\(C^1\)-regularity of the Aronsson equation in \(\mathbb{R}^2\)
This article deals with the Aronsson equation \[ \sum_{i,j}\frac{\partial H}{\partial p_i}(\nabla u)\;\frac{\partial H}{\partial p_j}(\nabla u)\;u_{ij}=0 \] on \(\mathbb{R}^2,\) where the Hamiltonian \(H:\mathbb{R}^2\rightarrow\mathbb{R}\) is assumed to be a nonnegative and uniformly convex function of class \(C^2\) such that \(H(0)=0.\) The following regularity property is proved: if \(u\in C^0(\Omega),\) \(\Omega\subset\mathbb{R}^2,\) is a viscosity solution of the Aronsson equation then \(u\) belongs to \(C^1(\Omega).\) This result generalizes the regularity result of Savin on viscosity solutions of the \(\infty\)-Laplace equation, where \(H(p)=|p|^2\) [\textit{O. Savin}, Arch. Ration. Mech. Anal. 176, No. 3, 351--361 (2005; Zbl 1112.35070)]. A bound on the oscillation of a viscosity solution of the Aronsson equation is then obtained, which leads the authors to the following Liouville property: if \(u\in C^0(\mathbb{R}^2)\) is a viscosity solution on \(\mathbb{R}^2\) such that \[ |u(x)|\leq C(1+|x|), \] then \(u\) is linear. The hypothesis of uniform convexity of \(H\) is finally slightly relaxed.
scientific article; zbMATH DE number 5306974
scientific article