里斯分数阶情形下的H收敛与Γ收敛：非线性情形
$H$-convergence and $Γ$-convergence in the Riesz fractional setting: the nonlinear case
AI总结：
研究里斯分数阶情形下非线性非局部单调算子的H收敛与相关非局部能量泛函的Γ收敛，证明其分别与相应局部算子及能量的收敛等价，借助新唯一性结果得出紧性，揭示了二者收敛的等价关系。
AI中文摘要：
本文关注通过里斯分数阶梯度和散度定义的非线性非局部单调算子的H收敛。我们证明了在这个非局部框架下的H收敛等同于相应局部算子的H收敛，从而得到一类合适的非局部单调算子的H紧性结果。接着研究了与保守单调算子子类相关的非局部能量泛函的Γ收敛，证明其等同于相应局部能量的Γ收敛。关键要素是局部和非局部泛函积分表示的新唯一性结果，还得到了所考虑的非局部能量类的Γ紧性。最后展示了非局部保守单调算子的H收敛与相关能量泛函的Γ收敛之间的等价性。
英文摘要：
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 $Γ$-convergence of nonlocal energy functionals associated with the subclass of \emph{conservative} monotone operators, proving that it is equivalent to the $Γ$-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 $Γ$-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 $Γ$-convergence of the associated energy functionals.
