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arXiv · 1210.5893

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

Patrick J. McKenna·Filomena Pacella·Michael Plum·Dagmar Roth
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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 the boundary of 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)x(0,1), 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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Patrick J. McKenna, Filomena Pacella, Michael Plum, Dagmar Roth. 2012-10-22. A Computer-Assisted Uniqueness Proof for a Semilinear Elliptic Boundary Value Problem. https://doi.org/10.1007/978-3-0348-0249-9_3

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