Temperature on rods with Robin boundary conditions

Abstract

We consider solutions uf to the one-dimensional Robin problem with the heat source f∈L1[−π,π] and Robin parameter α>0. For given m, M, and s, 0≤m<s<M, we identify the heat sources f0, such that uf0 maximizes the temperature gap max[−π,π]⁡uf−min[−π,π]⁡uf over all heat sources f such that m≤f≤M and ‖f‖L1=2πs. In particular, this answers a question raised by J. J. Langford and P. McDonald in [5]. We also identify heat sources, which maximize/minimize uf at a given point x0∈[−π,π] over the same class of heat sources as above and discuss a few related questions.