An Inverse Problem with Partial Neumann Data and $L^{n/2}$ Potentials

TL;DR

The paper proves uniqueness for the inverse Schrödinger problem with partial Neumann data for potentials in $L^{n/2}(\Omega)$. It develops an explicit Green's function and CGO solutions by constructing a global parametrix from small- and large-frequency components, together with boundary-correcting Poisson kernels, all in a flat, semiclassical framework. This approach yields a robust integral identity from CGO solutions, leading to $q_+=q_-$ under partial data and extending Calderón-type results to partial Neumann data. The methodology relies on boundary flattening, semiclassical pseudodifferential calculus, and careful Green's function construction, offering a direct route to uniqueness without Carleman estimates. The results have potential implications for practical inverse boundary problems where measurements are restricted to parts of the boundary and the conductivity is rough.

Abstract

We consider a partial data inverse problem with unbounded potentials. Rather than rely on functional analytic arguments or Carleman estimates, we construct an explicit Green's function with which we construct complex geometric optics (CGO) solutions and show unique determinability of potentials in $L^{n/2}$ for the Schrödinger equation with partial Neumann data.

Figures (2)

• Figure 1: An example of a valid set $\Omega$ meant as the interior of the drawn line. The arrows in the middle picture are select points on which the outward pointing normal vector has been sketched. Furthermore, the part of the boundary marked by dotted lines is where $\nu(y) \cdot y_n >0$, those marked by dashed lines is where $\nu(y) \cdot y_n <0$, and the part marked by a full line is where $\nu(y) \cdot y_n =0$. Finally, the right-most sketch depicts $\partial\Omega\setminus F = \Gamma_2\cup\Gamma_4$ and $\partial\Omega\setminus B = \Gamma_1\cup\Gamma_3$ where the $\Gamma_j$ are positively separated from each other. Note that the perfectly vertical part of the boundary on the left side is contained in both $F$ and $B$.
• Figure 2: To simplify sketching it, we consider the boundary $\partial\Omega$ to be flat, where $\Omega$ sits above the horizontal line. The sets $\Gamma$ and $\Gamma'$ are denoted by their endpoints The interior of the inner-most ellipse is the set in which $\chi = 1$, the area inside the next smallest ellipse is the support of $\chi$. The set inside the ellipse marked by the dashed line denotes $W$. Finally, the area above the thick line is where $y_n > g(y')$.

Theorems & Definitions (52)

• Theorem 1
• Corollary 2
• Proposition 3
• Theorem 4
• Lemma 5: CT20
• Lemma 6: CT20
• Lemma 7
• proof
• Lemma 8: CT20
• Lemma 9: CT20