the Poincaré s tye inequality obtained in [2]. Proof. Proosition 4.1from [2] yields the following inequality: λδ n u(x) dx χ δ (x, x ) u(x ) u(x) dx dx + δ n where λ = λ(n,, ) > 0 and n 1. g(x ) dx,

9 DIRICHLET S PRINCIPLE IN PERIDYNAMICS 9 Alying assumtion (2.8) we have the following: λδ n u(x) dx χ δ (x, x ) u(x ) u(x) dx dx + δ m g(x ) dx χ δ (x, x ) u(x ) u(x) dx dx + C g 1 C 0 µ(x, x ) u(x ) u(x) dx dx + C g = 1 C 0 G(u) L ( ) + C g Hence we have λ P ncr u L () G(u) L ( ) + C g, where (3.6) λ P ncr = λδ n C 0 > 0, C g = δ n g(x ) dx Nonlocal Dirichlet s Prinicile. Let us demonstrate that a solution (in the sense of distributions) of the boundary value roblem (1.1) can be characterized as the minimizer of an aroriate functional. Let b L q () and g L () where q = 1, and 1 <, q <. Define the functional: F[u] := 1 (3.7) u(x ) u(x) µ(x, x )dx dx + b(x)u(x)dx = 1 G(u)(x, x ) dx dx + b(x)u(x)dx, for u in the class of admissible functions (3.8) A = {u : R u = g + u 0, for some u 0 W}, which by convention will be denoted by g + W. This ensures that the admissible functions have their trace equal to g on. Theorem 3.8. (Dirichlet s Princile) (i) Assume u solves the nonlocal eridynamics roblem (1.1). Then (3.9) F[u] F[v] for every v A. (ii) Conversely, if u A satisfies (3.9) for every v A, then u solves the nonlocal eridynamics roblem (1.1). Proof. (i) Choose w A. For u a solution to (1.1) we see that u w = 0 over. Alying Lemma 2.2 we obtain 0 = (L u b)(u w)dx = G(u) 2 G(u) G(u w)dx dx b(u w)dx By exanding the integrands and alying Young s inequality we obtain

10 10 BRITTNEY HINDS, PETRONELA RADU = 1 G(u) dx dx + bu dx G(u) 2 G(w) G(u)dx dx + G(u) dx dx + 1 bw dx G(w) dx dx + Alying (2.1) and rearranging we see F[u] F[w], for every w A. bw dx. (ii) Conversely, suose (3.9) holds. Fix v W and write i(τ) := F[u + τv], where τ R. We have i(τ) = 1 G(u) + τg(v) dx dx + b(u + τv)dx Since u + τv A for each τ, the scalar function i( ) has a minimum at zero. Thus i (0) = 0, ( ) = d dτ, rovided the derivative exists. Alying Lemma 2.2 we have 0 = i (0) = G(u) 2 G(u) G(v)dx dx + bv dx = (L u b)v dx Since v was an arbitrarily assigned function in W, which is dense in every L when µ(x, x ) x x β with β < + n (since it contains the sace of C functions with comact suort on ), then L (u)(x) = b(x) in in the sense of distributions. Note that if β + n, then the equation is trivially satisfied since the function u W must be identically zero. Remark 3.9. Note that for the homoegeneous case, when the forcing b = 0, Dirichlet s Princile holds for oerators L with very general kernels µ that are only imosed to be symmetric and nonnegative, since our bound (2.8) on µ is not required for the roof. In the eridynamic setting this means that one may assume that the horizon of interaction H x may have an arbitrary radius δ that varies with x or may have a different geometry (other than radial) Existence of minimizers. In this section we will show that the minimizers to the functional F[u] exist, and they are unique. This will give the wellosedness of the system (1.1) by the Dirichlet s Princile. We will state the following lemma; its roof is a straightforward verification which we will omit. Lemma The maings u G[u] and u F[u] are strictly convex. The main result of this aer is contained in the following theorem. Theorem Let b L q () and g L (), where 1 <, q < are conjugate. If F[u] is defined as in (3.7) by F[u] = 1 G(u)(x, x ) dx dx + b(x)u(x)dx, then (3.10) inf{f[u] : u A}

11 DIRICHLET S PRINCIPLE IN PERIDYNAMICS 11 attains its minimum on a minimizer ū; furthermore, this minimizer is unique. Proof. Existence. Note that F[u] 0 for every u W. Write inf{f[u] : u A} = m. Observe that m < since the function F has a finite value for the function which is zero in and equals g on. Let {u ν } be a minimizing sequence so that F[u ν ] m. Thus there exists M > 0 such that F[u ν ] < M. We will show that the sequence u ν is bounded in L. We aly Young s inequality with constants deending on ε (to be chosen later) so we have M > 1 G(u ν ) dx dx + b(x)u ν (x)dx G(u ν) L [() 2 ] C(ε) b q L q () ε u ν L (), for > 1 finite and 1/ + 1/q = 1. Alying Lemma 3.5 we obtain (3.11) M û ν L µ[() 2 ] C(ε) b q L q () εa û ν L µ[() 2 ] where A > 0 deends on the Poincaré s constant λ P ncr. By choosing ε sufficiently small we can find γ > 0 such that (3.12) û ν L µ [() 2 ] γ. Therefore, we may extract a subsequence (unrelabeled) {û ν } such that û ν ū in L µ[( ) 2 ], for some ū L µ( ). By Mazur s Lemma, there exists ū ν = N j i=j λ(j) i u i, where for all j, λ (j) i = 1 and 0, such that ū ν ū in L µ[( ) 2 ]. Then emloying Lemma 3.10 we have λ (j) i F[ū ν ] = F[ N j i=j λ(j) i û i ] N j i=j λ(j) i F[û i ] ( N j i=j λ(j) i )F[û ν ] = F[û ν ]. By assing to the limit in the above inequality, since û ν is a minimizing sequence, we have lim ν F[ū ν] m. By Fatou s lemma F[ū] lim ν F[ū ν], hence, from the above inequalities we have that F[ū] m. Note also that ū satisfies the boundary conditions, since it is the limit of the minimizing sequence û ν where û ν = g on. This shows that ū is a minimizer of (3.10). Uniqueness. Assume that ū, v W are minimizers of F with ū v. Let w = ū + v. Notice that w W. Now by Lemma 3.10, we have 2 m F[ w] 1 2 F[ū] + 1 F[ v] = m. 2

12 12 BRITTNEY HINDS, PETRONELA RADU Hence w is a minimizer of F. This imlies that 1 2 F[ū] + 1 F[ v] F[ w] = 0. 2 Thus G(ū) + G( v) G( w) dx dx = However, by Lemma 3.10, the integrand is strictly ositive, a contradiction. Thus ū = v in, the desired uniqueness. References [1] R. A. Adams, Sobolev Saces. Academic Press, Second Edition, [2] B. Aksoylu, M. Parks, Variational theory and domain decomosition for nonlocal roblems. Alied Mathematics and Comutation 217, no. 14, (2011). [3] G. Aubert, P. Kornrobst, Can the nonlocal characterization of Sobolev saces by Bourgain et al. be useful for solving variational roblems?, SIAM Journal on Numerical Analysis, 47, no.2, (2009). [4] B. Alali, R. Liton, Multiscale analysis of heterogeneous media in eridynamic formulation. (To aear in Journal of Elasticity). [5] F. Andreu, J. Mazón, J. Rossi, J. Toledo, The Neumann roblem for nonlocal nonlinear diffusion equations. J. Evol. Eqns, 8(1), (2008) [6] F. Andreu, J. Mazón, J. Rossi, J. Toledo, A nonlocal -Lalacian evolution equation with Neumann boundary conditions. Journal de Mathematiques Pures et Aliques, 90, no. 2, (2008). [7] F. Andreu, J. Mazón, J. Rossi, J. Toledo, The limit as in a nonlocal Lalacian evolution equation. A nonlocal aroximation of a model for sandiles. Calc. Var. Partial Differential Equations, 35, (2009). [8] F. Andreu, J. Mazón, J. Rossi, J. Toledo, A nonlocal -Lalacian evolution equation with nonhomogeneous Dirichlet boundary conditions. SIAM Journal of Mathematical Analysis, 40, no. 5, (2009). [9] F. Andreu, J. Mazón, J. Rossi, J. Toledo, Nonlocal Diffusion Problems, American Mathematical Society and Real Sociedad Matemática Esañola (2010). [10] F. Bobaru, M. Duanganya, The eridynamic formulation for transient heat conduction. International Journal of Heat and Mass Transfer, 53, no , (2010). [11] N. Burch, R. Lehoucq, Classical, nonlocal, and fractional diffusion equations. (to aear in the International Journal for Multiscale Comutational Engineering) [12] C. Carrillo, P. Fife, Satial effects in discrete generation oulation models. Journal of Mathematical Biology, 50, (2005). [13] J. Coville, L. Duaigne, On a nonlocal equation arising in oulation dynamics. Proc. Roy. Soc. Edinburgh Sect. A (2007). [14] B. Dacorogna, Direct Methods in the Calculus of Variations, Sringer, [15] Q. Du, M. Gunzburger, R.B. Lehoucq, K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained roblems, and nonlocal balance laws. (2010) (Submitted) [16] Q. Du, K. Zhou, Mathematical analysis for the eridynamic nonlocal continuum theory, Mathematical Modeling and Numerical Analysis, 45, , (2011). [17] G. Gilboa, S. Osher, Nonlocal oerators with alications to image rocessing. Multiscale Modeling and Simulation, 7, no. 3, (2008). [18] G. Gilboa, S. Osher, Nonlocal image regularization and suervised segmentation. Multiscale Modeling and Simulation, 6, no. 2, (2007). [19] G.Gilboa, S. Osher, Nonlocal convex functionals for image regularization. UCLA CAM Reort, [20] M. Gunzburger, R.B. Lehoucq, A nonlocal vector calculus with alications to nonlocal boundary value roblems., Multiscale Modeling and Simulation, 8, (2010). [21] G. Mingione, The singular set of solutions to non-differentiable ellitic systems. Archive for Rational Mechanics and Analysis, 166, (2003).

13 DIRICHLET S PRINCIPLE IN PERIDYNAMICS 13 [22] A. Mogilner and Leah Edelstein-Keshet, A nonlocal model for a swarm, J. Math. Biol. 38, (1999) [23] P. Seleson, M. L. Parks, M. Gunzburger, and R. B. Lehoucq, Peridynamics as an Uscaling of Molecular Dynamics, Multiscale Modeling and Simulation, 8(1), (2009). [24] S.A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, Journal of the Mechanics and Physics of Solids, 48, (2000). University of Nebraska-Lincoln, Lincoln, NE address: s-bhinds1@math.unl.edu, radu@math.unl.edu