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$$L^{2,\mu} (\Omega)$$ estimate to the mixed boundary value problem for second order elliptic equations and its application in the thermistor problem. (English) Zbl 0815.35030
From the introduction: This paper deals with $$L^{2, \mu} (\Omega)$$ estimates for the solutions of mixed boundary value problems of second order elliptic differential equations in divergence form and the application of such estimates in studying the existence of $$C^ \alpha (\overline \Omega)$$ $$(0 \leq \alpha < 1)$$ solutions of the following nonlinear system which arises from the thermistor problem $\nabla \bigl( \sigma (u) \nabla \varphi \bigr) = 0, \quad \nabla \bigl( k(u) \nabla u \bigr) = - \sigma (u) | \nabla \varphi |^ 2, \quad \text{in } \Omega$ $\varphi = \varphi_ D \text{ on } \Gamma_ D, \quad {\partial \varphi \over \partial n} = 0, \text{ on } \Gamma_ N, \quad u = u_ D \text{ on } \Gamma_ D', \quad k(u) {\partial u \over \partial n} + \beta (u - \gamma) = 0 \text{ on } \Gamma_ N',$ where $$\partial \Omega = \Gamma_ D \cup \Gamma_ N = \Gamma_ D' \cup \Gamma_ N'$$, $$\varphi$$ and $$u$$ denote the electrical potential and the temperature distribution, respectively, $$\sigma$$ is the electrical conductivity and $$k$$ is the thermal conductivity. For many cases of practical interest, $$\sigma$$ and $$k$$ have the forms $$\sigma (u) = Au^ s \exp (-B/u)$$, $$A,B>0$$ and $$s \in [-1,1)$$, and $$k(u) = 1/(a + bu + cu^ 2)$$, $$a,b,c \geq 0$$ and $$a + b + c > 0$$.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35D10 Regularity of generalized solutions of PDE (MSC2000) 78A35 Motion of charged particles 35J55 Systems of elliptic equations, boundary value problems (MSC2000)
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##### References:
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