The free material design problem for the stationary heat equation on low dimensional structure

For a given balanced distribution of heat sources and sinks, Q, we find an optimal conductivity tensor field, Ĉ, minimizing the thermal compliance. We present Ĉ in a rather explicit form in term of the datum. Our solution is in a cone of non-negative tensor valued finite Borel measures. We present a series of examples with explicit solutions.

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