[Paper Review] Uniqueness for the inverse boundary value problem with singular potentials in 2D

Emilia Blåsten, Leo Tzou+1 more|arXiv (Cornell University)|Apr 21, 2017

Numerical methods in inverse problems3 citations

TL;DR

This paper establishes the global uniqueness of the inverse boundary value problem for the Schrödinger equation in two dimensions with singular potentials in $L^p(\Omega)$ for $p > 4/3$. By constructing complex geometrical optics solutions with phase $\Phi(z) = (z - z_0)^2$ and refining estimates for the conjugated Cauchy operator, the authors prove that the Dirichlet-to-Neumann map uniquely determines the potential, extending previous results from $p > 2$ to the critical range $p > 4/3$. The key contribution is a new $L^p$-based regularity improvement via the DN map, enabling the method of stationary phase to identify the potential difference.

ABSTRACT

In this paper we consider the inverse boundary value problem for the Schrödinger equation with potential in $L^p$ class, $p>4/3$. We show that the potential is uniquely determined by the boundary measurements.

Motivation & Objective

• To establish global uniqueness for the inverse boundary value problem of the Schrödinger equation in two dimensions with singular potentials in $L^p(\Omega)$ for $p > 4/3$.
• To extend the range of $p$ for which uniqueness holds beyond the previously known $p > 2$ and $p > 4/3$ cases, closing a gap in the theory of inverse problems with low-integrability potentials.
• To develop a new estimate for the conjugated Cauchy operator that enables the construction of complex geometrical optics solutions for $L^p$ potentials with $p > 1$.
• To show that the knowledge of the Dirichlet-to-Neumann map improves the integrability of the potential difference, ensuring $q_1 - q_2 \in L^2(\Omega)$ when $\Lambda_{q_1} = \Lambda_{q_2}$ and $3/4 < p < 2$.
• To provide a rigorous framework for the method of stationary phase in the context of $L^p$ potentials by deriving refined estimates in Alessandrini's identity.

Proposed method

• Construct complex geometrical optics solutions of the form $e^{i\tau\Phi}f_1$ and $e^{i\tau\overline{\Phi}}f_2$ with phase $\Phi(z) = (z - z_0)^2$ for $L^p$ potentials with $p > 1$.
• Use a Neumann series expansion to define $f_j(z) = \sum_{m=0}^\infty F_{j,m}(z)$, where $F_{j,m}$ are iteratively defined operators involving the potential and the conjugated Cauchy operator.
• Establish uniform $L^\infty$ bounds $\|F_{j,m}\|_\infty \leq (C\tau^{-\alpha})^m$ for large $\tau$, ensuring convergence of the series in $W^{1,2}(X)$.
• Apply the conjugated Cauchy operator $\overline{\partial}^{-1}$ and $\partial^{-1}$ to solve the associated Beltrami-type equations, leveraging boundedness of these operators from $L^p$ to $W^{1,p}$.
• Use the fact that $\Lambda_{q_1} = \Lambda_{q_2}$ implies $q_1 - q_2 \in L^2(\Omega)$ to improve integrability and dominate the leading term in Alessandrini's identity.
• Perform a stationary phase argument on the resulting integral identity after substituting the CGO solutions, using refined estimates to isolate the potential difference.

Experimental results

Research questions

• RQ1 Can the global uniqueness of the inverse boundary value problem for the Schrödinger equation be established for $L^p$ potentials with $p > 4/3$ in two dimensions?
• RQ2 Does the method of complex geometrical optics with phase $\Phi(z) = (z - z_0)^2$ extend to potentials in $L^p(\Omega)$ for $p > 1$?
• RQ3 Can the conjugated Cauchy operator be estimated with sufficient decay in $\tau$ to ensure convergence of the CGO solution series for $L^p$ potentials with $p > 4/3$?
• RQ4 How does the knowledge of the Dirichlet-to-Neumann map improve the integrability of the potential difference, and can this be used to dominate the leading term in Alessandrini's identity?
• RQ5 Is it possible to close the gap between the known uniqueness range $p > 2$ and the scale-invariant $L^2$-critical case $p = 4/3$ in 2D inverse problems?

Key findings

• The Dirichlet-to-Neumann map uniquely determines the potential $q$ in $L^p(\Omega)$ for $p > 4/3$ in two dimensions, extending previous results from $p > 2$.
• Complex geometrical optics solutions with phase $\Phi(z) = (z - z_0)^2$ exist for all $q \in L^p(\Omega)$ with $p > 1$, and the series $f_j(z) = \sum_{m=0}^\infty F_{j,m}(z)$ converges uniformly in $z \in X$ for large $\tau$.
• The authors establish a new estimate for the conjugated Cauchy operator that yields $\|F_{j,m}\|_\infty \leq (C\tau^{-\alpha})^m$ for some $\alpha > 0$, ensuring exponential decay and convergence of the series.
• The knowledge of $\Lambda_{q_1} = \Lambda_{q_2}$ implies $q_1 - q_2 \in L^2(\Omega)$ for $3/4 < p < 2$, which is crucial for dominating the leading term in Alessandrini's identity.
• The method of stationary phase successfully isolates the potential difference in the integral identity, proving $q_1 = q_2$ under the stated conditions.
• The result is consistent with the unique continuation property for $u \in H^{2,2}_{\text{loc}}$ when $q \in L^p_{\text{loc}}$ with $p > 4/3$, suggesting a natural threshold for the problem.