Application of conservation laws to Dirichlet problem for elliptic quasilinear systems

•The possibility of using of conservation laws to solve the Dirichlet problem for elliptic quasilinear systems is shown.•The integral representation of solution is obtained.•The problem with unknown boundary for elastic–plastic torsion of prismatic rods is solved.

In the present work we show the possibility of using of conservation laws to solve the Dirichlet problem for elliptic quasilinear systems. As a result the integral representation of solution is obtained. For the system of filtration of aerated fluid in porous medium and for system of elastic–plastic torsion of prismatic rods corresponding conservation laws are calculated in explicit form and the Dirichlet problems are solved.