Recovering the total singularity of a conormal potential from backscattering data

Joshi, Mark S.

Annales de l'Institut Fourier, Tome 48 (1998) no. 5, pp. 1513-1532.

On étudie le problème de la restitution de singularités d’un potentiel de la rétrodiffusion. Soit $\Omega$ un domaine précompact, convexe et $C^{\infty }$. Soit $V_i=v+w_i$ avec $v\in C_c^{\infty }({ℝ}^n)$ et $w_i$ conormale au bord de $\Omega$ et avec support dans $\overline{\Omega }$; si les données de la rétrodiffusion de $V_1$ et $V_2$ sont égaux, alors $V_1-V_2\in C^{\infty }$.

The problem of recovering the singularities of a potential from backscattering data is studied. Let $\Omega$ be a smooth precompact domain in ${ℝ}^n$ which is convex (or normally accessible). Suppose $V_i=v+w_i$ with $v\in C_c^{\infty }({ℝ}^n)$ and $w_i$ conormal to the boundary of $\Omega$ and supported inside $\overline{\Omega }$ then if the backscattering data of $V_1$ and $V_2$ are equal up to smoothing, we show that $w_1-w_2$ is smooth.

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