Isotropic p -harmonic systems in 2D Jacobian estimates and univalent solutions

The core result of this paper is an inequality (rather tricky) for the Jacobian determinant of solutions of nonlinear elliptic systems in the plane. The model case is the isotropic (rotationally invariant) p -harmonic system div |Dh|^p-2Dh = 0, h= (u, v) ε W^1,p(ω,ℝ²) , 1 < p < ∞, as opposed to a pair of scalar p -harmonic equations: (equations presented) Rotational invariance of the systems in question makes them meaningful, both physically and geometrically. An issue is to overcome the nonlinear coupling between δ_u and δ_v . In the extensive literature dealing with coupled systems various differential expressions of the form φ(δ_u ,δ_v) were subjected to thorough analysis. But the Jacobian determinant detDh = u_xv_y .u_yv_x was never successfully incorporated into such analysis. We present here new nonlinear differential expressions of the form φ(|Dh|, detDh) and show they are superharmonic, which yields much needed lower bounds for detDh. To illustrate the utility of such bounds we extend the celebrated univalence theorem of Radó-Kneser-Choquet on harmonic mappings ( p = 2 ) to the solutions of the coupled p -harmonic system.