On the existence of bounded temperature in the thermistor problem with degeneracy
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.
scientific article; zbMATH DE number 1491646