2024

Nurminen, Janne

This thesis focuses on studying inverse problems for nonlinear elliptic partial differential equations and in particular inverse problems for the minimal surface equation and semilinear elliptic equations. It is shown that one can recover information about the coefficients of the equation or some geometric information from boundary measurements of solutions. The main tool used is linearization, both first order and higher order linearization. The introduction describes inverse problems for partial differential equations in the context of the Calder´on problem and gives a survey of the literature related to the linearization methods. Main theorems of the included articles are presented and the methods to prove them are also discussed. The articles (A) and (C) focus on inverse problems for the minimal surface equation. In both articles we look at the minimal surface equation in Euclidean space that is equipped with a Riemannian metric. Then from boundary measurements we determine information about the metric. In (A) the metric is conformally Euclidean and in (C) the metric will be in a class of admissible metrics. The main method used in both articles is the higher order linearization method. The remaining articles (B) and (D) study inverse problems for semilinear elliptic equations. In (B) the equation has a power type nonlinearity and the aim is to determine an unbounded potential from boundary measurements. Also in (B) the method used is the higher order linearization method. In (D) the focus is on recovering a general zeroth order nonlinearity from boundary measurements. Here the first linearization is used and we improve previous results for this method in the case of semilinear equations.