G5 Doctoral dissertation (article)
Inverse problems for the minimal surface equation and semilinear elliptic partial differential equations (2024)
Nurminen, J. (2024). Inverse problems for the minimal surface equation and semilinear elliptic partial differential equations [Doctoral dissertation]. University of Jyväskylä. JYU Dissertations, 780. https://urn.fi/URN:ISBN:978-952-86-0159-3
JYU authors or editors
Publication details
All authors or editors: Nurminen, Janne
eISBN: 978-952-86-0159-3
Journal or series: JYU Dissertations
eISSN: 2489-9003
Publication year: 2024
Number in series: 780
Number of pages in the book: 1 verkkoaineisto (17 sivua, 77 sivua useina numerointijaksoina, 10 numeroimatonta sivua)
Publisher: University of Jyväskylä
Place of Publication: Jyväskylä
Publication country: Finland
Publication language: English
Persistent website address: https://urn.fi/URN:ISBN:978-952-86-0159-3
Publication open access: Openly available
Publication channel open access: Open Access channel
Abstract
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.
Keywords: partial differential equations; inverse problems
Free keywords: linearisaatio
Contributing organizations
Ministry reporting: Yes
