Fully nonlinear operators with Hamiltonian: Hölder regularity of the gradient
This paper deals with the regularity up to the boundary of bounded viscosity solutions \(u\) to the nonhomogeneous Dirichlet problem associated to the nonlinear elliptic equation \(|\nabla u|^\alpha F(D^2u) +|\nabla u|^\beta b=f\) in a bounded \(C^2\) domain \(\Omega\) of \(\mathbb{R}^N\). The authors establish that any bounded viscosity solution \(u\) belongs to \(C^{1,\gamma}(\overline{\Omega})\) for \(b,f\in C(\overline{\Omega})\), the function \(F\) being uniformly elliptic, the boundary datum belonging to \(C^{1,\gamma_0}(\partial\Omega)\) and \(0<\beta<1+\alpha\) and \(\alpha\geq 0\). The proof relies on the localization to \(B(x,r)\cap \{y_N\geq a(y')\}\) for any \(x\in B \cap \{y_N> a(y')\}\) with \(B(x,1)\subset B\) and any \(r<1\), by making recourse to the contradiction and recursive arguments. Also the Hölder regularity of the gradient is discussed for bounded viscosity solutions \(u\) to the Dirichlet problem associated to the generalized nonlinear elliptic equation \(|\nabla u|^\alpha F(D^2u) +\sum_{i=1}^{k} |\nabla u|^{\beta_i} b_i=f \) in \(B(0,1)\). The proof appeals to the contradiction argument once more.
scientific article; zbMATH DE number 6673667
scientific article