Computer-assisted proofs for fixed point problems in Sobolev spaces

Alain Schenkel, Jan Wehr, Peter Wittwer

Research output: Contribution to journalReview article

6 Scopus citations

Abstract

In this paper we extend the technique of computer-assisted proofs to fixed point problems in Sobolev spaces. Up to now the method was limited to the case of spaces of analytic functions. The possibility to work with Sobolev spaces is an important progress and opens up many new domains of applications. Our discussion is centered around a concrete problem that arises in the theory of critical phenomena and describes the phase transition in a hierarchical system of random resistors. For this problem we have implemented in particular the convolution product based on the Fast Fourier Transform (FFT) algorithm with rigorous error estimates.

Original languageEnglish (US)
Pages (from-to)1-67
Number of pages67
JournalMathematical Physics Electronic Journal
Volume6
StatePublished - Dec 1 2000

Keywords

  • Computer-assisted proofs
  • Constructive analysis in Sobolev spaces
  • Discrete convolutions are convolutions of splines
  • Phase transitions in random media

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability

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