A variable exponent Sobolev theorem for fractional integrals on quasimetric measure spaces

Abstract

Abstract We prove that the coefficients a i j ( x ) ${a_{ij}(x)}$ , q(x) and the domain Ω of a multidimensional fractional diffusion equation can be recovered uniquely from measurements u ( b , t ) ${u(b,t)}$ , t ∈ ( t 0 , t 1 ) ${t\in (t_0,t_1)}$ , at an arbitrary single point b inside a bounded domain Ω ⊂ ℝ n ${\Omega \subset \mathbb {R}^n}$ . From the measurements we first recover infinitely many spectral data ( λ m , ϕ m ( x ) ) ${(\lambda _m,\varphi _m(x))}$ of the elliptic operator associated with the fractional diffusion equation. Then, the coefficients a i j ( x ) ${a_{ij}(x)}$ and q(x) are found from linear algebraic systems of the form A y = b ${Ay = b}$ , where A is a generalized Wronskian of some set of eigenfunctions that can be shown to be nontrivial. The domain Ω is reconstructed using the first eigenfunction ϕ 1 ( x ) ${\varphi _1(x)}$ .