An inverse source problem for the Monge--Ampere equation from large boundary data

Technical Summary: An Inverse Source Problem for the Monge–Ampère Equation from Large Boundary Data

Problem Statement
This paper addresses the inverse source problem for the Monge–Ampère equation on a bounded, smooth, uniformly convex domain $\Omega \subset \mathbb{R}^n$ ($n \ge 2$). The equation is given by:
$$\det D^2 u = f(x) \quad \text{in } \Omega,$$
where the source term $f$ is a positive, smooth function. The objective is to determine whether the source $f$ is uniquely determined by the Dirichlet-to-Neumann (DN) map $\Lambda_f$, which maps boundary data $g$ to the normal derivative of the unique strictly convex solution $u_g$:
$$\Lambda_f(g) = \partial_\nu u_g|_{\partial\Omega}.$$
Specifically, the authors investigate whether $\Lambda_{f_1} = \Lambda_{f_2}$ implies $f_1 = f_2$ within the class of smooth, positive sources.

Methodology
The authors employ a "large-data" asymptotic approach, distinct from the standard higher-order linearization methods used for semilinear equations. The methodology proceeds through the following steps:

1. Large Cylindrical Boundary Data: Instead of perturbing around a zero or small solution, the authors prescribe a family of large boundary values parameterized by $t \gg 1$. For a fixed direction $\omega \in S^{n-1}$, the boundary data is defined as:
   $$g_{t,\omega} = t \phi_\omega|_{\partial\Omega}, \quad \text{where } \phi_\omega(x) = \frac{1}{2}|P_{\omega^\perp} x|^2.$$
   Here, $P_{\omega^\perp}$ is the orthogonal projection onto the hyperplane $\omega^\perp$. The profile $\phi_\omega$ is strictly convex in directions transverse to $\omega$ but degenerate (linear) in the $\omega$-direction.

2. Asymptotic Expansion: The solution $u_{t,\omega}$ corresponding to the large data $g_{t,\omega}$ is analyzed via an asymptotic expansion:
   $$u_{t,\omega} = t\phi_\omega + t^{1-n}w_\omega + r_{t,\omega},$$
   where $w_\omega$ is a "chordwise correction" term. The leading term $t\phi_\omega$ captures the transverse convexity, while the first correction $t^{1-n}w_\omega$ accounts for the missing longitudinal convexity required to satisfy the Monge–Ampère equation with source $f$.

3. Reduction to a 1D Problem: The correction term $w_\omega$ is shown to satisfy a one-dimensional Dirichlet problem along the chords of $\Omega$ parallel to $\omega$. Specifically, for a chord parameterized by $s$ (where $x = y + s\omega$), $w_\omega$ satisfies:
   $$\partial_s^2 w_\omega(y, s) = f(y + s\omega), \quad w_\omega(y, s_\pm(y)) = 0,$$
   where $s_\pm(y)$ are the endpoints of the chord.

4. Boundary Measurement Analysis: By analyzing the asymptotic behavior of the normal derivative $\partial_\nu u_{t,\omega}$ on the non-glancing boundary, the authors show that the DN map determines the boundary values of $\partial_\nu w_\omega$. Using the fundamental theorem of calculus on the chordwise equation, these boundary derivatives are directly related to the line integral of the source $f$ along the chord:
   $$\int_{s_-(y)}^{s_+(y)} f(y + s\omega) \, ds = \partial_s w_\omega(y, s_+(y)) - \partial_s w_\omega(y, s_-(y)).$$

5. Injectivity of the X-ray Transform: The equality of the DN maps for the large data family implies the equality of the Euclidean X-ray transforms (Radon transform restricted to lines) of the two sources $f_1$ and $f_2$. The authors utilize the standard Fourier-slice theorem to conclude that if the X-ray transforms are identical, the sources must be identical.

Key Contributions and Results

• Uniqueness Theorem (Theorem 1.1): The primary result establishes that if $\Lambda_{f_1}(g) = \Lambda_{f_2}(g)$ for all smooth boundary data $g$, then $f_1 = f_2$. Crucially, the proof demonstrates that uniqueness holds even if the DN maps are only known to be equal on the restricted family of large cylindrical boundary values $g_{t,\omega}$.
• Asymptotic Rigor (Theorem 3.3): The paper provides rigorous $L^\infty$ estimates for the difference between the true solution and the asymptotic ansatz $t\phi_\omega + t^{1-n}w_\omega$. It proves that the error is of order $O(t^{1-2n})$ in the bulk and $O(t^{1-n-n/2})$ for the normal derivative on compact subsets of the non-glancing boundary.
• Handling Glancing Points: A significant technical contribution is the analysis of the behavior of the chordwise correction $w_\omega$ near "glancing" points (where the chord is tangent to the boundary). The authors prove that despite the parametrization of endpoints becoming singular at these points, the function $w_\omega$ extends smoothly to the entire domain $\Omega$, ensuring the validity of the asymptotic expansion globally.
• Dimensional Generality: Unlike previous work restricted to two dimensions (e.g., [29]), this method is valid for all dimensions $n \ge 2$.

Significance and Claims
The paper claims to provide a new mechanism for solving inverse source problems for fully nonlinear equations of the Monge–Ampère type. The significance lies in:

1. Methodological Shift: Moving away from linearization at small solutions (which leads to linearized coefficients depending on the Hessian of the background solution) to a large-data asymptotic regime. This avoids the degeneracy issues often associated with linearizing at zero or small solutions in quasilinear problems.
2. Direct Geometric Reduction: The method reduces the nonlinear inverse problem directly to the injectivity of the Euclidean X-ray transform via the geometry of chords, bypassing the need to recover intermediate coefficients.
3. Completeness: The result confirms that the full DN map is not strictly necessary for uniqueness; a specific, physically motivated family of large boundary measurements suffices to recover the source uniquely.

The authors note that their approach is distinct from the method in [29], which linearizes at non-zero solutions and recovers the Hessian matrix before taking the determinant. In contrast, this paper's large-parameter approach extracts the source integrals directly from the first-order correction term.