Galerkin Approximation of the Fractional Hardy Constant

We establish sharp estimates for the discrete optimal constant of the fractional Hardy Inequality in dimension $N\geq 1$, with fractional exponent $s\in \left(0,\min\left\{1,\frac{N}{2}\right\}\right)$.

The convergence rates that we establish take place for the Galerkin approximation with piecewise linear elements, when the computations are carried out in a bounded, convex and smooth domain containing the origin, for which we employ a quasi-uniform and regular mesh.