A nonlocal-to-local transfer principle for parabolic Calderón problems

Abstract

We prove a nonlocal-to-local transfer principle for fractional powers of variable coefficient parabolic operators. Let $H_{\sigma }={\partial }_t-\nabla \cdot \sigma \nabla$, where $\sigma$ is a uniformly elliptic $C^2$ matrix-valued coefficient. For any $s\in (0,1)$ and any nonempty bounded Lipschitz exterior observation set $W$ with $\overline{W}\cap \overline{\Omega }=\varnothing$, we show that equality of the partial exterior DN maps for $H_{{\sigma }_j}^s$, $j=1,2$, together with the exterior agreement ${\sigma }_1={\sigma }_2$ in ${\Omega }_e$, implies equality of the full lateral local Cauchy data sets for $H_{{\sigma }_j}$ on $(\partial \Omega {)}_T$. The proof uses the parabolic Caffarelli--Silvestre extension and the averaging transform \[ v(t,x)=\int_0^\infty y^{1-2s}\widetilde u(t,x,y)\,dy. \] This transform sends fractional parabolic solutions to local parabolic energy solutions. The main analytic ingredient is a boundary density theorem: the lateral traces generated by these averaged exterior solutions are dense in the corresponding closed local energy trace space. Unlike the elliptic reduction, no exterior normalization to a constant is imposed; the common exterior coefficient may be variable. The result holds for all spatial dimensions $n\ge 2$. Consequently, local uniqueness results for parabolic Calderón problems transfer to the corresponding nonlocal problems. In particular, scalar leading coefficients are uniquely determined in $\Omega$ from partial exterior nonlocal measurements, relative to their exterior values.

43 pages