A microscopic convexity principle for nonlinear partial differential equations

By Baojun Bian, Pengfei Guan. 2008

117 citations recorded by the source corpus.

We study microscopic convexity property of fully nonlinear elliptic and parabolic partial differential equations. Under certain general structure condition, we establish that the rank of Hessian ∇2u is of constant rank for any convex solution u of equation F(∇2u,∇u,u,x)=0. The similar result is also proved for parabolic equations. Some of geometric applications are also discussed.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/6aa00b73be98f469ca882d829ca5e389b8a7b557

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

The J-flow and stability (2013, methodology): “By using the Legendre transform and the constant rank theorems of Korevaar-Lewis [11] and Bian-Guan [1] we show that this equation always has a solution.”
A Mean Curvature Type Flow in Space Forms (2013, methodology): “As in [3], we may assume at each point of computation, (Wij is diagonal.”
Convergence of the $J$-flow on toric manifolds (2014, methodology): “Using the Legendre transform and the constant rank theorem of Bian-Guan [1], this can be reduced to obtaining a priori C2,α estimates for convex solutions of the equation (6) ∆h+ d det(D2h) = 1.”
Isoperimetric type problems and Alexandrov-Fenchel type inequalities in the hyperbolic space (2013, methodology): “As in [3], in O × (t0 − δ, t0], the index set {1, · · · , n − 1} can be divided into two subsets B and G, where for i ∈ B, the eigenvalues λ̃i of W is small and for j ∈ G, λ̃j is uniformly positive away from 0.”
A generalization of Caffarelli’s contraction theorem via (reverse) heat flow (2010, unknown): “in [1,6,7,9,16,23] and the references therein, or in the classical book by Kawohl [31].”
Complete translating solitons to the mean curvature flow in ℝ3 with nonnegative mean curvature (2020, background): “Similarly if xm+1 = v(x1, . . . ,xm) is a graphical translating soliton in Rm+1, then for x= (x1, . . . ,xn) = (x′,xm+1, . . . ,xn), vλ(x) := λ2v ( x′ λ ) + n∑ k=m+1 αkxk, n∑ k=m+1 α2k = λ 2−1, λ≥ 1,(1.3) is a graphical translating soliton in Rn+1 =Rm×Rn−m×R. Conversely if xn+1 = u(x1, . . . ,xn) is”