EXISTENCE FOR NONLOCAL VARIATIONAL PROBLEMS IN ...

c 2014 Society for Industrial and Applied Mathematics 

SIAM J. MATH. ANAL. Vol. 46, No. 1, pp. 890–916

EXISTENCE FOR NONLOCAL VARIATIONAL PROBLEMS IN PERIDYNAMICS∗ ´ C. BELLIDO† AND CARLOS MORA-CORRAL‡ JOSE Abstract. We present an existence theory based on minimization of the nonlocal energies appearing in peridynamics, which is a nonlocal continuum model in solid mechanics that avoids the use of deformation gradients. We employ the direct method of the calculus of variations in order to find minimizers of the energy of a deformation. Lower semicontinuity is proved under a weaker condition than convexity, whereas coercivity is proved via a nonlocal Poincar´ e inequality. We cover Dirichlet, Neumann, and mixed boundary conditions. The existence theory is set in the Lebesgue Lp spaces and in the fractional Sobolev W s,p spaces, for 0 < s < 1 and 1 < p < ∞. Key words. peridynamics, nonlocal energy, minimum energy deformations AMS subject classifications. 26A33, 26B25, 35Q74, 45G15, 49J45, 74B20, 74G65 DOI. 10.1137/130911548

1. Introduction. Peridynamics is a nonlocal continuum model in solid mechanics introduced by Silling [45]. The main difference with the usual Cauchy–Green elasticity [9, 15] relies on the nonlocality, which is reflected in the fact that points separated by a positive distance exert a force upon each other. Mathematically, deformations are not assumed to be weakly differentiable, in contrast with classical continuum mechanics, and in particular hyperelasticity, where they are required to be Sobolev. This makes peridynamics a suitable framework for problems where discontinuities appear naturally, such as fractures, dislocations, or, in general, multiscale materials. Later developments and variants of the original peridynamic theory are to be found in [32, 43, 44]. The peridynamic equation of motion [45] is typically a second-order hyperbolic equation whose corresponding elliptic operator is nonlocal, of the type of a p-Laplacian. This is in contrast to nonlocal diffusion problems (see, e.g., [7]), which lead to parabolic equations, again with a corresponding elliptic operator being nonlocal and often taken as the p-Laplacian. This paper focuses on the variational formulation of equilibrium states in peridynamics. We thus ignore the time dependence, and make the fundamental object to be the energy (sometimes called macroelastic; see [45]) of the deformation u : Ω → Rd , to which external body and surface forces can be added. The macroelastic energy has typically the form   w(x − x , u(x) − u(x )) dx dx, (1.1) Ω

Ω

where Ω ⊂ R represents the body, and w : Rn × Rd → R is the pairwise potential n

∗ Received by the editors February 28, 2013; accepted for publication (in revised form) December 16, 2013; published electronically February 20, 2014. http://www.siam.org/journals/sima/46-1/91154.html † Department of Mathematics, E.T.S.I. Industriales, University of Castilla-La Mancha, E-13071 Ciudad Real, Spain ([email protected]). This author was supported by Project MTM201019739 of the Spanish Ministry of Economy and Competitivity. ‡ Faculty of Sciences, Department of Mathematics, University Aut´ onoma of Madrid, E-28049 Madrid, Spain ([email protected]). This author was supported by Project MTM2011-28198 of the Spanish Ministry of Economy and Competitivity, the ERC Starting Grant 307179, the “Ram´ on y Cajal” program, and the European Social Fund.

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function. Here n ∈ N is the space dimension, and d ∈ N the dimension of the target space; in real applications, d should coincide with n (and with 3), but since we also want to treat the antiplane case d = 1, we prefer to carry out the proofs for a general d. Expression (1.1) reflects the two main features of the peridynamic theory: the nonlocality (expressed as a double integral) and the absence of gradients, which are often replaced by weighted difference quotients. As nearby particles interact with a stronger force than distant ones, it is natural ˜ ) blows up to infinity at 0, for each y ˜ ∈ Rd . It is to assume that the function w(·, y also natural to assume that distant particles do not interact at all, so w(˜ x, ·) ≡ 0 if |˜ x| is larger than a so-called horizon, where the function w has been normalized so that its minimum value is 0. As a matter of fact, the assumption that w vanishes when |˜ x| is large adds a further difficulty to the mathematical analysis. The function w has to satisfy additional mathematical properties in order to meet some physical requirements such as objectivity (see [44, section 4]). In this paper we do not insist on those properties, but rather focus on the conditions on w that guarantee existence of minimizers for the total energy of the deformation. In truth, we work with functionals that are local perturbations of (1.1), corresponding to the addition of external forces. Our aim is to prove the existence of minimizers for those functionals under fairly general assumptions on the potential w; in particular, our existence theorems cover most of the existence results in peridynamics based on minimization, such as [2, 21, 28]. To that aim, we employ the direct method of the calculus of variations (e.g., [16]), so that semicontinuity and coercivity are the two main ingredients. While the available literature on existence covers mainly the linear case, so that w(˜ x, ·) is quadratic (see, e.g., [2, 21, 36]), only a few works deal with the nonlinear case (see [28]). We are not aware of an existing result dealing with w(˜ x, ·) when it is neither quadratic nor convex, which is the case of interest in the present paper. As a model case, and to anticipate the main results of this paper (Theorems 5.1, 5.2, 7.1, and 7.2), we prove existence of minimizers for energies of the form    w1 (u(x) − u(x ))  dx dx − f (x) · u(x) dx, w2 (x − x ) Ω Ω∩B(x,δ) Ω where w1 (˜ y) ∼ |˜ y|p and w2 (˜ x) ∼ |˜ x|α for some 1 < p < ∞ and some 0 ≤ α < n + p. In addition, w1 is assumed to satisfy a weak convexity assumption if 0 ≤ α ≤ n, but no convexity assumption whatsoever if n < α < n + p. The function f is so that the  integral Ω f · u is well defined. The first issue we find in our analysis is to determine the proper functional space to set the problem, and this depends on the growth conditions assumed on w. In this paper, they are quite general, and cover the cases of the papers mentioned above, in ˜ ) at 0 obeys an inverse power law of the form particular, when the singularity of w(·, y (1.2)

˜) ∼ w(˜ x, y

|˜ y |p |˜ x|α

˜∼0 for x

for some 1 < p < ∞ and 0 ≤ α < n + p. For this special growth, we distinguish the weakly singular case 0 ≤ α < n and the strongly singular case n < α < n + p. When 0 ≤ α < n, the analysis of the lower semicontinuity is reduced to the recent study carried out by Elbau [23] and lies in the functional framework of Lebesgue Lp spaces. The weak lower semicontinuity is proved in [23] to be equivalent to an interesting convexity property of the integrand w, of a different nature than those

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convexity properties equivalent to weak lower semicontinuity for local problems (see, e.g., [16, Chap. 8]); we will discuss this issue in section 3 in our particular peridynamics framework. The coercivity for the Dirichlet problem was proved by Andreu et al. [8] in their study of nonlocal diffusion problems, and later used by [3, 28] in the context of peridynamics. The coercivity for the Neumann and mixed problem was proved by Aksoylu and Mengesha [2] using a Poincar´e-type inequality proved by Ponce [40] in his study of nonlocal characterizations of Sobolev spaces (see also [13]). As a matter of fact, we shall need some adaptations of those results to our context. At this point, we ought to mention that Dirichlet and mixed boundary value problems have a slightly different meaning than for local problems, one the reasons being that Lp functions do not have traces of the boundary ∂Ω. In contrast, Dirichlet conditions in the context of peridynamics prescribe the value of the deformation in a set of positive measure. The lower semicontinuity in the case n < α < n + p is in fact trivial, since the functional framework is that of the fractional Sobolev spaces W s,p with s = α−n p , and weak convergence in W s,p implies (for a subsequence) convergence a.e. The coercivity, on the other hand, is a consequence of an improved Poincar´e-type inequality in fractional Sobolev spaces recently proved in Hurri-Syrj¨ anen and V¨ ah¨akangas [29]. It is worth mentioning that the need of improved Poincar´e-type inequalities is a result of the assumption that w(˜ x, ·) vanishes for |˜ x| large. The existence theory for the critical case α = n is also covered by reducing it to the case 0 ≤ α < n and to the functional framework of Lp spaces. In doing that, we do not provide a full characterization of the lower semicontinuity, so that our conditions on w may not be optimal. Nonlocal variational problems, of which (1.1) is a particular case, have attracted a great attention in the mathematical community in the last two decades, coming from fields such as statistical mechanics [5], abstract results involving nonlocality of gradients [37, 38], ferromagnetism [42], nonlocal p-Laplacian [7], imaging [12, 25, 30], characterization of Sobolev spaces [13, 14, 33, 40, 41], crystal dislocations [17, 26, 35] as well as, of course, peridynamics [2, 4, 21, 22, 27, 28]. The outline of the paper is as follows. In section 2 we present the mechanical model, make the general assumptions of the paper, and explain the notation used. In section 3 we prove the lower semicontinuity of the nonlocal energy in the weak topology of Lp , by means of a nonlocal convexity property of the integrand w. In section 4 we obtain the inequalities that allow us to prove the coercivity in Lp , for Dirichlet, mixed, and Neumann boundary conditions. Section 5 uses the results of sections 3 and 4 to show the existence of minimizers of the energy in the Lp context. Section 6 presents the key inequalities for the coercivity in W s,p , again for the three types of boundary conditions. Section 7 proves the existence theorems for deformations in W s,p , using the results of section 6. In section 8 we write down the Euler–Lagrange equations corresponding to the minimizers. 2. Model, notation, and general assumptions. In this section we present the mechanical model. We refer to the papers [2, 45, 43, 44] for further motivation and physical interpretation. In particular, our model follows closely that of Hinds and Radu [28]. Let Ω be a nonempty open bounded subset of Rn representing the reference configuration of the body. The peridynamics theory requires the distinction within Ω of an interior part Ω0 of the body and a nonlocal boundary Ω1 , so that Ω is the disjoint union of Ω0 and Ω1 ; this is in contrast to other nonlocal theories, where boundary conditions are often imposed on Rn \ Ω. When needed, we will assume that

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Ω0 + B(0, δ) ⊂ Ω for some δ > 0, where B(0, δ) denotes the open ball of center 0 and radius δ. This number δ represents the horizon of the potential w, and measures the distance after which there is no interaction between particles, i.e., w(˜ x, ·) = 0 if |˜ x| ≥ δ. Thus, we are making the natural assumption that the inner part Ω0 of the body does not interact with the exterior Rn \ Ω. In some studies (e.g., [2]), the stronger condition Ω0 + B(0, δ) = Ω is imposed, but this is not really needed in our model. Of course, Ω0 + B(0, δ) denotes the set of points in Rn that can be expressed as a sum of an element of Ω0 plus an element of B(0, δ). The set Ω can thus be regarded as a nonlocal closure of Ω0 , and the deformation u is defined on the whole Ω. In fact, we will assume u ∈ Lp (Ω, Rd ) for some 1 < p < ∞. The macroelastic energy of a deformation u is calculated through (1.1). Here ˜ is the set of x − x with ˜ × Rd → R is the pairwise potential function, and Ω w:Ω  ˜ x, x ∈ Ω. Clearly, Ω is open. Thanks to Fubini’s theorem, without loss of generality we may assume that (2.1)

˜) w(−˜ x, −˜ y) = w(˜ x, y

˜ and y ˜∈Ω ˜ ∈ Rd , for all x

˜ ) with just by substituting w(˜ x, y 1 ˜ ) + w(−˜ [w(˜ x, y x, −˜ y)] . 2 Equality (2.1) will be assumed throughout the paper, even though not explicitly stated; in turn, it is the realization in this context of Newton’s third law (see [45, ˜ × B(Rd )-measurable, eqs. (6) and (27)]). The function w is required to be Ln (Ω) i.e., Lebesgue measurable in the first n variables, and Borel measurable in the last d variables. This guarantees that the integrand in (1.1) is Lebesgue measurable. The expression a.e. for almost everywhere or almost every refers to the Lebesgue measure, which is denoted by Ln when the underlying space is Rn . External body and surfaces forces are added to the macroelastic energy to conform the total energy. Those external forces have the form  F (x, u(x)) dx (2.2) − Ω

for some potential function F : Ω×Rd → R assumed to be Ln (Ω)×B(Rd )-measurable. The part of the integral (2.2) in Ω0 corresponds to the body force, while the part in Ω1 corresponds to the surface force; we recall that, in the context of peridynamics, notions like boundary or surface have positive volume. The distinction between Ω0 and Ω1 is part of the mechanical model, but it hardly affects the mathematical analysis. In many practical cases, both body and surface forces are linear, so that F (x, y) = f (x) · y,

(2.3)

x ∈ Ω,

y ∈ Rd

for a given measurable f : Ω → Rd . Here · denotes the scalar (inner) product in Rd . Thus, the total energy of a deformation is    (2.4) I(u) := w(x − x , u(x) − u(x )) dx dx − F (x, u(x)) dx, Ω

Ω

Ω

so equilibrium solutions in the static theory correspond to critical points of I. In fact, this paper analyzes the existence of global minimizers of I.

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The nonlocal boundary conditions present some peculiarities (see [27, 20]). Dirichlet conditions prescribe u = b a.e. in a measurable subset ΩD of Ω1 , for a given b : ΩD → Rd . The Neumann part of the nonlocal boundary is Ω1 \ ΩD . A pure Dirichlet problem corresponds to Ln (ΩD ) = Ln (Ω1 ), a pure Neumann problem corresponds to Ln (ΩD ) = 0, while a mixed problem corresponds to 0 < Ln (ΩD ) < Ln (Ω1 ). In the particular but interesting case where F is of the form (2.3) with f ∈ L1 (Ω, Rd )  satisfying Ω f = 0, the energy I of (2.4) is invariant under translations. Hence, to avoid the trivial nonuniqueness  of the pure Neumann problem given by translations, the normalization condition Ω u = 0 is imposed. Now we say some words about the notation. We write x for the coordinates in ˜ the reference configuration Ω, and y in the deformed configuration Rd . We write x ˜ while y ˜ is reserved for coordinates of functions whose argument for coordinates in Ω, is typically a difference between two points in the deformed configuration. Thus, the ˜ ). Vector quantities are written in natural notation for the variables of w is (˜ x, y boldface. For 1 ≤ p ≤ ∞, the Lebesgue Lp spaces are defined in the usual way, and p denotes the conjungate exponent of p. We will always indicate the domain and target sets as in, for example, Lp (Ω, Rd ), except if the target space is R, in which case we p n − will simply write  L (Ω). Given a measurable subset A of R , the expression A u 1 indicates Ln (A) A u. We will also use fractional Sobolev spaces: for 0 < s < 1 and 1 ≤ p < ∞, the space W s,p (A, Rd ) is the set of functions u ∈ Lp (A, Rd ) such that the fractional Sobolev seminorm   |u|W s,p (A,Rd ) :=

A

A

|u(x) − u(x )| |x − x |

n+sp

p

 p1 dx dx 

is finite. The corresponding norm is defined as  p1  p p

u W s,p (A,Rd ) := u Lp (A,Rd ) + |u|W s,p (A,Rd ) . Weak convergence is indicated by , while strong or a.e. convergence is indicated by →. We recall that W s,p is a reflexive Banach space when 1 < p < ∞. For any real-valued function w, its positive and negative parts are denoted, respectively, by w+ := max{w, 0} and w− := − min{w, 0}. The characteristic function of a subset A of Rn is denoted by χA For the convenience of the reader, we write down the fractional Sobolev immersions that will be used in the paper. The following result is well known; proofs can be found, e.g., in [1, Chap. 7], [34, Chap. 14] or [18, sects. 6–8]. Proposition 2.1. Let Ω be a Lipschitz domain of Rn . Let 0 < s < 1 and 1 ≤ p < ∞. Then the following assertions hold. (i) If sp < n, define p∗ :=

np . n − sp

Then W s,p (Ω, Rd ) is continuously embedded in Lq (Ω, Rd ) for all q ∈ [1, p∗ ], and compactly embedded for all q ∈ [1, p∗ ). (ii) If sp = n, then W s,p (Ω, Rd ) is compactly embedded in Lq (Ω, Rd ) for all q ∈ [1, ∞).

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(iii) If sp > n, define α∗ :=

sp − n . p

Then W s,p (Ω, Rd ) is continuously embedded in C 0,α (Ω, Rd ) for all α ∈ (0, α∗ ], and compactly embedded for all α ∈ (0, α∗ ). In the statement above, the set C 0,α (Ω, Rd ) denotes the Banach space of H¨ older continuous functions (up to the boundary) of exponent α. 3. Weak lower semicontinuity in Lp . The first question that arises in an existence theory is the choice of the functional space. If the growth of w is of the form (1.2) with 0 ≤ α < n then Lp is the natural space to set the problem, as shown in the following inequality. Lemma 3.1. Let Ω be a bounded measurable subset of Rn . Let 1 ≤ p < ∞ and 0 ≤ α < n. Then there exist C1 , C2 > 0, depending only on n, p, α, and Ω, such that for any u ∈ Lp (Ω, Rd ),  p  p           |u(x) − u(x )|p      C1 dx dx ≤ C2 u(x) − − u dx ≤ u(x) − − u dx.  |α |x − x Ω Ω Ω Ω Ω Ω Proof. Using Jensen’s inequality, we find that for all x ∈ Ω,  p   p      u(x) − − u = − [u(x) − u(x )] dx  ≤ − |u(x) − u(x )|p dx     Ω Ω Ω   p |u(x) − u(x )| ≤ sup |x − x |α − dx , |x − x |α x ∈Ω Ω so, denoting by diam the diameter of a set, we have that  p     p   |u(x) − u(x )| u(x) − − u dx ≤ (diam Ω)α 1 dx dx,   Ln (Ω) Ω Ω |x − x |α Ω Ω which shows the first inequality. For the second, we have that  p   p      u(x) − −Ω u + u(x ) − −Ω u |u(x) − u(x )|p  p−1 dx dx ≤ 2 dx dx |x − x |α |x − x |α Ω Ω Ω Ω  p    u(x) − −Ω u p =2 dx dx. |x − x |α Ω Ω Now,  p     p       u(x) − −Ω u 1    u(x) − − u dx dx dx ≤ sup dx   α  α |x − x | x∈Ω Ω |x − x | Ω Ω Ω Ω and  sup

x∈Ω

Ω

1 dx ≤ |x − x |α

 B(0,diam Ω)

1 (diam Ω)n−α , dx = σ n |x|α n−α

where σn denotes the area of the unit sphere in Rn . This shows the second inequality.

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Lemma 3.1 is not actually used in the paper, but it explains why Lp is the correct space under the growth condition (1.2) with 0 ≤ α < n. The main results of this section show the lower semicontinuity of the macroelastic energy (1.1) under weak convergence in Lp . A useful tool will be Young measures. Since they are only used in this section, we will not explain them in detail, but rather we assume the reader to have some familiarity with them. We just mention their fundamental property, while we refer for the proofs and more background to [39], [10, section 4.3], and, in particular, [24, Chap. 8], whose notation is closely followed here. The abovementioned fundamental property is that if 1 ≤ p < ∞ and {uj }j∈N is a sequence bounded in Lp (Ω, Rd ) then, for a subsequence (not relabeled) there exists a Young measure (νx )x∈Ω such that {uj }j∈N converges to (νx )x∈Ω as j → ∞ in the sense of Young measures. This means that νx is a probability measure in Rd for a.e. x ∈ Ω with the property that for any Borel set E of Rd , the map x → νx (E) is measurable in Ω; moreover,   p (3.1) |y| dνx (y) dx < ∞ Ω

Rd

and for every continuous function ϕ : Rd → R of compact support,    lim ϕ(uj (x)) dx = ϕ(y) dνx (y) dx. j→∞

Ω

Ω

Rd

We say that {uj }j∈N generates (νx )x∈Ω . We start with an auxiliary result calculating the Young measure of the difference uj (x) − uj (x ), for which the following notation is useful. Definition 3.2. Given two probability measures μ1 and μ2 in Rd , we define its convolution difference μ1 μ2 as   (μ1 μ2 ) (E) := χE (y − y ) dμ1 (y) dμ2 (y ) Rd

Rd

for any Borel set E of Rd . Note that μ1 μ2 is a probability measure in Rd , and that for any continuous function ϕ : Rd → R of compact support,    (3.2) ϕ(˜ y) d (μ1 μ2 ) (˜ y) = ϕ(y − y ) dμ1 (y) dμ2 (y ). Rd

Rd

Rd

In fact, thanks to a standard approximation result, equality (3.2) holds true for continuous functions ϕ : Rd → R such that the left-hand side of (3.2) is finite. We employ the notation μ1 × μ2 for the (sometimes called tensorial or Cartesian) product of the measures μ1 and μ2 (see, e.g., [6, Thm. 1.74]). Lemma 3.3. Let 1 ≤ p < ∞. Let {uj }j∈N be a sequence bounded in Lp (Ω, Rd ) ¯ j ∈ Lp (Ω × Ω, Rd ) generating the Young measure (νx )x∈Ω . For each j ∈ N, define u as (3.3)

¯ j (x, x ) := uj (x) − uj (x ), u

(x, x ) ∈ Ω × Ω.

Then {¯ uj }j∈N generates the Young measure (νx νx )(x,x )∈Ω×Ω in Lp (Ω × Ω, Rd ). Proof. For each j ∈ N, define vj ∈ Lp (Ω × Ω, Rd × Rd ) as vj (x, x ) := (uj (x), uj (x )) ,

(x, x ) ∈ Ω × Ω.

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It was essentially proved in Pedregal [38, Prop. 2.3] (see also the proof of [23, Thm. 11]) that the sequence {vj }j∈N generates the Young measure (νx × νx )(x,x )∈Ω×Ω . Note that 1

¯ uj Lp (Ω×Ω,Rd ) ≤ 2 Ln (Ω) p uj Lp (Ω,Rd ) ,

j ∈ N,

and let (¯ ν(x,x ) )(x,x )∈Ω×Ω be the Young measure generated for a subsequence of {¯ uj }j∈N . We use the probabilistic representation formula for Young measures of Ball [11, p. 6], according to which for each x0 , x0 ∈ Ω, ¯ j (x, x ) ∈ E}) L2n ({(x, x ) ∈ B((x0 , x0 ), R) : u ,  R→0 j→∞ L2n (B((x0 , x0 ), R))

L2n ({(x, x ) ∈ B((x0 , x0 ), R) : vj (x, x ) ∈ E  }) × νx0 (E  ) = lim lim R→0 j→∞ L2n (B((x0 , x0 ), R)) ν¯(x0 ,x0 ) (E) = lim lim

(3.4)

νx 0

for any Borel set E ofRd , and any Borel set E  of R2d . Now, let E be an open set of Rd , and define E  := (y, y ) ∈ Rd × Rd : y − y ∈ E , which is easily seen to be an open set of R2d . Then ¯ j (x, x ) ∈ E} = {(x, x ) ∈ B((x0 , x0 ), R) : vj (x, x ) ∈ E  } {(x, x ) ∈ B((x0 , x0 ), R) : u for every R > 0, and χE (y − y ) = χE  (y, y ) for every y, y ∈ Rd . Consequently, using also (3.4) and Definition 3.2, we find that  

   χE  (y, y ) dνx0 (y) dνx0 (y ) ν¯(x0 ,x0 ) (E) = νx0 × νx0 (E ) = Rd Rd  

= χE (y − y ) dνx0 (y) dνx0 (y ) = νx0 νx0 (E). Rd

Rd

Thus, the two probability measures ν¯(x0 ,x0 ) and νx0 νx0 coincide in all open sets of Rd and, hence (see, e.g., [6, Prop. 1.8]), ν¯(x0 ,x0 ) = νx0 νx0 . By uniqueness, we conclude that the whole sequence {¯ uj }j∈N generates the Young measure (νx νx )(x,x )∈Ω×Ω . The following observation follows immediately from Fubini’s theorem, and will be ˜ then the property ˜∈Ω used throughout the paper: if a property P (˜ x) holds for a.e. x P (x − x ) holds for a.e. (x, x ) ∈ Ω × Ω. A characterization in terms of w of the lower semicontinuity of nonlocal functionals more general than (1.1) with respect to the weak topology of Lp was given in a recent paper by Elbau [23]. Unfortunately, his proof of [23, Thm. 11] has a gap, because at two different points he needs the equi-integrability of w. For the convenience of the reader, we have decided to write down a self-contained proof of the lower semicontinuity result of (1.1) (the sufficient condition) for the situation at hand, in which we have added assumption (a) below so as to guarantee the equi-integrability of w− . The equi-integrability of w+ is obtained by a common argument in Young measures (see, e.g., [24, Cor. 8.8]), according to which a sequence of functions bounded in Lp that generates a Young measure admits another sequence of functions that is p-equi-integrable and generates the same Young measure. We do not include here the necessity part, as we are only concerned with existence, but instead refer the interested reader to [23, Thm. 11] for the proof of that part. Nevertheless, it is worth emphasizing that the convexity property (d) below is equivalent to the sequential weak lower semicontinuity of the functional (1.1) in Lp . Proposition 3.4. Let Ω be a nonempty, bounded open subset of Rn . Let 1 < ˜ × Rd → R be Ln (Ω) ˜ × B(Rd )-measurable. Assume that p < ∞. Let w : Ω

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˜ with (a) there exist a1 ∈ L1 (Ω) a1 ≥ 0,

a1 (−˜ x) = a1 (˜ x),

˜ ˜∈Ω a.e. x

˜ and all y ˜∈Ω ˜ ∈ Rd , and 1 ≤ q < p such that for a.e. x ˜ ) ≤ a1 (˜ x, y x) (1 + |˜ y| ) ; w− (˜ q

˜ with a2 ≥ 0 and C > 0 such that for a.e. x ˜ and ˜∈Ω (b) there exist a2 ∈ L1loc (Ω) ˜ ∈ Rd , all y p

˜ ) ≤ a2 (˜ x, y x) + C |˜ y| ; w+ (˜ ˜ ∈ Ω; (c) w(˜ x, ·) is continuous for a.e. x (d) for a.e. x ∈ Ω and all u ∈ Lp (Ω, Rd ), the function  w(x − x , y − u(x )) dx y → Ω

is convex in Rd . For each j ∈ N, let uj , u ∈ Lp (Ω, Rd ) be such that uj  u in Lp (Ω, Rd ) as j → ∞. Then (3.5)        w(x − x , u(x) − u(x )) dx dx ≤ lim inf w(x − x , uj (x) − uj (x )) dx dx. Ω

j→∞

Ω

Ω

Ω

Proof. By passing to a subsequence, we can assume that the inferior limit in the right-hand side of (3.5) is in fact a finite limit, and that the sequence {uj }j∈N generates a Young measure ν = (νx )x∈Ω in Lp (Ω, Rd ). Thanks to a standard result in the theory of Young measures (see, e.g., [24, Cor. 8.8]), there exists a sequence {vj }j∈N in Lp (Ω, Rd ) generating the Young measure ν, and such that the sequence {|vj |p }j∈N is equi-integrable. By Lemma 3.3, the sequence {¯ uj }j∈N defined by (3.3) generates the Young measure (νx νx )(x,x )∈Ω×Ω in Lp (Ω×Ω, Rd ). We now show that the sequence of functions Ω × Ω (x, x ) → w− (x − x , uj (x) − uj (x ))

(3.6)

is equi-integrable. Indeed, using (a) we find that for a.e. (x, x ) ∈ Ω × Ω and all j ∈ N,

q 0 ≤ w− (x − x , uj (x) − uj (x )) ≤ a1 (x − x ) 1 + |uj (x) − uj (x )|

q  q ≤ a1 (x − x ) 1 + 2q−1 |uj (x)| + |uj (x )| . As the sum of equi-integrable sequences is equi-integrable, to show that (3.6) is equiintegrable, it suffices to show that each of the three sequences of functions (3.7) q q (x, x ) → a1 (x − x ), (x, x ) → a1 (x − x ) |uj (x)| , (x, x ) → a1 (x − x ) |uj (x )| is equi-integrable in Ω × Ω. As      a1 (x − x ) d(x, x ) = Ω×Ω

Ω

x−Ω

a1 (x ) dx dx ≤ Ln (Ω) a1 L1 (Ω) ˜ ,

EXISTENCE FOR NONLOCAL PROBLEMS IN PERIDYNAMICS

899

the first one is equi-integrable. For the second, we observe that for each t > 0 and j ∈ N,  q a1 (x − x ) |uj (x)| d(x, x ) {(x,x )∈Ω×Ω:a1 (x−x )|uj (x)|q >t}





|uj (x)|q

= Ω

{x ∈x−Ω:a1 (x )|uj (x)|q >t}





|uj (x)|

≤

q {x ∈x−Ω:a

Ω



1

(x )>t1/2 }

a1 (x ) dx dx

a1 (x ) dx 

+ {x ∈x−Ω:|uj (x)|q >t1/2 }

moreover,



 |uj (x)|q

sup j∈N

Ω



q sup uj Lq (Ω,Rd ) j∈N

≤ and



1

(x )>t1/2 }

˜ {x ∈Ω:a

1

Ω

{x ∈x−Ω:|uj (x)|q >t1/2 }



{x∈Ω:|uj (x)|q >t1/2 }

j∈N

|uj (x)|

j∈N

a1 (x ) dx

a1 (x − x ) dx dx

q Ω



≤ a1 L1 (Ω) ˜ sup

a1 (x ) dx dx

a1 (x ) dx dx



= sup



a1 (x ) dx dx;

(x )>t1/2 }

 |uj (x)|q

sup j∈N

{x ∈x−Ω:a



q

{x∈Ω:|uj (x)|

q

>t1/2 }

|uj (x)| dx,

˜ and the sequence {|uj |q }j∈N is which proves the equi-integrability, since a1 ∈ L1 (Ω) equi-integrable because q < p. For the third sequence in (3.7), we notice that, thanks to Fubini’s theorem and the symmetry of a1 , for each j ∈ N,  q a1 (x − x ) |uj (x )| d(x, x ) {(x,x )∈Ω×Ω:a1 (x−x )|uj (x )|q >t}



a1 (x − x ) |uj (x)| d(x, x ). q

= {(x,x )∈Ω×Ω:a

1

(x−x )|u

q

j (x)|

>t}

Thus, as the second sequence of (3.7) is equi-integrable, so is the third. Hence the sequence (3.6) is equi-integrable, so we can apply the main lower semicontinuity theorem for Young measures (e.g., [24, Thm. 8.6]) to obtain that    ˜ ) d (νx νx ) (˜ w(x − x , y y) dx dx Ω Ω Rd   (3.8) ≤ lim inf w(x − x , uj (x) − uj (x )) dx dx. j→∞

Ω

Recall from (3.2) that    (3.9)

Ω

˜ ) d (νx νx ) (˜ w(x − x , y y) dx dx Ω Ω     = w(x − x , y − y ) dνx (y) dνx (y ) dx dx. Rd

Ω

Ω

Rd

Rd

´ C. BELLIDO AND CARLOS MORA-CORRAL JOSE

900

Now, for each x ∈ Ω and each Young measure μ = (μx )x ∈Ω , define Φx,μ : Rn → R

as

 

(3.10)

Φx,μ (y) := Ω

Rd

w(x − x , y − y ) dμx (y ) dx .

Note that Φx,μ takes finite values thanks to the growth conditions Proposition 3.4(a)– (b) and to the integrability property (3.1). The definition is made so that (3.11)       Ω

Ω

Rd

Rd

w(x − x , y − y ) dνx (y) dνx (y ) dx dx =

Ω

Rd

Φx,ν (y) dνx (y) dx.

p d Note that, via the usual identification of a function u ∈ L (Ω, R ) with the Young measure δu(x) x∈Ω (where δ denotes Dirac’s delta), we have that



Φx,u (y) =

w(x − x , y − u(x )) dx .

Ω

Thus,  

w(x − x , u(x) − u(x )) dx dx =

(3.12) Ω



Ω

Φx,u (u(x)) dx. Ω

Also note that the symmetry (2.1) of w yields    (3.13) Φx,ν (u(x)) dx = Φx,u (y) dνx (y) dx. Ω

Ω

Rd

Now we show that for a.e. x ∈ Ω and all y ∈ Rn , the sequence of functions {fj,x,y }j∈N in Ω defined by fj,x,y (x ) := w(x − x , y − vj (x )),

x ∈ Ω,

− is equi-integrable. The sequence {fj,x,y }j∈N can be shown to be equi-integrable by applying the same argument that showed that (3.6) is equi-integrable. Let us show + }j∈N is equi-integrable. We have that {fj,x,y

p p

+ (x ) ≤ a2 (x − x ) + C |y − vj (x )| ≤ a2 (x − x ) + 2p−1 C |y|p + |vj (x )| . 0 ≤ fj,x,y As a2 (x−·) ∈ L1 (Ω) and the sequence {|vj |p }j∈N is equi-integrable, then the sequence + {fj,x,y }j∈N is equi-integrable. Therefore, when we define fx,y : Ω → R as  fx,y (x ) := w(x − x , y − y ) dνx (y ), x ∈ Ω, Rd

we have, by the fundamental theorem for Young measures (see, e.g., [24, Thm. 8.6]),     lim fj,x,y (x ) dx = fx,y (x ) dx . j→∞

Ω

Ω

Now for a.e. x ∈ Ω, each y1 , y2 ∈ Rd , 0 ≤ λ ≤ 1, and j ∈ N, assumption (d) shows that Φx,vj (λy1 + (1 − λ)y2 ) ≤ λ Φx,vj (y1 ) + (1 − λ) Φx,vj (y2 ),

EXISTENCE FOR NONLOCAL PROBLEMS IN PERIDYNAMICS

901

that is to say,        fj,x,λy1 +(1−λ)y2 (x ) dx ≤ λ fj,x,y1 (x ) dx + (1 − λ) fj,x,y2 (x ) dx . Ω

Ω

Ω

Taking limits as j → ∞, we find that        fx,λy1 +(1−λ)y2 (x ) dx ≤ λ fx,y1 (x ) dx + (1 − λ) fx,y2 (x ) dx , Ω

Ω

Ω

so Φx,ν (λy1 + (1 − λ)y2 ) ≤ λ Φx,ν (y1 ) + (1 − λ) Φx,ν (y2 ). Thus, Φx,ν is convex. The weak convergence of {uj }j∈N in Lp (Ω, Rd ) and its convergence in the sense of Young measures show that  y dνx (y) a.e. x ∈ Ω. u(x) = Rd

This and Jensen’s inequality imply that  Φx,ν (u(x)) ≤ Φx,ν (y) dνx (y), Rd

so

 

 Φx,ν (u(x)) dx ≤

(3.14)

a.e. x ∈ Ω

Ω

Ω

Rd

Φx,ν (y) dνx (y) dx.

Analogously, 

  Φx,u (u(x)) dx ≤

(3.15) Ω

Ω

Rd

Φx,u (y) dνx (y) dx.

Putting together the relations (3.8), (3.9), and (3.11)–(3.15) we obtain   w(x − x , u(x) − u(x )) dx dx Ω Ω    = Φx,u (u(x)) dx ≤ Φx,u (y) dνx (y) dx d Ω Ω R = Φx,ν (u(x)) dx ≤ Φx,ν (y) dνx (y) dx Ω Rd Ω    = w(x − x , y − y ) dνx (y) dνx (y ) dx dx Ω Ω Rd Rd    ˜ ) d (νx νx ) (˜ = w(x − x , y y) dx dx Ω Ω Rd   ≤ lim inf w(x − x , uj (x) − uj (x )) dx dx, j→∞

Ω

Ω

as desired. Note that the assumptions of Proposition 3.4 are slightly more general than necessary for applications in peridynamics, since the energy function w is usually assumed

´ C. BELLIDO AND CARLOS MORA-CORRAL JOSE

902

to be nonnegative. Thus, assumption (a) and, hence, the first part of the proof showing the equi-integrability of (3.6) can be dispensed with. In fact, continuity of w(˜ x, ·) can be relaxed to lower semicontinuity, as the following result shows. Proposition 3.5. Let Ω be a nonempty, bounded open subset of Rn . Let 1 ≤ ˜ × Rd → R be Ln (Ω) ˜ × B(Rd )-measurable. Assume that p < ∞. Let w : Ω 1 ˜ ˜ and all ˜∈Ω (a) there exist a ∈ Lloc (Ω) with a ≥ 0 and C > 0 such that for a.e. x d ˜∈R , y p

˜ ) ≤ a(˜ 0 ≤ w(˜ x, y x) + C |˜ y| ; ˜ ∈ Ω; (b) w(˜ x, ·) is lower semicontinuous for a.e. x (c) for a.e. x ∈ Ω and all u ∈ Lp (Ω, Rd ), the function  y → w(x − x , y − u(x )) dx Ω

is convex in Rd . For each j ∈ N, let uj , u ∈ Lp (Ω, Rd ) be such that uj  u in Lp (Ω, Rd ) as j → ∞. Then (3.5) holds. Proof. We apply a standard approximation procedure for lower semicontinuous ˜ × Rd → [0, ∞) as functions. For each k ∈ N, define wk : Ω  ˜ × Rd . ˜) ∈ Ω ˜ ) := inf w(˜ ˜ | : z˜ ∈ Rd , (˜ x, y wk (˜ x, y x, z˜) + k|˜ z−y ˜ × B(Rd )-measurable and wk (˜ x, ·) is Then (see, e.g., [6, Lemma 1.61]) wk is Ln (Ω) ˜ Moreover, wk (˜ ˜ ∈ Ω. continuous for a.e. x x, ·)  w(˜ x, ·) as k → ∞ pointwise in Rd ˜ Note that we can apply Proposition 3.4, even when p = 1, since the ˜ ∈ Ω. for all x assumption p > 1 was only used there in order to prove the equi-integrability of (3.6). Therefore, for each k ∈ N,   wk (x − x , u(x) − u(x )) dx dx Ω Ω   ≤ lim inf wk (x − x , uj (x) − uj (x )) dx dx j→∞ Ω Ω   ≤ lim inf w(x − x , uj (x) − uj (x )) dx dx, j→∞

Ω

Ω

whereas by the monotone convergence theorem we have that        lim wk (x − x , u(x) − u(x )) dx dx = w(x − x , u(x) − u(x )) dx dx, k→∞

Ω

Ω

Ω

Ω

which concludes inequality (3.5). If w(˜ x, ·) is convex, then the growth conditions of w can be relaxed as follows. Proposition 3.6. Let Ω be a nonempty, bounded open subset of Rn . Let 1 < ˜ × Rd → R be Ln (Ω) ˜ × B(Rd )-measurable. Assume that p < ∞. Let w : Ω 1 ˜ (a) there exist a ∈ L (Ω) with a ≥ 0,

a(−˜ x) = a(˜ x),

˜ ˜∈Ω a.e. x

˜ and all y ˜∈Ω ˜ ∈ Rd , and 1 ≤ q < p such that for a.e. x ˜ ) ≤ a(˜ w− (˜ x, y x) (1 + |˜ y |q ) ;

EXISTENCE FOR NONLOCAL PROBLEMS IN PERIDYNAMICS

903

˜ ∈ Ω, the function w(˜ (b) for a.e. x x, ·) is convex and lower semicontinuous in Rd . For each j ∈ N, let uj , u ∈ Lp (Ω, Rd ) be such that uj  u in Lp (Ω, Rd ) as j → ∞. Then (3.5) holds. Proof. As in Proposition 3.4, we can assume that the inferior limit in the righthand side of (3.5) is a finite limit, and that the sequence {uj }j∈N generates a Young measure ν = (νx )x∈Ω in Lp (Ω, Rd ). Moreover, there exists a sequence {vj }j∈N of functions in Lp (Ω, Rd ) generating the same Young measure ν, and such that the sequence {|vj |p }j∈N is equi-integrable. By Lemma 3.3, the sequence {¯ uj }j∈N defined by (3.3) generates the Young measure (νx νx )(x,x )∈Ω×Ω in Lp (Ω × Ω, Rd ). The argument of Proposition 3.4 shows that the sequence of functions (3.6) is equi-integrable. Therefore, (3.8) holds, and so does (3.9). For each x ∈ Ω and each Young measure μ = (μx )x ∈Ω , define Φx,μ : Rn → R ∪ {∞} as in (3.10). Then (3.11), (3.12), and (3.13) hold. Moreover, the convexity of ˜ ∈ Ω implies at once that Φx,μ is convex. Hence, the same argument w(˜ x, ·) for a.e. x of Proposition 3.4 shows that (3.14) and (3.15) hold, and the proof is concluded. We finish this section with some remarks about condition (d) of Proposition 3.4, referring to Elbau [23] for more insight, but, at the same time, admitting that a better ˜ (which is a nonrealistic understanding is still pending. If w does not depend on x assumption for peridynamics), condition (d) is easily seen to be equivalent to the ˜ , then condition (d) becomes a usual convexity of w, whereas if w does depend on x weaker condition than convexity, as will be shown in the following paragraph. This is in contrast with local variational problems, in which the convexity property for the integrand that is equivalent to weak lower semicontinuity of the functional only ˜ (see, e.g., [10, 16, 24]). We also refer to Pedregal [38] for a concerns the variable y characterization of the weak lower semicontinuity through a convexity property in a nonlocal but slightly different situation. We now present an example showing that condition (d) of Proposition 3.4 is weaker than the requirement of w(˜ x, ·) to be convex, even in dimension 1. Let n = 1, ˜ = (−1, 1). Additionally, let h : [−1, 1) → R be any d = 1, and Ω = (0, 1), hence Ω smooth function such that h(−1) < 0 and  t h ≥ 0, h(t) = h(−t) for all t ∈ (0, 1). −1+t

Define w : (−1, 1) × R → R as w(˜ x, y˜) := h(˜ x) y˜2 . Then w(˜ x, ·) is not convex in R for x ˜ in a neigborhood of −1, but for all x ∈ Ω and all u ∈ Lp (Ω), the function  y → w(x − x , y − u(x )) dx Ω

is convex in R, since its second derivative satisfies for all y ∈ R,   x  2 ∂ w      (x − x , y − u(x )) dx = 2 h(x − x ) dx = 2 h(t) dt ≥ 0. ˜2 x−1 Ω ∂y Ω 4. Coercivity in Lp . In the Lp framework, the inequality needed for the coercivity for the Dirichlet problem was proved by Andreu et al. [8, Prop. 2.5]. In the context of peridynamics it was used in Aksoylu and Parks [3, Prop. 4.1] and Hinds and Radu [28, Lemma 3.5]. In any of those papers, the following result is proved. Proposition 4.1. Let Ω be a bounded domain of Rn . Let Ω0 be a nonempty open subset of Ω for which there is a δ > 0 satisfying Ω0 + B(0, δ) ⊂ Ω. Let 1 ≤ p < ∞.

´ C. BELLIDO AND CARLOS MORA-CORRAL JOSE

904

Then there exists λ > 0 such that for all u ∈ Lp (Ω, Rd ), 

 

|u(x) − u(x )| dx dx + λ p

p

|u(x)| dx ≤ λ

(4.1) Ω

Ω



Ω∩B(x,δ)

p

|u(x)| dx. Ω\Ω0

Note that in the statement of Proposition 4.1, there is no mention of the exponent α, so it holds true for any α, but, in view of Lemma 3.1, it is especially suited for integrands w with the growth of the form (1.2) with 0 ≤ α < n. Note that the trivial bound     |u(x) − u(x )|p  p |u(x) − u(x )| dx dx ≤ δ α dx dx |x − x |α Ω Ω∩B(x,δ) Ω Ω∩B(x,δ) holds for any α ≥ 0. Regarding the coercivity inequality for the Neumann problem, it was observed by Aksoylu and Mengesha [2] that it can be easily obtained by invoking the Poincar´etype inequalities obtained by Ponce [40, 41] and, earlier, by Bourgain, Brezis, and Mironescu [13, 14] in their study of nonlocal characterizations of Sobolev spaces. The proof of [2, Cor. 3.4] is adapted as follows. Proposition 4.2. Let Ω be a Lipschitz domain of Rn , fix δ > 0, and let 1 ≤ p < ∞. Then there exists λ > 0 such that for all u ∈ Lp (Ω, Rd ), (4.2)

     p   u(x) − − u dx ≤ λ   Ω

Ω

Ω

|u(x) − u(x )| dx dx. p

Ω∩B(x,δ)

Proof. For each j ∈ N define the function ρj : [0, ∞) → [0, ∞) as ρj (t) :=

(n + p) j n+p p t χ[0,1] (jt), σn

t ≥ 0,

where σn denotes the surface area of the unit sphere of Rn . It is immediate to check that  ρj (|x|) dx = 1 for all j ∈ N Rn

and  lim

j→∞

Rn \B(0,η)

ρj (|x|) dx = 0

for all η > 0.

Moreover, when n = 1 we have that for all 0 < θ0 < 1,   inf ρj (|θt|) dt ≥ θ0p ρj (|t|) dt = θ0p . R θ∈[θ0 ,1]

R

Hence, by [40, Thms. 1.1 and 1.3], there exist c > 0 and j ∈ N such that for any u ∈ Lp (Ω, Rd ),

1 j

     p   |u(x) − u(x )|p u(x) − − u dx ≤ c ρj (|x − x |) dx dx.   |x − x |p Ω Ω Ω Ω

≤ δ and

EXISTENCE FOR NONLOCAL PROBLEMS IN PERIDYNAMICS

The definition of ρj and the inequality  

905

≤ δ show that

1 j

|u(x) − u(x )| ρj (|x − x |) dx dx p |x − x | Ω Ω   (n + p) j n+p p ≤ |u(x) − u(x )| dx dx, σn Ω Ω∩B(x,δ) p

which concludes the proof. As a remark, we mention that the proof shows that the dependence of λ in terms of δ obeys the law λ = O(δ −n−p ). It was again observed by Aksoylu and Mengesha [2] that the same techniques can also be used to prove the coercivity inequality for the mixed problem. The proof of the following result is, thus, a straightforward adaptation of [2, Cor. 3.4], who proved it for the case p = 2, with the variant given in Proposition 4.2 to cover the case of any 1 ≤ p < ∞. Proposition 4.3. Let Ω be a Lipschitz domain of Rn , fix δ > 0, and let 1 ≤ p < ∞. Let Ω0 be a nonempty open subset of Ω satisfying Ω0 + B(0, δ) ⊂ Ω. Let ΩD be a measurable subset of Ω \ Ω0 with positive measure. Then there exists λ > 0 such that for all u ∈ Lp (Ω, Rd ) with u = 0 a.e. in ΩD , we have    p p |u(x)| dx ≤ λ |u(x) − u(x )| dx dx. Ω

Ω

Ω∩B(x,δ)

We will in fact use Proposition 4.3 in the following form. Corollary 4.4. Let Ω be a Lipschitz domain of Rn , fix δ > 0, and let 1 ≤ p < ∞. Let Ω0 be a nonempty open subset of Ω satisfying Ω0 + B(0, δ) ⊂ Ω. Let ΩD be a measurable subset of Ω \ Ω0 with positive measure. Then there exists λ > 0 such that for all u ∈ Lp (Ω, Rd ),     p p p |u(x)| dx ≤ λ |u(x) − u(x )| dx dx + λ |u(x)| dx. Ω

Ω

Ω∩B(x,δ)

ΩD

5. Existence of minimizers in Lp . With the lower semicontinuity and coercivity results at hand, we pass now to show the existence of minimizers of the total energy in the functional setting of Lp spaces. As in sections 3 and 4, the exponent α is not mentioned in the statements, but, in view of Lemma 3.1, the results of this section are especially suited for integrands w with a growth (1.2) with 0 ≤ α < n. We start with the Dirichlet and mixed boundary conditions. Theorem 5.1. Let Ω be a Lipschitz domain of Rn , fix δ > 0, and let 1 < p < ∞. Let Ω0 be a nonempty open subset of Ω satisfying Ω0 + B(0, δ) ⊂ Ω. Let ΩD ˜ × Rd → R be be a measurable subset of Ω \ Ω0 with positive measure. Let w : Ω n ˜ d L (Ω) × B(R )-measurable. Assume that (a) there exists c0 > 0 such that p

˜ ) ≥ c0 χB(0,δ) (˜ w(˜ x, y x) |˜ y|

˜ and all y ˜∈Ω ˜ ∈ Rd ; for a.e. x

˜ and C > 0 such that (b) there exist a1 ∈ L1 (Ω) ˜ ) ≤ a1 (˜ w(˜ x, y x) + C |˜ y|

p

˜ and all y ˜∈Ω ˜ ∈ Rd ; for a.e. x

˜ ∈ Ω; (c) w(˜ x, ·) is lower semicontinuous for a.e. x

´ C. BELLIDO AND CARLOS MORA-CORRAL JOSE

906

(d) for a.e. x ∈ Ω and all u ∈ Lp (Ω, Rd ), the function  y → w(x − x , y − u(x )) dx Ω

is convex in Rd . Let F : Ω × Rd → R be Ln (Ω) × B(Rd )-measurable and satisfy that for a.e. x ∈ Ω, the function F (x, ·) is concave, upper semicontinuous, and F + (x, y) ≤ a2 (x) + c1 |y|q ,

(5.1)

F + (x, y) ≤ a2 (x) + a3 (x) · y

for a.e. x ∈ Ω and all y ∈ Rd , 

for some 1 ≤ q < p, some c1 > 0, some a2 ∈ L1 (Ω), and some a3 ∈ Lp (Ω, Rd ). Let b ∈ Lp (ΩD , Rd ) satisfy that   F − (x, 0) dx + F − (x, b(x)) dx < ∞. (5.2) Ω\ΩD

ΩD

Let A be the set of u ∈ Lp (Ω, Rd ) such that u = b a.e. on ΩD . Let I be as in (2.4). Then there exists a minimizer of I in A. Proof. Let u ∈ A. Assumption (a) and Corollary 4.4 yield the estimate   c0 p p

u Lp (Ω,Rd ) − c0 b Lp (ΩD ,Rd ) ≤ w(x − x , u(x) − u(x )) dx dx, (5.3) λ Ω Ω where λ is the constant of Corollary 4.4, while the first bound of (5.1) and H¨older’s inequality show that  q (5.4) F (x, u(x)) dx ≤ a2 L1 (Ω) + c2 u Lp (Ω,Rd ) Ω

for some c2 > 0 independent of u. Using Young’s inequality, we find that c2 u qLp (Ω,Rd ) ≤ c3 +

(5.5)

c0

u pLp (Ω,Rd ) 2λ

for some c3 > 0 independent of u. From (5.3), (5.4), and (5.5) we conclude that there exists c4 > 0, such that for all u ∈ A, I(u) ≥

(5.6)

c0 p

u Lp (Ω,Rd ) − c4 . 2λ

¯ ∈ Lp (Ω, Rd ) be the extension of b to Ω by zero. Assumption (b) shows that Let b   ¯ ¯  )) dx dx ≤ Ln (Ω) a1 1 ˜ + 2p CLn (Ω) b p w(x − x , b(x) − b(x L (ΩD ,Rd ) , L (Ω) Ω

Ω

while assumption (5.2) yields 

¯ F − (x, b(x)) dx < ∞,

Ω

¯ < ∞. hence, I(b) ¯ ∈ A. Estimate (5.6) shows that I is bounded below in Clearly, A = ∅, since b A, and, as just seen, I is not identically infinity. So let {uj }j∈N be a minimizing

EXISTENCE FOR NONLOCAL PROBLEMS IN PERIDYNAMICS

907

sequence of I in A. By (5.6), for a subsequence, there exists u ∈ Lp (Ω, Rd ) such that uj  u in Lp (Ω, Rd ) as j → ∞. Proposition 3.5 shows that (5.7)     w(x − x , u(x) − u(x )) dx dx ≤ lim inf

Ω

j→∞

Ω

w(x − x , uj (x) − uj (x )) dx dx,

Ω

Ω

while standard arguments on lower semicontinuity for convex functions (e.g., [24, Thm. 6.54]) show that   F (x, u(x)) dx ≥ lim sup F (x, uj (x)) dx. j→∞

Ω

Ω

In total, I(u) ≤ lim inf j→∞ I(uj ). Clearly, uj |ΩD  u|ΩD in Lp (ΩD , Rd ) as j → ∞, so u = b a.e. in ΩD . Hence u ∈ A and, thus, u is a minimizer of I in A. Note that the assumptions on F in Theorem 5.1 are satisfied when F (x, y) = f (x) · y for a given f ∈ Lr (Ω, Rd ) with r > p . This is typically the case for external forces, and even more so when dealing with Neumann boundary conditions, as presented in the following result, which remains true for nonlinear forces F satisfying the assumptions of Theorem 5.1. Theorem 5.2. Let Ω be a Lipschitz domain of Rn , fix δ > 0, and let 1 < p < ∞. ˜ × Rd → R be Ln (Ω) ˜ × B(Rd )-measurable. Let assumptions (a)–(d) of Let w : Ω p Theorem 5.1 hold. Let f ∈ L (Ω, Rd ). Let A be the set of u ∈ Lp (Ω, Rd ) such that  u = 0. Define I as Ω       w(x − x , u(x) − u(x )) dx dx − f (x) · u(x) dx, u ∈ A. (5.8) I(u) := Ω

Ω

Ω

Then there exists a minimizer of I in A. Proof. Assumption (a) and Proposition 4.2 yield the estimate   c0 p

u Lp (Ω,Rd ) ≤ w(x − x , u(x) − u(x )) dx dx, (5.9) λ Ω Ω where λ is the constant of Proposition 4.2, while H¨older’s and Young’s inequalities show that  c0 p

u Lp (Ω,Rd ) f (x) · u(x) dx ≤ f Lp (Ω,Rd ) u Lp (Ω,Rd ) ≤ c1 + (5.10) 2λ Ω for some c1 > 0 independent of u. From (5.9) and (5.10) we conclude that for all u ∈ A, c0 p

u Lp (Ω,Rd ) − c1 . (5.11) I(u) ≥ 2λ Assumption (b) shows that   w(x − x , 0) dx dx ≤ Ln (Ω) a2 L1 (Ω) ˜ , Ω

Ω

so I(0) < ∞. Clearly, A = ∅, since 0 ∈ A. Estimate (5.11) shows that I is bounded below in A, and, as just seen, I is not identically infinity. So let {uj }j∈N be a minimizing sequence of I in A. By (5.11), for a subsequence, there exists u ∈ Lp (Ω, Rd ) such that uj  u in Lp (Ω, Rd ) as j → ∞. 3.5 shows that inequality (5.7) holds, while weak convergence   yields  Proposition f · u → f · u as j → ∞. Therefore, I(u) ≤ lim inf I(u ). Clearly, u → j j→∞ j Ω Ω j Ω  u as j → ∞, so u = 0. Hence u ∈ A and u is a minimizer of I in A. Ω Ω

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6. Coercivity in W s,p . When the growth of w is of the form (1.2) with n < α < n + p, the natural functional spaces to set the problem are the fractional Sobolev spaces. We will need the following two coercivity inequalities, essentially proved in Hurri-Syrj¨ anen and V¨ ah¨ akangas [29]. Proposition 6.1. Let 0 < s < 1, 1 ≤ p < ∞, and δ > 0. Let Ω be a Lipschitz domain. Then there exists λ > 0 such that for all u ∈ W s,p (Ω, Rd ), (6.1)

 p       u(x) − − u dx ≤ λ   Ω

Ω

Ω

|u(x) − u(x )| dx dx |x − x |n+sp p

Ω∩B(x,δ)

and   (6.2) Ω

Ω

|u(x) − u(x )|p  dx dx ≤ λ |x − x |n+sp

  Ω

Ω∩B(x,δ)

|u(x) − u(x )|p  dx dx. |x − x |n+sp

Proof. Inequality (6.1) is a particular case of [29, Cor. 4.6]. As for (6.2), we notice that     p |u(x) − u(x )| 1 p  (6.3) dx dx ≤ |u(x) − u(x )| dx dx,  n+sp δ n+sp Ω Ω Ω Ω\B(x,δ) |x − x | while the inequality   p   p      p |u(x) − u(x )| ≤ 2p−1 u(x) − − u + u(x ) − − u , Ω

x, x ∈ Ω,

Ω

provides, by integration,  

|u(x) − u(x )| dx dx ≤ 2p Ln (Ω) p

(6.4) Ω

Ω

   p   u(x) − − u dx.   Ω

Ω

Putting together (6.3), (6.4), and (6.1), we obtain inequality (6.2), changing the value of the constant λ. We will need another Poincar´e-type inequality for the situation at hand. The result is well known, although it is difficult to trace back in the literature an explicit statement. Its proof, which we omit, is standard and based on the compact embeddings of Proposition 2.1. See, e.g., [19, Lemma 4.3], if necessary. Lemma 6.2. Let Ω be a Lipschitz domain and let ΩD be a measurable set of Ω of positive measure. Let 0 < s < 1 and 1 ≤ p < ∞. Then there exists λ > 0 such that for all u ∈ W s,p (Ω, Rd ) with u = 0 a.e. on ΩD , we have

u Lp (Ω,Rd ) ≤ λ |u|W s,p (Ω,Rd ) . We will use Lemma 6.2 in the following form. Corollary 6.3. Let Ω be a Lipschitz domain and let ΩD be a measurable set of Ω of positive measure. Let 0 < s < 1 and 1 ≤ p < ∞. Let b ∈ W s,p (Ω, Rd ). Then there exists λ > 0 such that for all u ∈ W s,p (Ω, Rd ) with u = b a.e. on ΩD , we have  

u W s,p (Ω,Rd ) ≤ λ |u|W s,p (Ω,Rd ) + b W s,p (Ω,Rd ) .

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EXISTENCE FOR NONLOCAL PROBLEMS IN PERIDYNAMICS

7. Existence of minimizers in W s,p . With the coercivity results at hand, we present the existence theorems in W s,p . In this case, no convexity conditions are needed on w or −F , since, as a consequence of Proposition 2.1, weak convergence in W s,p implies (for a subsequence) convergence a.e. As in section 6, the results of this section are especially suited for integrands w of the type (1.2) with α = n + sp. We start with Dirichlet and mixed boundary conditions. Theorem 7.1. Let 0 < s < 1 and 1 < p < ∞. Let Ω be a Lipschitz domain of Rn and fix δ > 0. Let Ω0 be a nonempty open subset of Ω satisfying Ω0 + B(0, δ) ⊂ Ω. ˜ × Rd → R Let ΩD be a measurable subset of Ω \ Ω0 with positive measure. Let w : Ω n ˜ d be L (Ω) × B(R )-measurable and satisfy the following conditions: (a) there exists c0 > 0 such that ˜ ) ≥ c0 w(˜ x, y

|˜ y |p χB(0,δ) (˜ x) |˜ x|n+sp

˜ and all y ˜∈Ω ˜ ∈ Rd ; for a.e. x

˜ the function w(˜ ˜ ∈ Ω, (b) for a.e. x x, ·) is lower semicontinuous. Let F : Ω × Rd → R be Ln (Ω) × B(Rd )-measurable and satisfy that for a.e. x ∈ Ω, the function F (x, ·) is upper semicontinuous, (7.1)

F + (x, y) ≤ a1 (x) + a2 (x) |y|

q

for a.e. x ∈ Ω and all y ∈ Rd ,

for some 1 ≤ q < p, some a1 ∈ L1 (Ω), and some a2 ∈ Lr (Ω) with r> (7.2)

p∗ , p∗ − q

p∗ :=

np n − sp r>1 r=1

if sp < n, if sp = n, if sp > n.

Let b ∈ W s,p (Ω, Rd ). Let A be the set of u ∈ W s,p (Ω, Rd ) such that u = b a.e. on ΩD . Let I be as in (2.4), and assume I(b) < ∞. Then there exists a minimizer of I in A. Proof. Note that A = ∅ and that I is not identically ∞ in A. Assumption (a), Proposition 6.1, and Corollary 6.3 show that there exist c1 , c2 > 0 such that for all u ∈ A,   p (7.3) c1 u W s,p (Ω,Rd ) ≤ w(x − x , u(x) − u(x )) dx dx + c2 . Ω

Ω

From this point, the proof is divided according to the cases sp < n, sp = n, and sp > n. Case sp < n. Using estimate (7.1) and H¨older’s inequality, we find that for all u ∈ A,  q 1 q F (x, u(x)) dx ≤ a1 L1 (Ω) + Ln (Ω) r − p∗ a2 Lr (Ω) u Lp∗ (Ω,Rd ) . (7.4) Ω

∗

Now, Young’s inequality, the fractional Sobolev immersion W s,p (Ω, Rd ) ⊂ Lp (Ω, Rd ) (see Proposition 2.1), and Corollary 6.3 show that there exists c3 > 0 such that for all u ∈ A, (7.5)

1

q

Ln (Ω) r − p∗ a2 Lr (Ω) u Lp∗ (Ω,Rd ) ≤ c3 + q

c1 p

u W s,p (Ω,Rd ) . 2

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Inequalities (7.3), (7.4) and (7.5) conclude that there exists c4 > 0 such that for all u ∈ A, I(u) ≥

(7.6)

c1

u pW s,p (Ω,Rd ) − c4 . 2

Hence I is bounded below in A. Take a minimizing sequence {uj }j∈N of I in A. Due to bound (7.6), for a subsequence, there exists u ∈ W s,p (Ω, Rd ) such that uj  u in W s,p (Ω, Rd ) and uj → u a.e. as j → ∞, thanks to the compact immersion W s,p (Ω, Rd ) ⊂ Lp (Ω, Rd ) (see Proposition 2.1). In particular, u = b a.e. in ΩD , and, hence, u ∈ A. By Fatou’s lemma, (7.7)     w(x − x , u(x) − u(x )) dx dx ≤ lim inf

Ω

w(x − x , uj (x) − uj (x )) dx dx,

j→∞

Ω

and



Ω

F − (x, u(x)) dx ≤ lim inf

(7.8)



j→∞

Ω

Ω

F − (x, uj (x)) dx.

Ω ∗

On the other hand, the sequence {uj }j∈N is bounded in Lp (Ω, Rd ), again by Proposition 2.1, hence {a2 |uj |q }j∈N is bounded in Ls (Ω, Rd ) with 1 q 1 + ∗ = , r p s so s > 1. Therefore, the sequence {a2 |uj |q }j∈N is equi-integrable. Bound (7.1) and Vitali’s convergence theorem show that  F + (x, u(x)) dx ≥ lim sup F + (x, uj (x)) dx. (7.9) j→∞

Ω

Inequalities (7.7), (7.8), and (7.9) show that I(u) ≤ lim inf I(uj ).

(7.10)

j→∞

Hence u is a minimizer of I in A. Case sp = n. Choose 1 < p∗ < ∞ big enough so that r>

p∗ . p∗ − q ∗

As the compact immersion W s,p (Ω, Rd ) ⊂ Lp (Ω, Rd ) holds (see Proposition 2.1), one can repeat the proof of the previous case. Case sp > n. Using estimate (7.1), we find that  q F + (x, u(x)) dx ≤ a1 L1 (Ω) + a2 L1 (Ω) u L∞ (Ω,Rd ) . Ω

Now, Young’s inequality, the fractional Sobolev immersion W s,p (Ω, Rd ) ⊂ L∞ (Ω, Rd ) (see Proposition 2.1), and Corollary 6.3 show that there exists c3 > 0 such that for all u ∈ A, q

a2 L1 (Ω) u L∞ (Ω,Rd ) ≤ c3 +

c1 p

u W s,p (Ω,Rd ) . 2

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As before, we conclude that there exists c4 > 0 such that bound (7.6) holds for all u ∈ A, hence I is bounded below in A. Take a minimizing sequence {uj }j∈N of I in A. Then, for a subsequence, there exists u ∈ W s,p (Ω, Rd ) such that uj  u in W s,p (Ω, Rd ) and uj → u uniformly as j → ∞ (see Proposition 2.1). In particular, u = b a.e. in ΩD , and, hence, u ∈ A. By Fatou’s lemma, inequalities (7.7) and (7.8) hold. It is easy to check that the sequence {a2 |uj |q }j∈N is equi-integrable. As before, inequalities (7.9) and (7.10) hold. Hence u is a minimizer of I in A. For Neumann boundary conditions, the result is as follows. As in Theorem 5.2, we present it with linear forces, but the result is also true for nonlinear forces F satisfying the assumptions of Theorem 7.1. Theorem 7.2. Let 0 < s < 1 and 1 < p < ∞. Let Ω be a Lipschitz domain and ˜ × B(Rd )-measurable and satisfy conditions ˜ × Rd → R be Ln (Ω) fix δ > 0. Let w : Ω r (a)–(b) of Theorem 7.1. Let f ∈ L (Ω, Rd ) with r = (p∗ ) ,

p∗ :=

np n − sp r>1 r=1

if sp < n, if sp = n, if sp > n.

 Let A be the set of u ∈ W s,p (Ω, Rd ) such that Ω u = 0. Let I be as in (5.8), and assume that I is not identically infinity in A. Then there exists a minimizer of I in A. Proof. Note that A = ∅ since 0 ∈ A. In addition, we are assuming that I is not identically ∞ in A. Assumption (a) of Theorem 7.1 and Proposition 6.1 show that there exists c1 > 0 such that for all u ∈ A,   p (7.11) c1 u W s,p (Ω,Rd ) ≤ w(x − x , u(x) − u(x )) dx dx. Ω

Ω

From this point, the proof is divided according to the cases sp < n, sp = n, and sp > n. Case sp < n. Using H¨older’s and Young’s inequalities, as well as Proposition 2.1, we obtain that there exists c2 > 0 such that for all u ∈ A,  c1 p

u W s,p (Ω,Rd ) , (7.12) f (x) · u(x) dx ≤ f L(p∗ ) (Ω,Rd ) u Lp∗ (Ω,Rd ) ≤ c2 + 2 Ω which, together with (7.11), yields I(u) ≥

c1 p

u W s,p (Ω,Rd ) − c2 . 2

Hence I is bounded below in A. Take a minimizing sequence {uj }j∈N of I in A. Then, d s,p d for a subsequence, there exists u ∈ W s,p (Ω,  R ) such that uj  u in W (Ω, R ) and uj → u a.e. as j → ∞. In particular, Ω u = 0, and, hence,  Fatou’s  u ∈ A. By lemma, inequality (7.7) holds, while by weak convergence, Ω f · uj → Ω f · u as j → ∞. Therefore, inequality (7.10) holds and u is a minimizer of I in A. Case sp = n. Instead of (7.12), we have  c1 p

u W s,p (Ω,Rd ) , f (x) · u(x) dx ≤ f Lr (Ω,Rd ) u Lr (Ω,Rd ) ≤ c2 + 2 Ω and we can repeat the argument of the previous case.

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Case sp > n. Instead of (7.12), we have  Ω

f (x) · u(x) dx ≤ f L1 (Ω,Rd ) u L∞ (Ω,Rd ) ≤ c2 +

c1 p

u W s,p (Ω,Rd ) , 2

and we can repeat the argument of the previous cases. 8. Euler–Lagrange equations. In this section we write out the Euler–Lagrange equations satisfied by the minimizers of I. The results are divided according to the spaces and boundary conditions used. The following result, which is an immediate consequence of the differentiation under the integral sign (see, e.g., [31, Chap. 13, section 2, Lemma 2.2]), shows the derivative of the energy under growth conditions compatible with Lp . Lemma 8.1. Let Ω be a bounded open set of Rn . Let 1 ≤ p < ∞ and u, v ∈ ˜ a2 ∈ Lp (Ω), ˜ b1 ∈ L1 (Ω), and b2 ∈ Lp (Ω). Let w : Lp (Ω, Rd ). Let a1 ∈ L1 (Ω), d n d ˜ × R → R be L (Ω) ˜ × B(R )-measurable, and F : Ω × Rd → R be Ln (Ω) × B(Rd )Ω ˜ with derivative ˜ ∈ Ω measurable. Suppose w(˜ x, ·) is differentiable in Rd for a.e. x D2 w(˜ x, ·), and F (x, ·) is differentiable in Rd for a.e. x ∈ Ω with derivative D2 F (x, ·). ˜ and all y ˜∈Ω ˜ ∈ Rd , Assume that there exists C > 0 for which for a.e. x p ˜ )| ≤ a1 (˜ |w(˜ x, y x) + C |˜ y| ,

(8.1)

˜ )| ≤ a2 (˜ |D2 w(˜ x, y x) + C |˜ y|p−1 ,

and for a.e. x ∈ Ω and all y ∈ Rd , |F (x, y)| ≤ b1 (x) + C |y|p , (8.2)

|D2 F (x, y)| ≤ b2 (x) + C |y|p−1 .

Let I be as in (2.4). Then    d  I(u + τ v) = D2 w(x − x , u(x) − u(x )) · (v(x) − v(x )) dx dx dτ τ =0 Ω Ω (8.3)  D2 F (x, u(x)) · v(x) dx. − Ω

For growth conditions compatible with W s,p , the corresponding result is the following, where the embeddings of Proposition 2.1 are used. Lemma 8.2. Let Ω be a Lipschitz domain in Rd . Let 0 < s < 1 and 1 ≤ p < ∞. ˜ × Rd → R be Ln (Ω) ˜ × B(Rd )-measurable, and Let u, v ∈ W s,p (Ω, Rd ). Let w : Ω d n d F : Ω × R → R be L (Ω) × B(R )-measurable. Suppose w(˜ x, ·) is differentiable in Rd ˜ ˜ ∈ Ω with derivative D2 w(˜ for a.e. x x, ·), and F (x, ·) is differentiable in Rd for a.e. x ∈ Ω with derivative D2 F (x, ·). ˜ a2 ∈ L(p∗ ) (Ω), ˜ (i) If sp < n, assume that there exist C > 0, a1 ∈ L1 (Ω), ∗  np 1 (p ) ∗ ˜ ˜ ∈ Ω, b1 ∈ L (Ω), and b2 ∈ L (Ω), with p := n−sp , such that for a.e. x

EXISTENCE FOR NONLOCAL PROBLEMS IN PERIDYNAMICS

˜ ∈ Rd , a.e. x ∈ Ω, and all y ∈ Rd , all y

 ∗ ˜ )| ≤ a1 (˜ |w(˜ x, y x) + C |˜ y |p + 

˜ )| ≤ a2 (˜ x, y x) + C |D2 w(˜

|˜ y|

p∗ −1

 p∗ |F (x, y)| ≤ b1 (x) + C |y| +  |D2 F (x, y)| ≤ b2 (x) + C

|y|

p∗ −1

|˜ y| |˜ x| +

,

n+sp

|˜ y|

|y| |x|



p

|˜ x|

913



p−1

,

n+sp



p

n+sp p−1

,

|y| + |x|n+sp

 .

˜ a2 ∈ Lr (Ω), ˜ b1 ∈ (ii) If sp = n, assume that there exist C > 0, a1 ∈ L1 (Ω),  1 r ˜ all ˜ ∈ Ω, L (Ω), and b2 ∈ L (Ω), with 1 < r < ∞, such that for a.e. x d d ˜ ∈ R , a.e. x ∈ Ω, and all y ∈ R , y   |˜ y|p ˜ )| ≤ a1 (˜ |w(˜ x, y x) + C |˜ y|r + n+sp , |˜ x|   p−1 |˜ y| r−1 ˜ )| ≤ a2 (˜ x, y x) + C |˜ y| + , |D2 w(˜ n+sp |˜ x|  p  |y| r |F (x, y)| ≤ b1 (x) + C |y| + n+sp , |x|   p−1 |y| r−1 + . |D2 F (x, y)| ≤ b2 (x) + C |y| n+sp |x| (iii) If sp > n, assume that for each compact K ⊂ Rd ,       sup |w(x − x , y)| dx dx < ∞, sup |D2 w(x − x , y)| dx dx < ∞, Ω Ω y∈K Ω Ω y∈K   sup |F (x, y)| dx < ∞, sup |D2 F (x, y)| dx < ∞. Ω y∈K

Ω y∈K

Then equality (8.3) holds, where I is as in (2.4). The Euler–Lagrange equations are as follows. Theorem 8.3. Let Ω be a bounded open set of Rn , and let 1 ≤ p < ∞. Let d d ˜ ˜ w : Ω×R → R be Ln (Ω)×B(R )-measurable, and F : Ω×Rd → R be Ln (Ω)×B(Rd )˜ with derivative ˜ ∈ Ω measurable. Suppose w(˜ x, ·) is differentiable in Rd for a.e. x D2 w(˜ x, ·), and F (x, ·) is differentiable in Rd for a.e. x ∈ Ω with derivative D2 F (x, ·). The following assertions hold. (1) Let ΩD be a measurable subset of Ω of positive measure. Assume either of the following: (a) let b ∈ Lp (ΩD , Rd ), and let A be the set of u ∈ Lp (Ω, Rd ) such that ˜ u = b a.e. on ΩD . Assume the bounds (8.1)–(8.2) for some a1 ∈ L1 (Ω),  ˜ b1 ∈ L1 (Ω), and b2 ∈ Lp (Ω); a2 ∈ Lp (Ω), (b) let 0 < s < 1 and suppose that Ω is a Lipschitz domain. Let b ∈ W s,p (Ω, Rd ), and let A be the set of u ∈ W s,p (Ω, Rd ) such that u = b a.e. on ΩD . Suppose, further, that Ω \ ΩD coincides a.e. with an open set. Assume any of conditions (i)–(iii) of Lemma 8.2.

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Let I be as in (2.4), and let u be a minimizer of I in A. Then  D2 w(x−x , u(x)−u(x )) dx = D2 F (x, u(x)) a.e. x ∈ Ω\ΩD . (8.4) 2 Ω

(b) Assume either of the following:  (a) let A be the set of u ∈ Lp (Ω, Rd ) such that Ω u = 0. Assume the ˜ a2 ∈ Lp (Ω), ˜ b1 ∈ L1 (Ω), and bounds (8.1)–(8.2) for some a1 ∈ L1 (Ω),  p b2 ∈ L (Ω); (b) let 0 < s < 1 and suppose that Ω is a Lipschitz domain. Let A be the set of u ∈ W s,p (Ω, Rd ) such that Ω u = 0. Assume any of the conditions (i)–(iii) of Lemma 8.2. Let I be as in (2.4), and let u be a minimizer of I in A. Then there exists a ∈ Rd such that  D2 w(x − x , u(x) − u(x )) dx = D2 F (x, u(x)) + a a.e. x ∈ Ω. (8.5) 2 Ω

Proof. Assume first case (1). Let v belong to Lp (Ω, Rd ) or to W s,p (Ω, Rd ), according to whether option (1)(a) or (1)(b) holds. In addition, assume that v = 0 a.e. on ΩD . Then, u + τ v ∈ A for all τ ∈ R. As u is a minimizer, we apply Lemmas 8.1 or 8.2 (according to whether option (1)(a) or (1)(b) holds), and obtain that    D2 w(x−x , u(x)−u(x ))·(v(x) − v(x )) dx dx− D2 F (x, u(x))·v(x) dx = 0. Ω

Ω

Ω

Changing the order of integration, using the symmetry ˜) x, −˜ y) = −D2 w(˜ x, y D2 w(−˜

˜ and all y ˜∈Ω ˜ ∈ Rd for a.e. x

coming from (2.1), and applying the boundary condition v = 0 a.e. on ΩD , we arrive at        D2 w(x − x , u(x) − u(x ))dx − D2 F (x, u(x)) · v(x) dx = 0. 2 Ω\ΩD

Ω

In case (1)(a), this implies at once that (8.4) holds. In case (1)(b), this also implies equality (8.4), thanks to a classic approximation result (see, e.g., [16, Thm. 3.40]), using that an a.e. representant of Ω \ ΩD is open. d Now we assume case (2). Let v belong to Lp (Ω, Rd ) or to W s,p (Ω,  R ), according to whether option (2)(a) or (2)(b) holds. In addition, assume that Ω v = 0. Then, u + τ v ∈ A for all τ ∈ R. As before, we arrive at the equality     (8.6) D2 w(x − x , u(x) − u(x ))dx − D2 F (x, u(x)) · v(x) dx = 0. 2 Ω

Ω

Hence the function inside the square brackets of (8.6) is orthogonal to the closed hyperplane of Lp (Ω, Rd ) or of W s,p (Ω, Rd ) formed by the functions with zero integral in Ω. Thus, there exists a ∈ Rd such that (8.5) holds. We finish this paper with a brief mention that the Euler–Lagrange equation (8.4) can be given an interpretation in terms of Neumann boundary conditions. Indeed, let Ω0 , Ω1 be nonempty measurable disjoint subsets with union Ω, such that ΩD ⊂ Ω1 . Define ΩN := Ω1 \ ΩD , and, for definiteness, assume Ln (ΩN ) > 0. Equality (8.4) is

EXISTENCE FOR NONLOCAL PROBLEMS IN PERIDYNAMICS

915

split into two: for a.e. x ∈ Ω0 and for a.e. x ∈ ΩN . The equality for a.e. x ∈ Ω0 corresponds to the equation satisfied in the inner part of the body. The equality for a.e. x ∈ ΩN , on the other hand, corresponds to the equation satisfied on the Neumann part of the nonlocal boundary of the body, and can be given an interpretation of a nonlocal flux through the boundary of ΩN , thus mimicking what happens for the local equations. This nonlocal calculus is developed in [21, 20, 27], to which we refer for further explanation. Acknowledgments. We thank Pablo Pedregal for his suggestion to study this problem, Javier C´arcamo for explaining to us a probabilistic interpretation of Lemma 3.3, and the referees for their suggestions in the presentation of the paper. REFERENCES [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed., Pure Appl. Math. 140, Academic Press, Amsterdam, 2003. [2] B. Aksoylu and T. Mengesha, Results on nonlocal boundary value problems, Numer. Funct. Anal. Optim., 31 (2010), pp. 1301–1317. [3] B. Aksoylu and M. L. Parks, Variational theory and domain decomposition for nonlocal problems, Appl. Math. Comput., 217 (2011), pp. 6498–6515. [4] B. Alali and R. Lipton, Multiscale dynamics of heterogeneous media in the peridynamic formulation, J. Elasticity, 106 (2012), pp. 71–103. [5] G. Alberti and G. Bellettini, A nonlocal anisotropic model for phase transitions. I. The optimal profile problem, Math. Ann., 310 (1998), pp. 527–560. [6] L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Oxford University Press, New York, 2000. ´ n, J. D. Rossi, and J. J. Toledo-Melero, Nonlocal Diffusion [7] F. Andreu-Vaillo, J. M. Mazo Problems, Math. Surveys Monogr. 165, AMS, Providence, RI, 2010. ´ n, J. D. Rossi, and J. Toledo, A nonlocal p-Laplacian evolution [8] F. Andreu, J. M. Mazo equation with nonhomogeneous Dirichlet boundary conditions, SIAM J. Math. Anal., 40 (2009), pp. 1815–1851. [9] S. S. Antman, Nonlinear Problems of Elasticity, Appl. Math. Sci. 107, Springer, New York, 1995. [10] H. Attouch, G. Buttazzo, and G. Michaille, Variational Analysis in Sobolev and BV Spaces, MPS/SIAM Ser. Optim. 6, SIAM, Philadelphia, 2006. [11] J. M. Ball, A version of the fundamental theorem for Young measures, in PDEs and Continuum Models of Phase Transitions, Lecture Notes in Phys. 344, Springer, Berlin, 1989, pp. 207–215. [12] J. Boulanger, P. Elbau, C. Pontow, and O. Scherzer, Non-local functionals for imaging, in Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Optim. Appl. 49, Springer, New York, 2011, pp. 131–154. [13] J. Bourgain, H. Brezis, and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations, J. L. Menaldi, E. Rofman, and A. Sulem, eds., IOS Press, Amsterdam, 2001, pp. 439–455. [14] J. Bourgain, H. Brezis, and P. Mironescu, Limiting embedding theorems for W s,p when s ↑ 1 and applications, J. Anal. Math., 87 (2002), pp. 77–101. [15] P. G. Ciarlet, Mathematical Elasticity. Vol. I, Stud. Math. Appl. 20, North-Holland, Amsterdam, 1988. [16] B. Dacorogna, Direct Methods in the Calculus of Variations, 2nd ed., Appl. Math. Sci. 78, Springer, New York, 2008. [17] S. Dipierro, G. Palatucci, and E. Valdinoci, Dislocation Dynamics in Crystals: A Macroscopic Theory in a Fractional Laplace Setting, http://cvgmt.sns.it/paper/2241/ (2013). [18] E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), pp. 521–573. [19] Q. Du, M. Gunzburger, R. B. Lehoucq, and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), pp. 667–696. [20] Q. Du, M. Gunzburger, R. B. Lehoucq, and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci., 23 (2013), pp. 493–540.

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