Quasilinear elliptic systems with nonlinear physical data

Abstract

Using the theory of Young measures, we prove the existence of weak solutions to the following quasilinear elliptic system:

$$A\left(u\right)=f\left(x\right)+\text{div }{\sigma }_0\left(x,u\right),$$

where $A\left(u\right)=-\text{div }\sigma \left(x,u,Du\right)$ and $f\in W^{-1}{L}_{\overline{M}}\left(\mathrm{\Omega };{ℝ}^m\right)$. This problem corresponds to a diffusion phenomenon with a source $f$ in a moving and dissolving substance, where the motion is described by ${\sigma }_0$.