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

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 H\"older 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.