## Abstract

Could the location of the maximum point for a positive

solution of a semilinear Poisson equation on a convex domain be

independent of the form of the nonlinearity? Cima and Derrick

found certain evidence for this surprising conjecture.

We construct counterexamples on the half-disk, by working with

the torsion function and first Dirichlet eigenfunction. On an isosceles right triangle the conjecture fails again. Yet the conjecture has

merit, since the maxima of the torsion function and eigenfunction

are unexpectedly close together. It is an open problem to quantify

this closeness in terms of the domain and the nonlinearity

solution of a semilinear Poisson equation on a convex domain be

independent of the form of the nonlinearity? Cima and Derrick

found certain evidence for this surprising conjecture.

We construct counterexamples on the half-disk, by working with

the torsion function and first Dirichlet eigenfunction. On an isosceles right triangle the conjecture fails again. Yet the conjecture has

merit, since the maxima of the torsion function and eigenfunction

are unexpectedly close together. It is an open problem to quantify

this closeness in terms of the domain and the nonlinearity

Original language | English (US) |
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Pages (from-to) | 81-88 |

Number of pages | 8 |

Journal | Bulletin of the Irish Mathematical Society |

Issue number | 78 |

State | Published - 2016 |