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A Computer-Assisted Uniqueness Proof for a Semilinear Elliptic Boundary Value Problem

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Inequalities and Applications 2010

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 161))

Abstract

A wide variety of articles, starting with the famous paper (Gidas, Ni and Nirenberg in Commun. Math. Phys. 68, 209–243 (1979)), is devoted to the uniqueness question for the semilinear elliptic boundary value problem −Δu=λu+u p in Ω, u>0 in Ω, u=0 on ∂Ω, where λ ranges between 0 and the first Dirichlet Laplacian eigenvalue. So far, this question was settled in the case of Ω being a ball and, for more general domains, in the case λ=0. In (McKenna et al. in J. Differ. Equ. 247, 2140–2162 (2009)), we proposed a computer-assisted approach to this uniqueness question, which indeed provided a proof in the case Ω=(0,1)2, and p=2. Due to the high numerical complexity, we were not able in (McKenna et al. in J. Differ. Equ. 247, 2140–2162 (2009)) to treat higher values of p. Here, by a significant reduction of the complexity, we will prove uniqueness for the case p=3.

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Notes

  1. 1.

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Acknowledgements

The authors are grateful to two anonymous referees for their helpful remarks and suggestions.

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Authors and Affiliations

  1. Dep. of Mathematics, University of Connecticut, Storrs, CT, 06269-3009, MSB 328, USA

    Patrick J. McKenna

  2. Dipartimento di Matematica, Università di Roma “La Sapienza”, P.le A. Moro 2, 00185, Roma, Italy

    Filomena Pacella

  3. Institut für Analysis, Karlsruher Institut für Technologie (KIT), 76128, Karlsruhe, Germany

    Michael Plum & Dagmar Roth

Authors
  1. Patrick J. McKenna
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  2. Filomena Pacella
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  3. Michael Plum
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  4. Dagmar Roth
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Corresponding author

Correspondence to Michael Plum .

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Editors and Affiliations

  1. Institute of Mathematics, University of Basel, Rheinsprung 21, Basel, 4051, Switzerland

    Catherine Bandle

  2. Faculty of Informatics, University of Debrecen, Pf. 12, Debrecen, 4010, Hungary

    Attila Gilányi

  3. Department of Economic Analysis, and Information Technology for Business, University of Debrecen, Kassai út 26, Debrecen, 4028, Hungary

    László Losonczi

  4. , Institute of Analysis, KIT, Kaiserstr. 12, Karlsruhe, 76128, Germany

    Michael Plum

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Dedicated to the memory of Wolfgang Walter

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McKenna, P.J., Pacella, F., Plum, M., Roth, D. (2012). A Computer-Assisted Uniqueness Proof for a Semilinear Elliptic Boundary Value Problem. In: Bandle, C., Gilányi, A., Losonczi, L., Plum, M. (eds) Inequalities and Applications 2010. International Series of Numerical Mathematics, vol 161. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0249-9_3

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