Elenbaas Problem of Electric Arc Discharge

The Elenbaas problem of electric discharge origination is considered. The mathematical model is an elliptic boundary-value problem with a parameter and discontinuous nonlinearity. The nontrivial solutions of the problem determine the free boundaries separating different phase states. A survey of results obtained for this problem is given. The greatest lower bound $\lambda_{\min}$ of the values of the parameter $\lambda$ for which the electric discharge is possible is obtained. The fact that the discharge domain appears for any $\lambda \ge \lambda_{\min}$ is proved. The range of the parameter values for which the boundary of the discharge domain is of two-dimensional Lebesgue measure zero is determined. An unsolved problem is formulated.