On the integrability of ``finite energy'' solutions for \(p\)-harmonic equations (Q1884687)

The author considers equations of the type: \[ \text{div }H(x,Du) =0\text{ in }\Omega,\tag{1} \] where \(u:\Omega\to\mathbb R^m\), \(\Omega \subset\mathbb{R}^n\) is a domain, \(H:\Omega\times\mathbb R^{m\times n}\to \mathbb R^{m\times n}\). Under the suitable assumptions on the data of (1), the author proves a higher integrability result for the solutions of (1). The main tool in the proof is an approximation argument and a priori estimates for solutions of associated to (1) regular equations. In the case of \(m=1\), the author obtains that every finite energy solution to (1) is a weakly monotone.