The existence of multiple topologically distinct solutions to σ_2,p-energy - Taghavi, Mojgan; Shahrokhi-Dehkordi, Mohammad S. (Topological Methods in Nonlinear Analysis, 2023)
Let ^n be a bounded Lipschitz domain and consider the -energy functional equation* F_[u; ] := _ |^2 abla u|^p dx, equation* with p]1, ∞] over the space of measure preserving maps equation* A_p() ={u W^1,2p(, ^n) : u|_ = x, abla u =1 for L^n-a.e. in }. equation* In this article we address the question of multiplicity versus uniqueness for extremals and strong local minimizers of the -energy funcional F_[·; ] in A_p(). We use a topological class of maps referred to as generalised twists and examine them in connection with the Euler-Lagrange equations associated with -energy functional over A_p(). Most notably, we prove the existence of a countably infinite of topologically distinct twisting solutions to the later system in all even dimensions by linking the system to a set of nonlinear isotropic ODEs on the Lie group SO(n). In sharp contrast in odd dimensions the only solution is the map ux. The result relies on a careful analysis of the full versus the restricted Euler-Lagrange equations. Indeed, an analysis of curl-free vector fields generated by symmetric matrix fields plays a pivotal role.