Symmetry and monotonicity of solutions to elliptic and parabolic fractional p-equations

In this article we first establish the maximum principle of the antisymmetric functions for parabolic fractional p-equations. Then we use it and the parabolic inequalities to provide a different proof of symmetry and monotonicity for solutions to elliptic fractional p-equations with gradient terms. Finally, base on suitable initial value, by the maximum principle of the antisymmetric functions for parabolic fractional p-equations, we attain symmetry and monotonicity of positive solutions in each finite time to nonlinear parabolic fractional p-equations on the whole space and bounded domains. We believe that the maximum principle and parabolic inequalities obtained here can be utilized to many elliptic and parabolic problems involving nonlinear nonlocal operators.