On a Model of Nonlocal Continuum Mechanics Part I: Existence and Regularity

In this work we assume that the total stored energy functional for a body $\B$ depends not only on the local strain field, but also on the spatial average of the strain field over the body weighted with an influence kernel. We investigate the problem of minimizing the total stored energy subject to a given bulk displacement $\Delta\geq 0$. After the general setup for this problem is reviewed, we give sufficient conditions for an energy minimizing strain field $e(\cdot)$ to satisfy an integro-differential Euler--Lagrange equation. The result is general and applies to material energies that display a wide variety of singular behavior. Through analysis of this Euler--Lagrange equation for a special class of influence kernels, we arrive at a regularity theorem which ensures that energy minimizing strain fields must be periodic, piecewise smooth, and possess a finite number of simple discontinuities. We then combine this with a well-known existence result for relaxed minimization problems to arrive at a genera...