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A priori estimates for solutions of fully nonlinear special Lagrangian equations. (English. French summary) Zbl 0988.35058
Summary: We derive an a priori $C^{2,\alpha}$ estimate in dimension three for the equation $F(D^2u)= \arctan\lambda_1 +\arctan\lambda_2 + \arctan \lambda_3=c$, where $\lambda_1,\lambda_2, \lambda_3$ are the eigenvalues of the Hessian $D^2u$. For $-\pi/2 <c<\pi/2$, the $c$-level set of $F(D^2u)$ fails the convexity condition. Note that for any solution $u$ of the above equation, $(x,\nabla u(x))$ is a minimizing graph in $\bbfR^6$. For $c=0$, $\pm\pi$, the equation is equivalent to $\Delta u=\det D^2u$.

35J60Nonlinear elliptic equations
35B45A priori estimates for solutions of PDE
Full Text: DOI Numdam EuDML
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