The Dirichlet problem for a class of Hessian type equations (Q2396882)

This paper deals with the regularity and existence for solutions of the following equation \[ \begin{cases} f(\lambda[D^2u+\gamma\Delta u I])=\phi, \quad\text{in}\;\Omega,\\ u=\varphi\quad\text{on}\;\partial\Omega. \end{cases}\tag{1} \] Here \(\Omega \) is a smooth bounded domain in \(\mathbb{R}^{n}\), \(\gamma> 0\) is a constant, \(I\) is the unit matrix and \(\lambda[D^2u+\gamma\Delta u I]\) denotes the eigenvalues of the matrix \(D^2u+\gamma\Delta u I\). Under assumptions on \(f\), \(\phi\) and \(\varphi\), the authors proved the existence of a smooth solution \(u\in C^{\infty}(\bar{\Omega})\) for (1).