Regularity of the solution and well-posedness of a mixed problem for an elliptic system with quadratic nonlinearity in gradients

A boundary value problem for an elliptic system of equations is studied that arises in the analysis of a new hydrodynamic model describing charge transport in a planar semiconductor MESFET (metal semiconductor field effect transistor). The problem has a number of features, specifically, the equations of the system involve squared components of the gradients of the unknown functions; the boundary conditions are of a mixed character, i.e., Dirichlet and Neumann conditions are set on different portions of the boundary; and the boundary of the domain is a nonsmooth curve, namely, a rectangle. Under a certain optimal condition, the $C^{1,\alpha}$-regularity of a weakened solution of the problem is justified and its existence is proved, while its uniqueness is shown under additional constraints. The results are used to justify the stabilization method as applied to finding approximate stationary solutions of the hydrodynamic model.