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Abstract We prove two compactness results for function spaces with finite Dirichlet energy of half‐space nonlocal gradients. In each of these results, we provide sufficient conditions on a sequence of kernel functions that guarantee the asymptotic compact embedding of the associated nonlocal function spaces into the class of square‐integrable functions. Moreover, we will demonstrate that the sequence of nonlocal function spaces converges in an appropriate sense to a limiting function space. As an application, we prove uniform Poincaré‐type inequalities for sequence of half‐space gradient operators. We also apply the compactness result to demonstrate the convergence of appropriately parameterized nonlocal heterogeneous anisotropic diffusion problems. We will construct asymptotically compatible schemes for these type of problems. Another application concerns the convergence and robust discretization of a nonlocal optimal control problem. 

Abstract We extend the Calderón–Zygmund theory for nonlocal equations tostrongly coupled system of linear nonlocal equations ${\mathcal{ℒ}}_A^{s}u=f$ {\mathcal{L}^{s}_{A}u=f}, where the operator ${\mathcal{ℒ}}_A^s$ {\mathcal{L}^{s}_{A}}is formally given by ${\mathcal{ℒ}}_A^{s}u={\int }_{{ℝ}^n}\frac{A(x,y)}{{\left|x-y\right|}^{n+2s}}\frac{\left(x-y\right)\otimes \left(x-y\right)}{{\left|x-y\right|}^2}\left(u\left(x\right)-u\left(y\right)\right)𝑑y.$ \mathcal{L}^{s}_{A}u=\int_{\mathbb{R}^{n}}\frac{A(x,y)}{|x-y|^{n+2s}}\frac{(x-%y)\otimes(x-y)}{|x-y|^{2}}(u(x)-u(y))\,dy. For $0<s<1$ {0and $A:{ℝ}^n\times {ℝ}^n\to ℝ$ {A:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}}taken to be symmetric and serving asa variable coefficient for the operator, the system under consideration is the fractional version of the classical Navier–Lamé linearized elasticity system. The study of the coupled system of nonlocal equations is motivated by its appearance in nonlocal mechanics, primarily in peridynamics. Our regularity result states that if $A(\cdot ,y)$ {A(\,\cdot\,,y)}is uniformly Holder continuous and ${inf}_{x\in {ℝ}^n}A(x,x)>0$ {\inf_{x\in\mathbb{R}^{n}}A(x,x)>0}, then for $f\in L_{\mathrm{loc}}^p$ {f\in L^{p}_{\rm loc}}, for $p\ge 2$ {p\geq 2}, the solution vector $u\in H_{\mathrm{loc}}^{2s-\delta ,p}$ {u\in H^{2s-\delta,p}_{\rm loc}}for some $\delta \in (0,s)$ {\delta\in(0,s)}. 

The goal of this paper is to study the [Formula: see text]-solvability of the strongly coupled nonlocal system [Formula: see text] where [Formula: see text] is a linear nonlocal coupled vector-valued operator associated with a kernel [Formula: see text] comparable to [Formula: see text] for [Formula: see text], satisfying certain ellipticity and cancellation conditions. For any [Formula: see text], [Formula: see text], the existence of a unique strong solution [Formula: see text] is proved via the method of continuity. To apply this method, we establish the continuity of the operator [Formula: see text] and the necessary a priori estimates. These are obtained through the study of the corresponding parabolic system. The proof strategy follows and extends recent ideas developed for the scalar setting, combining commutator estimates, Sobolev embeddings, a level set estimate and a bootstrap argument. 

We study optimal design problems where the design corresponds to a coefficient in the principal part of the state equation. The state equation, in addition, is parameter dependent, and we allow it to change type in the limit of this (modeling) parameter. We develop a framework that guarantees asymptotic compatibility, that is unconditional convergence with respect to modeling and discretization parameters to the solution of the corresponding limiting problems. This framework is then applied to two distinct classes of problems where the modeling parameter represents the degree of nonlocality. Specifically, we show unconditional convergence of optimal design problems when the state equation is either a scalar-valued fractional equation, or a strongly coupled system of nonlocal equations derived from the bond-based model of peridynamics. 

We prove the well posedness in weighted Sobolev spaces of certain linear and nonlinear elliptic boundary value problems posed on convex domains and under singular forcing. It is assumed that the weights belong to the Muckenhoupt class with ). We also propose and analyze a convergent finite element discretization for the nonlinear elliptic boundary value problems mentioned above. As an instrumental result, we prove that the discretization of certain linear problems are well posed in weighted spaces. 

We study a non-local optimal control problem involving a linear, bond-based peridynamics model. In addition to existence and uniqueness of solutions to our problem, we investigate their behavior as the horizon parameter 𝛿, which controls the degree of nonlocality, approaches zero. We then study a finite element-based discretization of this problem, its convergence, and the so-called asymptotic compatibility as the discretization parameter h and the horizon parameter 𝛿 tend to zero simultaneously.