Existence and multiplicity proofs for semilinear elliptic boundary value problems by computer assistance

M. Plum, University of Karlsruhe, Germany

Wednesday, March 26
at 16.15
in HG E1.2

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, a method will be proposed which exploits the knowledge of a ``good'' numerical approximate solution, in order to provide a rigorous proof of an exact solution close to the approximate one; the expression "close to" is explicitly quantified in form of an error bound. 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. The method is used to prove existence and multiplicity statements for some specific examples, including cases where purely analytical methods had not been successful.