This thesis explores a nonlocal nonlinear thermal diffusion law inspired by peridynamics, a nonlocal reformulation of classical mechanics. The primary objective is
to establish an existence theory for nonlocal equilibrium states based on the Dirichlet principle of minimum energy. By rigorously defining and analyzing nonlocal
analogues of gradient and divergence operators, we prove the existence and uniqueness of nonlocal equilibrium states, characterizing them as weak solutions to a nonlocal p-Laplacian law. Dual formulations are derived using Kelvin’s principle of
minimum complementary energy, and their well-posedness is established through
the Ladyzhenskaya-Babuška-Brezzi theory and Fenchel-Rockafellar duality. Furthermore, we demonstrate the convergence of nonlocal equilibrium states to their local
counterparts as the nonlocal interaction horizon vanishes, drawing on relevant results
from Bourgain-Brezis-Mironescu and Ponce.
The scope narrows down to linear diffusion, specifically focusing on the formulation and analysis of nonlocal optimal control and obstacle problems. Interestingly,
the analysis of linear-quadratic optimal distributed control problems closely mirrors
the corresponding local analysis. However, for locally ill-posed nonlinear control in
conductivity, nonlocal analogues yield practical solutions without additional regularization. Similarly, we analyze nonlocal obstacle problems with minimal assumptions
on the obstacles. We argue that nonlocal modeling proves advantageous in considering discontinuous conductivities and obstacles, which pose challenges in the local
theory.
Lastly, the thesis investigates the numerical approximation of equilibrium states.
We demonstrate the extension of the finite element method to the nonlocal case,
albeit with increased computational costs. Rigorous analysis confirms the convergence of finite element approximations to the nonlocal equilibrium state, and this
is supported by a series of numerical experiments. Additionally, we apply the nonlocal finite element approximations to solve nonlocal optimal control and obstacle
problems