A regularity result for a class of non-uniformly elliptic operators (Q2136117)

The authors investigate the regularity of viscosity solutions of the equation \[ \mathcal{M}_{\mathbf{a}}(D^2u) + H(\nabla u) = f(x) ~\text{in}~ \Omega, \] where \(\Omega\) is a domain in \(\mathbb{R}^N\), \(f\) is continuous, the operator \(\mathcal{M}_{\mathbf{a}}\) has the form \[ \mathcal{M}_{\mathbf{a}}(D^2u) = \sum_{i=1}^N a_i \lambda_i(D^2u) \] with \(a_1,a_N>0\) and all other \(a_i\) are nonnegative, and the function \(H:\mathbb{R}^N \to \mathbb{R}\) is continuous and satisfies \[ |H(p+q) - H(p)| \leq C (1+|p|+|q|)|q| ~\text{for all}~ p,q \in \mathbb{R}^N. \] Clearly, if some \(a_i=0\), then \(\mathcal{M}_{\mathbf{a}}\) is not uniformly elliptic. The main result asserts that any continuous viscosity solution of this equation belogs to the space \(C_{\text{loc}}^{0,\beta}(\Omega)\), where \(\beta \in (0,1)\) is explicitly quantified as \[ \beta = 1 - \frac{a_1+a_N}{(\sqrt{a_1}+\sqrt{a_N})^2}, \] This is a generalization and a refinement of a local Hölder regularity result from [the first author and \textit{A. Vitolo}, Adv. Nonlinear Stud. 20, No. 1, 213--241 (2020; Zbl 1437.35294)]. Finally, the authors show that without the assumption \(a_1,a_N>0\) the Hölder regularity cannot be guaranteed, by constructing a corresponding counterexample.