We study Planck 2015 cosmic microwave background (CMB) anisotropy data using the energy density inhomogeneity power spectrum generated by quantum fluctuations during an early epoch of inflation in the non-flat $\Lambda$CDM model. Unlike earlier analyses of non-flat models, which assumed an inconsistent power-law power spectrum of energy density inhomogeneities, we find that the Planck 2015 data alone, and also in conjunction with baryon acoustic oscillation measurements, are reasonably well fit by a closed $\Lambda$CDM model in which spatial curvature contributes a few percent of the current cosmological energy density budget. In this model, the measured Hubble constant and non-relativistic matter density parameter are in good agreement with values determined using most other data. Depending on parameter values, the closed $\Lambda$CDM model has reduced power, relative to the tilted, spatially-flat $\Lambda$CDM case, and can partially alleviate the low multipole CMB temperature anisotropy deficit and can help partially reconcile the CMB anisotropy and weak lensing $\sigma_8$ constraints, at the expense of somewhat worsening the fit to higher multipole CMB temperature anisotropy data. Our results are interesting but tentative; a more thorough analysis is needed to properly gauge their significance.

Planck 2015 constraints on the non-flat $\Lambda$CDM inflation model

We construct Delaunay-type solutions for the fractional Yamabe problem with an isolated singularity 

$$\begin{aligned} (-\Delta )^\gamma w= c_{n, {\gamma }}w^{\frac{n+2\gamma }{n-2\gamma }}, w>0 \quad \text{ in } \quad \mathbb R^n \backslash \{0\}. \end{aligned}$$

We follow a variational approach, in which the key is the computation of the fractional Laplacian in polar coordinates.

Delaunay-type singular solutions for the fractional Yamabe problem

For singular $SU(3)$ Toda systems, we prove that the limit of energy concentration is a finite set. In addition, for fully bubbling solutions we use Pohozaev identity to prove a uniform estimate. Our results extend previous results of Jost-Lin-Wang on regular $SU(3)$ Toda systems.

/pdf/classification-of-blowup-limits-for-su-3-singular-toda-47az5o21oe.pdf

Classification of blowup limits for SU(3) singular Toda systems

We construct Delaunay-type solutions for the fractional Yamabe problem with an isolated singularity $(-\Delta)^\gamma w = c_{n, \gamma} w^{\frac{n+2\gamma}{n-2\gamma}}, w>0 \ \mbox{in} \ \mathbb{R}^n \backslash \{0\}$ We follow a variational approach, in which the key is the computation of the fractional Laplacian in polar coordinates.

Delaunay-type singular solutions for the fractional Yamabe Problem

The Keller–Segel system u t = D Δ u − D χ ∇ ⋅ ( u v ∇ v ) , x ∈ Ω , t > 0 , v t = D Δ v − v + u , x ∈ Ω , t > 0 , is considered in a bounded domain Ω ⊂ R n , n ≥ 2 , with smooth boundary, where χ > 0 and D > 0 . The main results identify a condition on the parameters χ 2 n and D > 0 , essentially reducing to the assumption that χ 2 D be suitably small, under which for all reasonably regular and positive initial data the corresponding classical solution of an associated Neumann initial–boundary value problem, known to exist globally according to previous findings, approaches the homogeneous steady state ( u ¯ 0 , u ¯ 0 ) at an exponential rate with respect to the norm in ( L ∞ ( Ω ) ) 2 as t → ∞ , where u ¯ 0 ≔ 1 | Ω | ∫ Ω u ( ⋅ , 0 ) . As a particular consequence, this entails a convergence statement of the above flavor in the normalized system with D = 1 and fixed χ 2 n , provided that Ω satisfies a certain smallness condition.

Stabilization in the logarithmic Keller–Segel system

We consider the problem of constructing solutions to the fractional Yamabe problem which are singular at a given smooth submanifold, for which we establish the classical gluing method of Mazzeo and Pacard (J. Differential Geom., 1996) for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of a fractional-order ordinary differential equation (ODE). Thus, our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of nonlocal ODEs. Note, however, that no traditional phase-plane analysis is available here. Instead, we first provide a rigorous construction of radial fast-decaying solutions by a blowup argument and a bifurcation method. Then, second, we use conformal geometry to rewrite this nonlocal ODE, giving a hint of what a nonlocal phase-plane analysis should be. Third, for the linear theory, we use complex analysis and some non-Euclidean harmonic analysis to examine a fractional Schrodinger equation with a Hardy-type critical potential. We construct its Green’s function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a 2-dimensional kernel as in the second-order case.

/pdf/on-higher-dimensional-singularities-for-the-fractional-2l73s710v1.pdf

On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard program

We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, and we establish the classical gluing method of Mazzeo and Pacard for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of an ODE, and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. No traditional phase-plane analysis is available here. Instead, first, we provide a rigorous construction of radial fast-decaying solutions by a blow-up argument and a bifurcation method. Second, we use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be. Third, for the linear theory, we examine a fractional Schrodinger equation with a Hardy type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a two-dimensional kernel as in the second order case.

On higher dimensional singularities for the fractional Yamabe problem: a non-local Mazzeo-Pacard program

We consider the following nonlinear Schrodinger equation $$\begin{aligned} \left\{ \begin{array}{l} \Delta u-(1+\delta V)u+f(u)=0 \ \ \hbox { in }\mathbb {R}^N,\\ u>0 \ \hbox {in} \ \mathbb {R}^N, u\in H^1(\mathbb {R}^N) \end{array} \right. \end{aligned}$$
 where $$V$$
 is a continuous potential and $$ f(u)$$
 is a nonlinearity satisfying some decay condition and some non-degeneracy condition, respectively. Using localized energy method, we prove that there exists a $$\delta _0$$
 such that for $$0<\delta <\delta _0$$
 , the above problem has infinitely many positive solutions. This generalizes and gives a new proof of the results by Cerami et al. (Comm. Pure Appl. Math. 66, 372–413, 2013). The new techniques allow us to establish the existence of infinitely many positive bound states for elliptic systems.

Infinitely many positive solutions for nonlinear equations with non-symmetric potentials

An optimal bound on the number of interior spike solutions for the Lin–Ni–Takagi problem

For the generalized surface quasi-geostrophic equation $$\left\{ \begin{aligned} & \partial_t \theta+u\cdot \nabla \theta=0, \quad \text{in } \mathbb{R}^2 \times (0,T), \\ & u=\nabla^\perp \psi, \quad \psi = (-\Delta)^{-s}\theta \quad \text{in } \mathbb{R}^2 \times (0,T) , \end{aligned} \right. $$ $0<s<1$, we consider for $k\ge1$ the problem of finding a family of $k$-vortex solutions $\theta_\varepsilon(x,t)$ such that as $\varepsilon\to 0$ $$ \theta_\varepsilon(x,t) \rightharpoonup \sum_{j=1}^k m_j\delta(x-\xi_j(t)) $$ for suitable trajectories for the vortices $x=\xi_j(t)$. We find such solutions in the special cases of vortices travelling with constant speed along one axis or rotating with same speed around the origin. In those cases the problem is reduced to a fractional elliptic equation which is treated with singular perturbation methods. A key element in our construction is a proof of the non-degeneracy of the radial ground state for the so-called fractional plasma problem $$(-\Delta)^sW = (W-1)^\gamma_+, \quad \text{in } \mathbb{R}^2, \quad 1<\gamma < \frac{1+s}{1-s}$$ whose existence and uniqueness have recently been proven in \cite{chan_uniqueness_2020}.

Weiwei Ao

Papers

On higher-dimensional singularities for the fractional Yamabe problem: A nonlocal Mazzeo–Pacard program

On higher dimensional singularities for the fractional Yamabe problem: a non-local Mazzeo-Pacard program

Infinitely many positive solutions for nonlinear equations with non-symmetric potentials

An optimal bound on the number of interior spike solutions for the Lin–Ni–Takagi problem

Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation