The Free Material Design problem for stationary heat equation on low dimensional structures

For a given balanced distribution of heat sources and sinks, $Q$, we find an optimal conductivity tensor field, $\widehat{C}$, minimizing the thermal compliance. We present $\widehat{C}$ in a rather explicit form in terms 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.49J20, %Singular parabolic equations secondary: 49K20, 80M50