Examples of weak minimizers with continuous singularities
The author considers a functional (1) \(\int F(\nabla u, x) dx\), \(F(\xi, x)= |M(x) \xi|^p\), \(x\in \mathbb{R}^n\), \(n\geq 2\), \(1< p\leq n\), on an open set \(\Omega\subset \mathbb{R}^n\) and the corresponding Euler equation \[ -\text{div } A_M(\nabla u, x)= 0,\quad A_M(\xi, x)= |M\xi|^{p- 2} M^* M\xi \] under some natural restriction on the function \(M\). The class of minimizers of (1) coincides with the class of the \(A_M\)-harmonic functions, i.e., functions \(u\in W^{1, p}_{\text{loc}}(\Omega)\) satisfying \(\int_{\Omega'} A_M(\nabla u, x)\cdot \nabla v dx= 0\), \(\forall \Omega'\subset\subset \Omega\), and \(v\in W^{1, p}_0(\Omega')\). Weakening some of these assumptions, in particular, assuming only \(u\in W^{1, 1}_{\text{loc}}(\Omega)\), one arrives at the definition of weak minimizers, weakly \(A\)-harmonic functions, etc.. Examples are known of weakly \(A\)-harmonic functions that are not \(A\)-harmonic. The author gives, for general \(p\)'s, such examples which are also Hölder continuous.
scientific article; zbMATH DE number 849123