### 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 language | English (US) |
---|---|

Pages (from-to) | 1-67 |

Number of pages | 67 |

Journal | Mathematical Physics Electronic Journal |

Volume | 6 |

State | Published - 2000 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematical Physics

### Cite this

*Mathematical Physics Electronic Journal*,

*6*, 1-67.

**Computer-assisted proofs for fixed point problems in Sobolev spaces.** / Schenkel, Alain; Wehr, Jan; Wittwer, Peter.

Research output: Contribution to journal › Article

*Mathematical Physics Electronic Journal*, vol. 6, pp. 1-67.

}

TY - JOUR

T1 - Computer-assisted proofs for fixed point problems in Sobolev spaces

AU - Schenkel, Alain

AU - Wehr, Jan

AU - Wittwer, Peter

PY - 2000

Y1 - 2000

N2 - 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.

AB - 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.

KW - Computer-assisted proofs

KW - Constructive analysis in Sobolev spaces

KW - Discrete convolutions are convolutions of splines

KW - Phase transitions in random media

UR - http://www.scopus.com/inward/record.url?scp=15944366874&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=15944366874&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:15944366874

VL - 6

SP - 1

EP - 67

JO - Mathematical Physics Electronic Journal

JF - Mathematical Physics Electronic Journal

SN - 1086-6655

ER -