Abstract: Given is a bounded domain C Rn, and a vector-valued function defined on ∂(depicting temperature distributions from different sources), our objective is to study the mathematical model of a physical problem of enclosing ∂with a specific volume of insulating material to reduce heat loss in a stationary scenario. Mathematically, this task involves identifying a vector-valued function u = (u1, um) (m ? 1) that represents the temperature within and gives rise to a free boundary, somehow reminiscent of, but not equivalent to, th Bernoulli free boundary problem.