Title: Variational methods for steady-state Darcy/Fick flow in swollen and poroelastic solids,
Abstract: Existence of steady states in elastic media at small strains with diffusion
of a solvent or fluid due to Fick's or Darcy's laws is proved by combining
usage of variational methods inspired from static situations with Schauder's
fixed-point arguments. In the plain variant, the problem consists in the force
equilibrium coupled with the continuity equation, and the underlying operator
is non-potential and non-pseudomonotone so that conventional methods are not
applicable. In advanced variants, electrically-charged multi-component flows
through an electrically charged elastic solid are treated, employing critical
points of the saddle-point type. Eventually, anisothermal variants involving
heat-transfer equation are treated, too.