$H$-convergence and $\Gamma$-convergence in the Riesz fractional setting: the nonlinear case

This paper concerns the $H$-convergence of nonlinear nonlocal monotone operators defined through the Riesz fractional gradient and divergence.

We show that the $H$-convergence in this nonlocal framework is equivalent to the $H$-convergence of the corresponding local one.

As a consequence, we obtain a $H$-compactness result for a suitable class of nonlocal monotone operators.

We then study the $\Gamma$-convergence of nonlocal energy functionals associated with the subclass of \emph{conservative} monotone operators, proving that it is equivalent to the $\Gamma$-convergence of the corresponding local energies.

A key ingredient is a new uniqueness result for the integral representation of both local and nonlocal functionals.

As a by-product, we obtain the $\Gamma$-compactness of the class of nonlocal energies under consideration.

Finally, we show the equivalence between the $H$-convergence of nonlocal conservative monotone operators and the $\Gamma$-convergence of the associated energy functionals.