A weakly coupled system of p-Laplace type in a heat conduction problem

Given is a bounded domain <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:math> {Ωⁿ} , and a vector-valued function defined on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:math> {∂Ω} (depicting temperature distributions from different sources), our objective is to study the mathematical model of a physical problem of enclosing <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:math> {∂Ω} with a specific volume of insulating material to reduce heat loss in a stationary scenario. Mathematically, this task involves identifying a vector-valued function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝐮</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>u</m:mi> <m:mn>1</m:mn> </m:msup> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msup> <m:mi>u</m:mi> <m:mi>m</m:mi> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {u=(u¹,,uᵐ)} ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {m≥ 1} ) that represents the temperature within Ω and gives rise to a free boundary, somehow reminiscent of, but not equivalent to, the Bernoulli free boundary problem.