Lu Xu — Page 6 | Searcharxiv

arXiv2013

KAM for Hamiltonian partial differential equations with weaker Spectral Asymptotics

In this paper, we establish an abstract infinite dimensional KAM theorem dealing with normal frequencies in weaker spectral asymptotics Ω_{i}(ξ)=i^d+o(i^{d})+o(i^δ), where $d>0, δ<0$, which can be applied to a large class of Hamiltonian partial differential equations in high dimensions. As a consequence, it is proved that there exist many invariant tori and thus quasi-periodic solutions for Schrödinger equations, the Klein-Gordon equations with exponential nonlinearity and other equations of any spatial dimension.

math.DS↗

arXiv2010

Convexity estimates for level sets of quasiconcave solutions to fully nonlinear elliptic equations

We establish a geometric lower bound for the principal curvature of the level surfaces of solutions to $F(D^2u, Du, u, x)=0$ in convex ring domains, under a refined structural condition introduced by Bianchini-Longinetti-Salani in \cite{BLS}. We also prove a constant rank theorem for the second fundamental form of the convex level surfaces of these solutions.

math.AP↗