Liouville theorems for entire local minimizers of energies defined on the class L log L and for entire solutions of the stationary Prandtl-Eyring fluid model

摘要

If u: ? ~n → ? ~M locally minimizes the energy with density, then we show that the boundedness of the function u already implies its constancy. The same is true in case n = M = 2 for entire solutions of the equations modelling the stationary flow of a so-called Prandtl-Eyring fluid. Moreover, in the variational setting we will present various extensions of the above mentioned Liouville theorem for entire local minimizers valid in any dimensions n and M.