Optimal Shape Design of 1D Fractional Order Heat Equation in Large Time

In this paper, we consider an optimal design problem for the 1D nonlocal heat equations involving the fractional Laplacian $$(-\frac{d^2}{dx^2})^s (0<s<1)$$ . The control here is the shape of the interval on which heat diffuses. We work in the geometric setting introduced by Šverák in [1] where the intervals under consideration are assumed to have a limited number of holes. Based on a $$\Gamma$$ -convergence approach, we can prove when $$s>\frac{1}{2}$$ , the nonlocal parabolic optimal designs converge, in the complementary Hausdorff topology, to an optimal design for the corresponding stationary nonlocal elliptic equation when time tends to infinity.