Linearity of homogeneous order‐one solutions to elliptic equations in dimension three

A Bernstein problem for special Lagrangian equations

“…The only difference is that, instead of relying on Proposition 2.1, we use the fact that any non-parametric minimal cone of dimension three must be flat, see Theorem 2.3], [B, Theorem]. For a quick PDE proof of this fact, see [HNY,p.2].…”

The Weyl problem in warped product spaces

“…In the Euclidean space, we give two proofs only using maximum principle. The idea comes from [15] and [18]. For hyperbolic and spherical space cases, we use Beltrami map to extend the infinitesimal rigidity in Euclidean space to space forms.…”

A proof of Alexandrov's uniqueness theorem for convex surfaces in \( R^{3} \)

“…The first inequality in (10) follows (8) and (12). The proof for φ E follows the same argument and the following facts: Proof.…”

“…Acknowledgement: The first author would like to thank Professor Louis Nirenberg for stimulation conversations. Our initial proof was the global maximum principle for C 3 surfaces Lemma 5 and Corollary 6 (we only realized the connection of the result of [12] to Theorem 1 afterward). It was Professor Louis Nirenberg who brought our attention to the paper of [14] and suggested using the unique continuation theorem of [8].…”