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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 $\Delta u=\det D^2u$.

##### MSC:
 35J60 Nonlinear elliptic equations 35B45 A priori estimates for solutions of PDE
##### Keywords:
eigenvalues of the Hessian; minimizing graph
Full Text:
##### References:
 [1] Caffarelli, L. A.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. math. 130, 189-213 (1989) · Zbl 0692.35017 [2] Caffarelli, L. A.; Cabré, X.: Fully nonlinear elliptic equations. American mathematical society colloquium publications 43 (1995) [3] Caffarelli, L. A.; Crandall, M. G.; Kocan, M.; Świech, A.: On viscosity solutions of fully nonlinear equations with measurable ingredients. Comm. pure appl. Math. 49, 365-397 (1996) · Zbl 0854.35032 [4] Caffarelli, L. A.; Nirenberg, L.; Spruck, J.: The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian. Acta math. 155, 261-301 (1985) · Zbl 0654.35031 [5] Caffarelli L.A., Yuan Y., A Priori estimates for solutions of fully nonlinear equations with convex level set, Indiana Univ. Math. J., to appear · Zbl 0965.35045 [6] Calabi, E.: Minimal immersions of surfaces in Euclidean spheres. J. differential geom. 1, 111-125 (1967) · Zbl 0171.20504 [7] Chiarenza, F.; Frasca, M.; Longo, P.: W2,p solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans. amer. Math. soc. 336, 841-853 (1993) · Zbl 0818.35023 [8] Evans, L. C.: Classical solutions of fully nonlinear, convex, second-order elliptic equations. Comm. pure appl. Math. 35, No. 3, 333-363 (1982) · Zbl 0469.35022 [9] Gilbarg, D.; Trudinger, N. S.: Elliptic partial differential equations of second order. (1983) · Zbl 0562.35001 [10] Harvey, R.; Jr., H. B. Lawson: Calibrated geometry. Acta math. 148, 47-157 (1982) [11] Huang Q.-B., On the regularity of solutions to fully nonlinear elliptic equations via Liouville property, Proc. Amer. Math. Soc., to appear [12] Krylov, N. V.: Boundedly nonhomogeneous elliptic and parabolic equations. Izv. akad. Nauk SSSR ser. Mat. 46, No. 3, 487-523 (1982) · Zbl 0511.35002 [13] Simon, L.: Lectures on geometric measure theory. Proc. C. M. A., austr. Nat. univ. 3 (1983) · Zbl 0546.49019