6533b839fe1ef96bd12a62d5

RESEARCH PRODUCT

Nonlinear anisotropic heat conduction in a transformer magnetic core

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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.