Temperature on rods with Robin boundary conditions

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 $\Vert f{\Vert }_{L^1}=2\pi s$. 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.