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arXiv · 2211.15991

Temperature on rods with Robin boundary conditions

Dimitrios Betsakos·Alexander Yu. Solynin
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Abstract

We consider solutions $u_f$ to the one-dimensional Robin problem with the heat source $f\in L^1[-π,π]$ and Robin parameter $α>0$. 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 $\max_{[-π,π]}u_f -\min_{[-π,π]}u_f$ over all heat sources $f$ such that $m\le f\le M$ and $\|f\|_{L^1}=2πs$. 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 $x_0\in [-π,π]$ over the same class of heat sources as above and discuss a few related questions.

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Dimitrios Betsakos, Alexander Yu. Solynin. 2022-11-29. Temperature on rods with Robin boundary conditions. https://arxiv.org/abs/2211.15991

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