Stable Determination and Reconstruction of a Quasilinear Term in an Elliptic Equation

TL;DR

The paper addresses the inverse problem of recovering a quasilinear term $\gamma$ in the elliptic equation $\nabla \cdot (\gamma(u) A(x) \nabla u)=0$ from conormal boundary data, proving Hölder stability with a single boundary measurement for $n\ge 2$ and establishing 1D uniqueness and Hölder stability under weaker regularity. The authors introduce a representation via $\Gamma(s)=\int_0^s\gamma(t)dt$ and derive key energy identities that link boundary measurements to interior properties, without relying on linearization. For $n=1$, a variational approach yields an optimal Hölder exponent $\frac{p-1}{2p-1}$ (and $\frac{1}{2}$ when $p=\infty$), while $n\ge 2$ achieves stability with a single Dirichlet excitation. Numerically, they implement Tikhonov regularization with an $H^1$ penalty and demonstrate stable reconstructions in both 2D and 1D under noisy data, validating the theoretical results and suggesting practical applicability in diffusion-type problems such as heat conduction. The work bridges theoretical stability with practical reconstruction, offering a framework for stable identification of quasilinear diffusion laws from boundary measurements.

Abstract

In this work, we investigate the inverse problem of determining a quasilinear term appearing in a nonlinear elliptic equation from the measurement of the conormal derivative on the boundary. This problem arises in several practical applications, e.g., heat conduction. We derive novel Hölder stability estimates for both multi- and one-dimensional cases: in the multi-dimensional case, the stability estimates are stated with one single boundary measurement, whereas in the one-dimensional case, due to dimensionality limitation, the stability results are stated for the Dirichlet boundary condition varying in a space of dimension one. We derive these estimates using different properties of solution representations. We complement the theoretical results with numerical reconstructions of the quasilinear term, which illustrate the stable recovery of the quasilinear term in the presence of data noise.

Figures (3)

• Figure 1: Numerical results for the 2D case.
• Figure 2: Numerical results in the 1D case.
• Figure 3: A schematic illustration of the minimizer $f_0(s)$.

Theorems & Definitions (18)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• proof : Proof of Theorem \ref{['thm:main theoretical result']}
• Proposition 4.1
• proof
• Proposition 4.2
• proof
• proof : Proof of Theorem \ref{['thm:1D-W1,1']}
• proof : Proof of Theorem \ref{['thm:1D-stability']}