Vasily V. Zhikov: On passage to the limit in nonlinear elliptic and parabolic problems

The talk will focus on existence results for nonlinear equations. For a given nonlinear equation we introduce an approximate (or regularized) equations and try to pass to the limit in approximate solutions. Here we face a fundamental problem of proving that the approximate fluxes converge to the flux of the original equation. We define flux as the vector under the divergence sign. Then our goal is to characterize the limit of approximate fluxes and to show that it coincides with the flux of the original equation. A number of applications will be considered. Namely,

• elliptic equations with non-standard growth conditions,
• equations with p(x)-Laplacian;
• termistor problem;
• Navier-Stokes equation for electrorheological fluids.