We construct a radially smooth positive ancient solution for energy critical semi-linear heat equation in $mathbb{R}^n$, $ngeq 7$. It blows up at the origin with the profile of multiple Talenti bubbles in the backward time infinity.
We consider half-harmonic maps from $mathbb{R}$ (or $mathbb{S}$) to $mathbb{S}$. We prove that all (finite energy) half-harmonic maps are non-degenerate. In other words, they are integrable critical points of the energy functional. A full description
of the kernel of the linearized operator around each half-harmonic map is given. The second part of this paper devotes to studying the quantitative stability of half-harmonic maps. When its degree is $pm 1$, we prove that the deviation of any map $boldsymbol{u}:mathbb{R}to mathbb{S}$ from Mobius transformations can be controlled uniformly by $|boldsymbol{u}|_{dot H^{1/2}(mathbb{R})}^2-deg boldsymbol{u}$. This result resembles the quantitative rigidity estimate of degree $pm 1$ harmonic maps $mathbb{R}^2to mathbb{S}^2$ which is proved recently. Furthermore, we address the quantitative stability for half-harmonic maps of higher degree. We prove that if $boldsymbol{u}$ is already near to a Blaschke product, then the deviation of $boldsymbol{u}$ to Blaschke products can be controlled by $|boldsymbol{u}|_{dot H^{1/2}(mathbb{R})}^2-deg boldsymbol{u}$. Additionally, a striking example is given to show that such quantitative estimate can not be true uniformly for all $boldsymbol{u}$ of degree 2. We conjecture similar things happen for harmonic maps ${mathbb R}^2to {mathbb S}^2$.
Suppose $uin dot{H}^1(mathbb{R}^n)$. In a seminal work, Struwe proved that if $ugeq 0$ and $|Delta u+u^{frac{n+2}{n-2}}|_{H^{-1}}:=Gamma(u)to 0$ then $dist(u,mathcal{T})to 0$, where $dist(u,mathcal{T})$ denotes the $dot{H}^1(mathbb{R}^n)$-distance of
$u$ from the manifold of sums of Talenti bubbles. Ciraolo, Figalli and Maggi obtained the first quantitative version of Struwes decomposition with one bubble in all dimensions, namely $delta (u) leq C Gamma (u)$. For Struwes decomposition with two or more bubbles, Figalli and Glaudo showed a striking dimensional dependent quantitative estimate, namely $delta(u)leq C Gamma(u)$ when $3leq nleq 5$ while this is false for $ ngeq 6$. In this paper, we show that [dist (u,mathcal{T})leq Cbegin{cases} Gamma(u)left|log Gamma(u)right|^{frac{1}{2}}quad&text{if }n=6, |Gamma(u)|^{frac{n+2}{2(n-2)}}quad&text{if }ngeq 7.end{cases}] Furthermore, we show that this inequality is sharp.
In this paper we classify all positive extremal functions to a sharp weighted Sobolev inequality on the upper half space, which involves divergent operators with degeneracy on the boundary. As an application of the results, we can derive a sharp Sobo
lev type inequality involving Baouendi-Grushin operator, and classify certain extremal functions for all $tau>0$ and $m e2 $ or $ n e1$.
The Han-Li conjecture states that: Let $(M,g_0)$ be an $n$-dimensional $(ngeq 3)$ smooth compact Riemannian manifold with boundary having positive (generalized) Yamabe constant and $c$ be any real number, then there exists a conformal metric of $g_0$
with scalar curvature $1$ and boundary mean curvature $c$. Combining with Z. C. Han and Y. Y. Lis results, we answer this conjecture affirmatively except for the case that $ngeq 8$, the boundary is umbilic, the Weyl tensor of $M$ vanishes on the boundary and has a non-zero interior point.
We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension $ngeq 3$. We prove the existence of such conformal metrics
in the cases of $n=6,7$ or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be $1$, there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to $+infty$.
We employ three different methods to prove the following result on prescribed scalar curvature plus mean curvature problem: Let $(M^n,g_0)$ be a $n$-dimensional smooth compact manifold with boundary, where $n geq 3$, assume the conformal invariant $Y
(M,partial M)<0$. Given any negative smooth functions $f$ in $M$ and $h$ on $partial M$, there exists a unique conformal metric of $g_0$ such that its scalar curvature equals $f$ and mean curvature curvature equals $h$. The first two methods are sub-super-solution method and subcritical approximation, and the third method is a geometric flow. In the flow approach, assume another conformal invariant $Q(M,pa M)$ is a negative real number, for some class of initial data, we prove the short time and long time existences of the so-called prescribed scalar curvature plus mean curvature flows, as well as their asymptotic convergence. Via a family of such flows together with some additional variational arguments, under the flow assumptions we prove existence and uniqueness of positive minimizers of the associated energy functional and also the above result by analyzing asymptotic limits of the flows and the relations among some conformal invariants.
