Isotropic \(p\)-harmonic systems in 2D Jacobian estimates and univalent solutions (Q268238)

The central result of the interesting paper under review is an inequality for the Jacobian determinant of solutions to nonlinear elliptic systems in the plane. The model case is the isotropic (rotationally invariant) \(p\)-harmonic system NEWLINENEWLINE\[NEWLINE \operatorname{div}(| Dh| ^{p-2}Dh)=0,\quad h=(u,v)\in W^{1,p}(\Omega,\mathbb R^2),\quad p\in(1,\infty)NEWLINE\]NEWLINE as opposed to a pair of scalar \(p\)-harmonic equations NEWLINE\[NEWLINE\operatorname{div}(| Du| ^{p-2}Du)=0\quad \text{and}\quad \operatorname{div} (| Dv| ^{p-2}Dv)=0.NEWLINE\]NEWLINE Rotational invariance of the systems considered makes them meaningful, both physically and geometrically. An issue is to overcome the nonlinear coupling between \(\nabla u\) and \(\nabla v\). In the extensive literature dealing with coupled systems, various differential expressions of the form \(\Phi(\nabla u,\nabla v)\) were subjected to thorough analysis, but the Jacobian determinant \(\det Du= u_xv_y-u_yv_x\) was never successfully incorporated into such analysis. The authors present here new nonlinear differential expressions of the form \(\Phi(| Dh|,\det Dh)\) and show they are superharmonic, which yields much needed lower bounds for \(\det Dh\). To illustrate the utility of such bounds, the celebrated univalence theorem of Radó-Kneser-Choquet is extended on harmonic mappings \((p=2)\) to the solutions of the coupled \(p\)-harmonic system.