arXiv subjects
Publications and source records attributed to Fengbo Hang.
We prove all the maximizers of the sharp Hardy-Littlewood-Sobolev inequality are smooth. More generally, we show all the nonnegative critical functions are smooth, radial with respect to some points and strictly decreasing in the radial direction. In particular, this resolves all the cases left open by previous works of Chen, Li and Ou on the corresponding integral systems.
We study the existence of a metric with zero scalar curvature maximizing the isoperimetric ratio among all zero scalar curvature metrics in a fixed conformal class of metrics on a compact manifold with boundary. The question may be reduced to an extremal problem for the harmonic extension of functions and the related nonlinear integral equation is discussed.
We prove some rigidity results for compact manifolds with boundary. In particular for a compact Riemannian manifold with nonnegative Ricci curvature and simply connected mean convex boundary, it is shown that if the sectional curvature vanishes on the boundary, then the metric must be flat.
We will show that in the conformal class of the standard metric $g_{S^n}$ on $S^n$, the scaling invariant functional $(μ_g(S^n))^{\frac{2m-n}{n}}\int_{S^n}Q_{2m,g}dμ_g$ maximizes at $g_{S^n}$ when $n$ is odd and $m=\frac{n+1}{2}$ or $\frac{n+3}{2}$. For $n$ odd and $m\geq\frac{n+5}{2}$, $g_{S^n}$ is not stable and the functional has no local maximizer. Here $Q_{2m,g}$ is the $2m$th order $Q $-curvature.
We identify all the weak sequential limits of smooth maps in $W^{1,2}(M,N)$. In particular, this implies a necessary and sufficient topological condition for smooth maps to be weakly sequentially dense in $W^{1,2}(M,N)$.
Motivared by Carleman's proof of the isoperimetric inequality in the plane, we study some sharp integral inequalities for harmonic functions on the upper halfspace. We also derive the regularity for nonnegative solutions of the associated integral system and some Liouville type theorems.
In 1992, P. Poláčik\cite{P2} showed that one could linearly imbed any vector fields into a scalar semi-linear parabolic equation on $Ω$ with Neumann boundary condition provided that there exists a smooth vector field $Φ=(ϕ_{1},...,ϕ_{n}) $ on $\barΩ$ such that \[ \left\{\begin{array} [c]{l}% \operatorname*{rank}(Φ(x) ,\partial_{1}Φ(x) ,...,\partial_{n}Φ(x)) =n\text{for all}x\in\barΩ, \frac{\partialΦ}{\partialν}=0\text{on}\partialΩ\text{.}% \end{array} \right. \] In this short note, we give a classification of all the domains on which one may find such type of vector fields.