Existence and Regularity of Minimizers for Nonlocal Functionals

We present a theory combining two fields; calculus of variations and the theory of nonlocal calculus. In this framework, we consider a functional modeled in the nonlocal theory such as peridynamics or nonlocal diffusion models which involve natural discontinuities. We derive necessary conditions, in the form of Euler-Lagrange equations, for minimizers of such functionals under some imposed growth conditions imposed. Then, we establish connections between this nonlocal result and the Euler-Lagrange equations associated with the standard functional considered in the classical theory. We also show existence results which come from a converse of the Euler-Lagrange equations as well as another existence theorem with lower bounds for the integrand of the functional. Furthermore, we show regularity of solutions to nonlocal equations which are some of the first results concerning this topic. Lastly, we provide several examples applying the developed theory to show existence and regularity of solutions to various nonlocal equations