A boundary value problem for an elliptic equation in two variables with asymmetric tensor coefficients (Q1920767)

The author investigates the equations \[ {\partial J_x\over \partial x}+ {\partial J_y\over \partial y}= Q,\quad - {\partial E_x\over \partial y}+ {\partial E_y\over \partial x}= G,\tag{1} \] where \(\vec E= (E_x, E_y)\) is the electric field strength, \(\vec J\) is the current density (\(\vec J\) and \(\vec E\) are connected by Ohm's law) and the corresponding boundary condition is of the form \[ J_n- {\partial\over \partial\ell} (A(\ell) E_\ell)\Biggl|_{\Gamma}= q(\ell),\tag{2} \] where the normal and the tangent components of vectors relative to the boundary are marked with indices \(n\) and \(\ell\). Condition (2) is satisfied at all smooth points of the boundary. At the break points it is replaced by the condition of continuity: \(I(\ell)= A(\ell)\cdot E_\ell\). The author reformulates the above problem to a problem with a symmetric positive definite operator. A minimum principle for the quadratic energy functional is obtained. Existence and uniqueness of a generalized solution is proved.