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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.

##### MSC:
 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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