Nonlinear anisotropic heat conduction in a transformer magnetic core

Abstract

In this chapter we deal with a quasilinear elliptic problem whose classical formulation reads:

Find $u \in {C^1}\left( {\bar \Omega } \right)$ such that u|_Ω ∈ C ²(Ω) and

$$ - div\left( {A\left( { \cdot ,u} \right)grad\;u} \right) = f\quad in\;\Omega $$

((9.1))

$$ u = \bar u\quad on\;{\Gamma _1} $$

((9.2))

$$ \alpha u + {n^T}A\left( { \cdot ,u} \right)grad\;u = g\quad on\;{\Gamma _2} $$

((9.3))

where Ω ∈ L, n = (n ₁, ..., n _d )^T is the outward unit normal to ∂Ω, d ∈ {1, 2, ...,}, Γ₁ and Γ₂ are relatively open sets in the boundary ∂Ω, ${\overline \Gamma _1} \cup {\overline \Gamma _2} = \partial \Omega ,\,{\Gamma _1} \cap {\Gamma _2} = \phi$, $A = \left( {{a_{ij}}} \right)_{i,j = 1}^d$ is a uniformly positive definite matrix, α ≥ 0. Let the functions A, α, f, ū and g be sufficiently smooth for the time being (precise assumptions on these functions will be given later). The boundary condition (9.3) is called the Newton boundary condition.