On the second boundary value problem for special Lagrangian curvature potential equation (Q2163309)

The aim of this paper is to study the special Lagrange curvature potential equation \(F_{\tau}(\lambda(D^2u))=c\) with \(F_{\tau}(\lambda) =\sum_{i=1}^n \arctan k_i =c\) in \(\Omega\) with \(\tau=\pi/2\) associated with the second boundary value problem \(\mathcal{D}_u(\Omega )= \widetilde{\Omega}\) where \(k_1,\dots,k_n\) are the principal curvatures of the graph \(\Gamma= \{(x, u(x)), x\in \Omega\}\), \(c\) is a constant to be determined and \(\Omega \) and \(\widetilde{\Omega}\) are uniformly convex bounded domains with smooth boundary in \(\mathbb{R}^n\). The main theorem states that if \(\Omega \) and \(\widetilde{\Omega}\) are uniformly convex bounded domains with smooth boundary in \(\mathbb{R}^n\), then there exists a uniformly convex solution \(u\in \mathcal{C}^\infty (\widetilde \Omega)\) and a unique constant \(c\) solving the equation and \(u\) is unique up to a constant. In order to obtain existence result the authors use the method of continuity by carrying out a-priori estimates on the solutions of the equation mentioned above. The work is presented as follows. In the first section the authors present the structural conditions for the operator \(F\) with the related operator \(G\) and the fundamental formulas for the principal curvatures of the graph \(\Gamma\). In the third section, the strict estimate is carried out, by restating the boundary condition through a boundary defining function, whose construction can be seen in [\textit{J. Urbas}, J. Reine Angew. Math. 487, 115--124 (1997; Zbl 0880.35031)]. In the fourth section the authors obtain the second derivative estimate by considering the operator on the manifold as in [\textit{J. Urbas}, Math. Z. 240, No. 1, 53--82 (2002; Zbl 1130.35341)]. In the fifth section the proof of the theorem is completed.