Liouville type theorems and Hessian estimates for special Lagrangian equations

Abstract

In this paper, we get a Liouville type theorem for the special Lagrangian equation with a certain 'convexity' condition, where Warren-Yuan first studied the condition in [29]. Based on Warren-Yuan's work, our strategy is to show a global Hessian estimate of solutions via the Neumann-Poincaré inequality on special Lagrangian graphs, and mean value inequality for superharmonic functions on these graphs. Moreover, we derive interior Hessian estimates in terms of the linear exponential dependence on the gradient of the solutions to the equation with this 'convexity' condition or with supercritical phase. Here, the linear exponential dependence is optimal.

Minimal graphs of arbitrary codimension in Euclidean space with bounded 2-dilation

“…which means that f is superharmonic on M in the distribution sense. Now we can follow the argument of De Giorgi-Nash-Moser iteration (see Theorem 3.2 in [21] for instance) and finish the proof. Note that the constant δ * in (8.4) is obtained from the dimension n and the exponent of the Sobolev inequality, which implies that δ * only depends on n.…”

“…From the Sobolev inequality [42], nonnegative subharmonic functions on stationary varifolds admit the mean value inequality on sptT (see [32] for instance). Since Neumann-Poincaré inequality (5.23) holds on a stationary indecomposable component T of T, by De Giorgi-Nash-Moser iteration (see [43][44], or [39], or Theorem 3.2 in [21] for instance) there holds the mean value inequality for superharmonic functions on sptT . Hence, we get Harnack's inequality for weakly harmonic functions on sptT as follows.…”

“…Appendix IIAnalog to Lemma 4.3 in[21], we have the following multiplicity one convergence for Lipschitz minimal graphs. Let Ω be a domain in R n with countably (n − 1)rectifiable boundary ∂Ω.Lemma 10.1.…”

Minimal hypersurfaces in manifolds of Ricci curvature bounded below

“…where c ′ k is a constant depending only on k and H k (X). With the famous De Giorgi-Nash-Moser iteration, we get the following mean value inequality on metric balls in X(refer to Theorem 3.2 in [15]). Lemma 6.2.…”