Temperature on rods with Robin boundary conditions

TL;DR

This paper characterizes heat sources that maximize temperature variations and pointwise temperature at a given location for the one-dimensional Robin heat problem, answering a question posed by Langford and McDonald.

Contribution

It explicitly identifies extremal heat sources for temperature gap and pointwise temperature in Robin boundary conditions, extending previous understanding.

Findings

Maximizers of temperature gap are characterized.

Extremal heat sources for pointwise temperature are identified.

Results answer a previously posed open question.

Abstract

We consider solutions $u_f$ to the one-dimensional Robin problem with the heat source $f\in L^1[-\pi,\pi]$ and Robin parameter $\alpha>0$. For given $m$, $M$, and $s$, $0\le m<s<M$, we identify the heat sources $f_0$, such that $u_{f_0}$ maximizes the temperature gap $\max_{[-\pi,\pi]}u_f -\min_{[-\pi,\pi]}u_f$ over all heat sources $f$ such that $m\le f\le M$ and $\|f\|_{L^1}=2\pi s$. In particular, this answers a question raised by J.~J.~Langford and P.~McDonald in \cite{LM}. We also identify heat sources, which maximize/minimize $u_f$ at a given point $x_0\in [-\pi,\pi]$ over the same class of heat sources as above and discuss a few related questions.