Notes on optimal flux fields

For a given region, and specified boundary flux and density rate of an extensive property, the optimal flux field that satisfies the balance conditions is considered. The optimization criteria are the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> -norm and a Sobolev-like norm of the flux field. Finally, the capacity of the region to accommodate various boundary fluxes and density rates is defined and analyzed.