A Fourier-Legendre spectral method for approximating the minimizers of $\sigma _{2,p}$-energy

This paper proposes a Fourier-Legendre spectral method to find the minimizers of a variational problem, called $\sigma _{2,p}$-energy, in polar coordinates. Let ${\mathbb {X}}\subset \mathbb {R}^n$ be a bounded Lipschitz domain and consider the energy functional $(1.1)$ whose integrand is defined by ${\mathbf {W}}(\nabla u(x))≔(\sigma _2(u))^{\frac {p}{2}}+\Phi (\det \nabla u)$ over an appropriate space of admissible maps, $\mathcal {A}_p({\mathbb {X}})$. Using Fourier and Legendre interpolation errors, we obtain an error estimate for the energy functional and prove a convergence theorem for the proposed method. Furthermore, we apply the gradient descent method to solve a nonlinear algebraic system which is obtained by discretizing the Euler-Lagrange equations. The numerical experiments are performed to demonstrate the accuracy and effectiveness of our method.