CRM CAMP in Nonlinear Analysis

Computer-Assisted Mathematical Proofs in Nonlinear Analysis

July 7, 2020 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

Computer-assisted existence and multiplicity proofs for semilinear elliptic problems on bounded and unbounded domains

Seminar presented by Michael Plum (Karlsruhe Institute of Technology, Germany)

Many boundary value problems for semilinear elliptic partial differential equations allow very stable numerical computations of approximate solutions, but are still lacking analytical existence proofs. In this lecture, we propose a method which exploits the knowledge of a "good" numerical approximate solution, in order to provide a rigorous proof of existence of an exact solution close to the approximate one. This goal is achieved by a fixed-point argument which takes all numerical errors into account, and thus gives a mathematical proof which is not "worse" than any purely analytical one. A crucial part of the proof consists of the computation of eigenvalue bounds for the linearization of the given problem at the approximate solution. The method is used to prove existence and multiplicity statements for some specific examples, including cases where purely analytical methods had not been successful.