### Abstract

Original language | English |
---|---|

Number of pages | 48 |

Volume | arXiv:1711.02779 |

Publication status | Published - 8 Nov 2017 |

### Cite this

*Non-concavity of Robin eigenfunctions*.

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**Non-concavity of Robin eigenfunctions.** / Clutterbuck, Julie Faye; Hauer, Daniel; Andrews, Ben .

Research output: Working paper › Working Paper › Other

TY - UNPB

T1 - Non-concavity of Robin eigenfunctions

AU - Clutterbuck, Julie Faye

AU - Hauer, Daniel

AU - Andrews, Ben

PY - 2017/11/8

Y1 - 2017/11/8

N2 - On a convex bounded Euclidean domain, the ground state for the Laplacian with Neumann boundary conditions is a constant, while the Dirichlet ground state is log-concave. The Robin eigenvalue problem can be considered as interpolating between the Dirichlet and Neumann cases, so it seems natural that the Robin ground state should have similar concavity properties. In this paper we show that this is false, by analysing the perturbation problem from the Neumann case. In particular we prove that on polyhedral convex domains, except in very special cases (which we completely classify) the variation of the ground state with respect to the Robin parameter is not a concave function. We conclude from this that the Robin ground stat is not log-concave (and indeed even has some superlevel sets which are non-convex) for small Robin parameter on polyhedral convex domains outside a special class, and hence also on arbitrary convex domains which approximate these in Hausdorff distance.

AB - On a convex bounded Euclidean domain, the ground state for the Laplacian with Neumann boundary conditions is a constant, while the Dirichlet ground state is log-concave. The Robin eigenvalue problem can be considered as interpolating between the Dirichlet and Neumann cases, so it seems natural that the Robin ground state should have similar concavity properties. In this paper we show that this is false, by analysing the perturbation problem from the Neumann case. In particular we prove that on polyhedral convex domains, except in very special cases (which we completely classify) the variation of the ground state with respect to the Robin parameter is not a concave function. We conclude from this that the Robin ground stat is not log-concave (and indeed even has some superlevel sets which are non-convex) for small Robin parameter on polyhedral convex domains outside a special class, and hence also on arbitrary convex domains which approximate these in Hausdorff distance.

M3 - Working Paper

VL - arXiv:1711.02779

BT - Non-concavity of Robin eigenfunctions

ER -