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Register a new userOne dimensional conformal metric flow II
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In this paper we continue our studies of the one dimensional conformal metric flows, which were introduced in [8]. In this part we mainly focus on evolution equations involving fourth order derivatives. The global existence and exponential convergence of metrics for the 1-Q and 4-Q flows are obtained.
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We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere $S^2$, begin{align*} u_t & = Delta u + | abla u|^2 u quad text{in } Omegatimes(0,T) u &= varphi quad text{on } partial Omegatimes(0,T) u(cdot,0) &= u_0 quad text{in } Omega , end{align*} where $Omega$ is a bounded, smooth domain in $mathbb{R}^2$, $u: Omegatimes(0,T)to S^2$, $u_0:barOmega to S^2$ is smooth, and $varphi = u_0big|_{partialOmega}$. Given any points $q_1,ldots, q_k$ in the domain, we find initial and boundary data so that the solution blows-up precisely at those points. The profile around each point is close to an asymptotically singular scaling of a 1-corrotational harmonic map. We build a continuation after blow-up as a $H^1$-weak solution with a finite number of discontinuities in space-time by reverse bubbling, which preserves the homotopy class of the solution after blow-up.
We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere $S^2$, begin{align*} u_t & = Delta u + | abla u|^2 u quad text{in } Omegatimes(0,T) u &= u_b quad text{on } partial Omegatimes(0,T) u(cdot,0) &= u_0 quad text{in } Omega , end{align*} with $u(x,t): bar Omegatimes [0,T) to S^2$. Here $Omega$ is a bounded, smooth axially symmetric domain in $mathbb{R}^3$. We prove that for any circle $Gamma subset Omega$ with the same axial symmetry, and any sufficiently small $T>0$ there exist initial and boundary conditions such that $u(x,t)$ blows-up exactly at time $T$ and precisely on the curve $Gamma$, in fact $$ | abla u(cdot ,t)|^2 rightharpoonup | abla u_*|^2 + 8pi delta_Gamma text{ as } tto T . $$ for a regular function $u_*(x)$, where $delta_Gamma$ denotes the Dirac measure supported on the curve. This the first example of a blow-up solution with a space-codimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng.
In this note, we study Liouville type theorem for conformal Gaussian curvature equation (also called the mean field equation) $$ -Delta u=K(x)e^u, in R^2 $$ where $K(x)$ is a smooth function on $R^2$. When $K(x)=K(x_1)$ is a sign-changing smooth function in the real line $R$, we have a non-existence result for the finite total curvature solutions. When $K$ is monotone non-decreasing along every ray starting at origin, we can prove a non-existence result too. We use moving plane method and moving sphere method.
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with smooth boundary $partial M$. Suppose that $(M,g)$ admits a scalar-flat conformal metric. We prove that the supremum of the isoperimetric quotient over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality in the Euclidean space, and consequently is achieved, if either (i) $nge 12$ and $partial M$ has a nonumbilic point; or (ii) $nge 10$, $partial M$ is umbilic and the Weyl tensor does not vanish at some boundary point.
We consider a fixed angle inverse scattering problem in the presence of a known Riemannian metric. First, assuming a no caustics condition, we study the direct problem by utilizing the progressing wave expansion. Under a symmetry assumption on the metric, we obtain uniqueness and stability results in the inverse scattering problem for a potential with data generated by two incident waves from opposite directions. Further, similar results are given using one measurement provided the potential also satisfies a symmetry assumption. This work extends the results of [23,24] from the Euclidean case to certain Riemannian metrics.
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