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Abstract

We study recovery of a domain from the entire boundary heat flux at one positive time, for Dirichlet evolution with initial temperature one. Near a ball, after fixing volume and centroid, the flux defect in C^(m−1,α) controls the radial deformation in C^(m,α), where m≥3 and 0<α<1. Dividing the flux by its physical boundary mean gives one stability constant and one shape neighborhood valid for every observation time T≥ T₀>0. This normalization removes the exponential late-time loss of the absolute flux signal; it does not remove sensitivity to fixed absolute measurement noise. The proof combines strict positivity of the spherical heat multipliers above translations with uniform shape-differentiable convergence to the normalized first-eigenfunction flux. We also give a moving-plane proof that constant signed flux on every boundary component at one time forces a bounded connected smooth domain to be a ball, without convexity or connected boundary. In contrast, a moving proper observation cap admits infinitely many nearby normalized nonballs with exactly the ball flux at the prescribed time.