zbMATH — the first resource for mathematics

Existence and regularity of solutions of a nonlinear nonuniformly elliptic system arising from a thermistor problem. (English) Zbl 0830.35045
The author studies the steady state thermistor problem, which consists of an elliptic system \[ \nabla (\sigma (u) \nabla \varphi) = 0,\quad (2) - \nabla (k(u) \nabla u) = \sigma (u) |\nabla \varphi |^2,\quad \text{in } \Omega,\tag{1} \] and boundary conditions (3) \(u = \varphi_0\) on \(\Gamma^\varphi_D\), \(\partial_n \varphi = 0\) on \(\Gamma^\varphi_N \equiv \partial \Omega \backslash \overline {\Gamma^\varphi_D}\), (4) \(u = u_0\) on \(\Gamma^u_D\), \(\partial_nu + h(x,u) = 0\) on \(\Gamma^u_N \equiv \partial \Omega \backslash \overline {\Gamma^u_D}\), where \(\Omega\) is a domain in \(\mathbb{R}^N\), \(\partial_n\) is the outward normal derivative and \(\Gamma^\varphi_D\), \(\Gamma^u_D\) are smooth surfaces. This problem has many applications such as current regulation, switching, thermal conductivity analysis, etc. In the paper, \(\sigma(u)\) is not necessarily uniformly bounded from above and away from zero as \(u \to \infty\), (this causes the main difficulty to study (1)–(4), together with the quadratic growth in (2)). Under certain additional restrictions on \(\sigma (u)\) and \(h(x,u)\), it is shown the existence and Hölder continuity of the solution. The proofs use the classical Nash-Moser iteration (to establish \(L^\infty\) bounds) and some results from a method exposed by M. Giaquinta [Multiple integrals in the calculus of variations and nonlinear elliptic systems (Princeton, New Jersey) (1983; Zbl 0516.49003)]. The paper finishes with a non-existence and nonuniqueness example.
Reviewer: A.Canada (Granada)

35J70 Degenerate elliptic equations
35J55 Systems of elliptic equations, boundary value problems (MSC2000)