Dual extremum principles relating to cooling fins

Under consideration is a differential equation ${\left ( {pu’} \right )’} = qu$ of the Sturm—Liouville type where the function $q\left ( x \right ) > 0$ is given. The problem is to find a function $p\left ( x \right ) > 0$ in $0 \le x < b$, a constant $b$ and a solution $u\left ( x \right )$ of the corresponding differential equation such that the energy functional $\int _0^b {\left [ {p{{\left ( {u’} \right )}^2} + q{u^2}} \right ]} dx$ is maximized when $p\left ( x \right )$ is subject to the constraint $\int _0^b {{p^\rho }dx \le {K^\rho }}$ and $u$ is subject to the boundary conditions $u = 1$ at $x = 0$ and $p\left ( {du/dx} \right ) = 0$ at $x = b$. Here $K > 0$ and $\rho \ge 1$ are constants. A key relation $\left | {du/dx} \right | = \lambda {p^{\left ( {\rho - 1} \right )/2}}$, where $\lambda$ is a positive constant, is found. This criterion leads to explicit solution of the problem. A further consequence of this criterion together with a pair of dual extremum principles is a “duality inequality” giving sharp upper and lower estimates of the maximum value of the energy functional.