A Bernstein problem for special Lagrangian equations

By Yu Yuan. 2002

142 citations recorded by the source corpus.

We derive a Bernstein type result for the special Lagrangian equation, namely, any global convex solution must be quadratic. In terms of minimal surfaces, the result says that any global minimal Lagrangian graph with convex potential must be a hyper-plane.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/faf960f833e66f25ddd226e1c6e7dd7d13ec34b1

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In Applied Mathematics

1. Dake Wang
2. Yu Yuan
3. Jingyi Chen
4. Weiyong He
5. Micah W. Warren
6. Albert Chau
7. Zhisu Li
8. Dongsheng Li

In Computational Theory and Mathematics

1. Dake Wang
2. Yu Yuan
3. Micah W. Warren
4. Jingyi Chen
5. Zhisu Li
6. Dongsheng Li
7. Albert Chau
8. Weiyong He

Citations in the corpus

• Lagrangian mean curvature flow for entire Lipschitz graphs (2009, methodology): “But the Bernstein theorem in [15] contradicts |D(3)u∞(0)| = 1.”
• Hessian estimates for special Lagrangian equations with critical and supercritical phases in general dimensions (2011, background): “on on the boundary data. Another quick consequence is a Liouville type result for global solutions with quadratic growth to (1.1) with |Θ| = (n−2)π/2,namely any such a solution must be quadratic (cf. [Y1], [Y2] where other Liouville type results for convex solutions to (1.1) and Bernstein type resul”
• On graphic Bernstein type results in higher codimension (2002, background): “The special Lagrangian case is studied in [4], [10], [8], [14], and [11].”
• A Bernstein problem for special Lagrangian equations in exterior domains (2017, methodology): “…(cid:7) -regularity in the geometric measure theory, the minimal cone ( (cid:19) x, Dv ( (cid:19) x )) is in fact smooth away from its vertex (cf [35]), then the solution v to the analytic special Lagrangian equation is analytic away from the origin It follows from Nadirashvili-Vladut [21] v is…”
• Hessian and gradient estimates for three dimensional special Lagrangian equations with large phase (2008, methodology): “Urbas [U2][U3], also Bao and Chen [ BC ] obtained (pointwise) Hessian estimates in term of certain integrals of the Hessian, for �k equations and special Lagrangian equation (1.1) with n = 3, � = � respectively.”
• Hessian estimates for the sigma‐2 equationin dimension 3 (2009, background): “[ Y1 ], [Y2]). Another consequence is the regularity (analyticity) of the C0 viscosity solutions to (1.1) or (1.2) with”
• Singular solutions to special Lagrangian equations with subcritical phases and minimal surface systems (2011, methodology): “manifold is still a graph in the rotated new coordinate system, thus a valid equation (1.1) to work on. 4 DAKE WANG AND YU YUAN (Earlier development of those U (n) rotations for (1.1) can be found in [Y2] [Y3].) Here our elementary analytic justification for the “inversion” π 2 rotation (Proposition ”