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On the existence of bounded temperature in the thermistor problem with degeneracy. (English) Zbl 0964.35005
One of the two results is concerning sufficient conditions for the existence of a capacity solution \((v,\phi)\) for the problem \[ -\text{div}(a(v)\nabla v)= a(v)|{\nabla \varphi}|^2, \quad \text{div}(a(v)\nabla \varphi)=0\quad \text{in }\Omega, \] \[ v= v_0\quad \text{on }\Gamma_D^u, \qquad -a(v){{\partial v} \over {\partial \nu }}=H(x,v)\quad \text{on }\Gamma _N^u, \] \[ \varphi =\varphi_0\quad \text{on }\Gamma_D^\varphi, \qquad {{\partial \varphi } \over {\partial \nu }}=0\quad \text{on }\Gamma _N^\varphi, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\); \(\Gamma_D^u\), \(\Gamma_D^\varphi\) are two nonempty open subsets of \(\partial \Omega\) with smooth bounderies, \(\Gamma_N^u=\partial \Omega\setminus \overline{\Gamma_D^u},\) \(\Gamma_N^\varphi=\partial \Omega \setminus\overline{\Gamma_D^\varphi}.\) The following theorem is proved:
Assume \(a\in C(\mathbb{R})\) is such that \(0<a(s)\leq M\) for some \(M>0\), and there exist \(0<m_2\leq M_2,\) \(L_1>0\) with \(m_2\leq a(s+\tau)\) and \(a(s)\leq M_2\) for all \(s\in(-\infty,\infty)\), \(\tau\in[-L_1,L_1].\) Suppose \(H(x,v)\) is measurable in \(x\) for all \(v\in \mathbb{R}\) and continuous in \(v\) for a.e. \(x\in\Gamma_N^u.\) Moreover, assume that there exist \(m_3<M_3\) such that \(H(x,v)\geq 0\) for \(v\geq M_3\), \(H(x,v)\leq 0\) for \(v\leq m_3\) and \(H(x,v)\) is bounded on bounded sets, \(v_0\in W^{1,2}(\Omega)\cup L^{\infty}(\Omega),\) \(\varphi_0 \in W^{1,4/(N+2)}(\Omega)\cup L^{\infty}(\Omega)\) if \(N\geq 3\) and \(\varphi_0 \in W^{1,p}(\Omega)\cup L^{\infty}(\Omega)\) for some \(p>2\) if \(N=2.\) Then there exists a capacity solution \((v,\varphi)\) with \(v, \varphi\) being locally Hölder continuous in \(\Omega .\) The notion of a capacity solution was first introduced by the author in 1993.

35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35J60 Nonlinear elliptic equations
78A35 Motion of charged particles
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