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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. 

In this note, we prove an estimate on the level sets of a function with (𝑝,𝑞) growth that depends on the difference quotient of a bounded weak solution to a nonlocal double-phase equation. This estimate is related to a self-improving property of these solutions. 

Nonlocal vector calculus, which is based on the nonlocal forms of gradient, divergence, and Laplace operators in multiple dimensions, has shown promising applications in fields such as hydrology, mechanics, and image processing. In this work, we study the analytical underpinnings of these operators. We rigorously treat compositions of nonlocal operators, prove nonlocal vector calculus identities, and connect weighted and unweighted variational frameworks. We combine these results to obtain a weighted fractional Helmholtz decomposition which is valid for sufficiently smooth vector fields. Our approach identifies the function spaces in which the stated identities and decompositions hold, providing a rigorous foundation to the nonlocal vector calculus identities that can serve as tools for nonlocal modeling in higher dimensions. 

This work aims to prove a Hardy-type inequality and a trace theorem for a class of function spaces on smooth domains with a nonlocal character. Functions in these spaces are allowed to be as rough as an [Formula: see text]-function inside the domain of definition but as smooth as a [Formula: see text]-function near the boundary. This feature is captured by a norm that is characterized by a nonlocal interaction kernel defined heterogeneously with a special localization feature on the boundary. Thus, the trace theorem we obtain here can be viewed as an improvement and refinement of the classical trace theorem for fractional Sobolev spaces [Formula: see text]. Similarly, the Hardy-type inequalities we establish for functions that vanish on the boundary show that functions in this generalized space have the same decay rate to the boundary as functions in the smaller space [Formula: see text]. The results we prove extend existing results shown in the Hilbert space setting with p = 2. A Poincaré-type inequality we establish for the function space under consideration together with the new trace theorem allows formulating and proving well-posedness of a nonlinear nonlocal variational problem with conventional local boundary condition. 

We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic "phases" that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work "Nonlocal self-improving properties" Analysis & PDE, 8(1):57–114 for the specific nonlinear setting under investigation in this manuscript.