## 解析学火曜セミナー

開催情報 火曜日　16:50～18:20　数理科学研究科棟(駒場) 128号室 石毛 和弘, 坂井 秀隆, 伊藤 健一 https://www.ms.u-tokyo.ac.jp/seminar/analysis/

### 2020年02月18日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室

Alessio Porretta 氏 (Tor Vergata university of Rome)
Long time behavior of mean field games systems (English)
[ 講演概要 ]
I will review several aspects related to the long time ergodic behavior of mean field game systems: the turnpike property, the exponential rate of convergence, the role of monotonicity of the couplings, the convergence of u up to translations, the limit of the vanishing discounted problem, the long time behavior of the master equation. All those aspects have independent interest and are correlated at the same time.

### 2020年01月14日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室
Erik Skibsted 氏 (オーフス大学)
Scattering near a two-cluster threshold (English)
[ 講演概要 ]
For a one-body Schr\"odinger operator with an attractive slowly decaying potential the scattering matrix is well-defined at the energy zero, and the structure of its singularities is well-studied. The usual (non-relativistic) model for the Hydrogen atom is a particular example of such Schr\"odinger operator.
Less is known on scattering at a two-cluster threshold of an $N$-body Schr\"odinger operator for which the effective interaction between the two bound clusters is attractive Coulombic. An example of interest is scattering at a two-cluster threshold of a neutral atom/molecule. We present results of an ongoing joint work with X.P. Wang on the subject, including a version of the Sommerfeld uniqueness result and its applications.
We shall also present general results on spectral theory at a two-cluster threshold (not requiring the effective interaction to be attractive Coulombic). This includes a general structure theorem on the bound and resonance states at the threshold as well as a resolvent expansion in weighted spaces above the threshold (under more restrictive conditions). Applications to scattering theory will be indicated.

### 2019年12月10日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室
Tobias Barker 氏 (École Normale Supérieure)
Vorticity alignment vs vorticity creation at the boundary (English)
[ 講演概要 ]
The Navier-Stokes are used as a model for viscous incompressible fluids such as water. The question as to whether or not the equations in three dimensions form singularities is an open Millennium prize problem. In their celebrated paper in 1993, Constantin and Fefferman showed that (in the whole plane) if the vorticity is sufficiently well aligned in regions of high vorticity then the Navier-Stokes equations remain smooth. For the half-space it is commonly assumed that viscous fluids stick' to the boundary, which generates vorticity at the boundary. In such a setting, it is open as to whether Constantin and Fefferman's result remains to be true. In my talk I will present recent results in this direction. Joint work with Christophe Prange (CNRS, Université de Bordeaux)

### 2019年11月26日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室

Accurate lower bounds for eigenvalues of electronic Hamiltonians (Japanese)
[ 講演概要 ]
Electronic Hamiltonians are differential operators depending on relative positions of nuclei as parameters. When we regard an eigenvalues of an electronic Hamiltonian as a function of relative positions of nuclei, minimum points correspond to shapes of molecules. Upper bounds for eigenvalues are obtained by variational methods. However, since the physical information as minimum points does not change when a reference point of energy changes, physical information can not be obtained by variational methods only. Combining lower and upper bounds physical information is obtained.
In this talk we discuss the Weinstein-Arnszajn intermediate problem methods for lower bounds of eigenvalues based on comparison of operators. A method for lower bounds of one-electronic Hamiltonians is also introduced. Some computations for two kinds of hydrogen molecule-ion are shown.

### 2019年11月19日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室
Wenjia Jing 氏 (清華大学)
Quantitative homogenization for the Dirichlet problem of Stokes system in periodic perforated domain - a unified approach (English)
[ 講演概要 ]
We present a new unified approach for the quantitative homogenization of the Stokes system in periodically perforated domains, that is domains outside a periodic array of holes, with Dirichlet data at the boundary of the holes. The method is based on the (rescaled) cell-problem and is adaptive to the ratio between the typical distance and the typical side length of the holes; in particular, for the critical ratio identified by Cioranescu-Murat, we recover the “strange term from nowhere”termed by them, which, in the context of Stokes system, corresponds to the Brinkman’s law. An advantage of the method is that it can be systematically quantified using the periodic layer potential technique. We will also report some new correctors to the homogenization problem using this approach. The talk is based on joint work with Yong Lu and Christophe Prange.

### 2019年11月12日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室

Mould展開を用いたResurgence構造の解析 (Japanese)
[ 講演概要 ]
Mould解析はJ. Ecalle氏により考案された解析手法であり, ベクトル場の標準形の構成や多重ゼータ値などに応用されている. Mould 解析では word による展開を用いるが, その後Ecalle氏によりtreeを用いた展開も導入された. 2017年に, F. Fauvet氏とF. Menous氏により, このtreeによる展開のConnes-Kreimer Hopf代数を用いた明確な定式化が与えられた. 本講演では, この定式化に則り, ベクトル場の線形化問題に現れるStokes現象のResurgence構造に関して議論を行う.

### 2019年11月05日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室
Ngô Quốc Anh 氏 (ベトナム国家大学ハノイ校 / 東京大学)
Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space (English)
[ 講演概要 ]
This talk concerns entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity $\Delta^m u = \pm u^\alpha$ in $\mathbb R^n$ with $n \geqslant 1$, $m \geqslant 1$, and $\alpha \in \mathbb R$. For small $m$, the above equations arise in many physical phenomena and applied mathematics. They also arise from several prescribing geometric curvture problems in conformal geometry such as the Yamabe problem, the scalar curvature problem, and the Q-curvature problem for the Paneitz operator. Higher-order cases also arise from the Q-curvature problem for the GJMS operator. In this talk, I will present a complete picture of the existence and non-existence of solutions to the above equations in the full rage of the parameters $n$, $m$, and $\alpha$. This is joint work with V.H. Nguyen, Q.H. Phan, and D. Ye.

### 2019年06月11日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室
Antonio De Rosa 氏 (クーラン数理科学研究所)
Solutions to two conjectures in branched transport: stability and regularity of optimal paths (English)
[ 講演概要 ]
Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. The transportation cost in these models is proportional to a concave power $\alpha \in (0,1)$ of the intensity of the flow. We focus on the stability of the optimal transports, with respect to variations of the source and target measures. The stability was known when $\alpha$ is bigger than a critical threshold, but we prove it for every exponent $\alpha \in (0,1)$ and we provide a counterexample for $\alpha=0$. Thus we completely solve a conjecture of the book Optimal transportation networks by Bernot, Caselles and Morel. Moreover the robustness of our proof allows us to get the stability for more general lower semicontinuous functional. Furthermore, we prove the stability for the mailing problem, which was completely open in the literature, solving another conjecture of the aforementioned book. We use the latter result to show the regularity of the optimal networks. (Joint works with Maria Colombo and Andrea Marchese)

### 2019年04月09日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室
Fabio Punzo 氏 (ミラノ工科大学)
The Poisson equation on Riemannian manifolds (English)
[ 講演概要 ]
The talk is concerned with the existence of solutions to the Poisson equation on complete non-compact Riemannian manifolds. In particular, the interplay between the Ricci curvature and the behaviour at infinity of the source function will be discussed. This is a joint work with G. Catino and D.D. Monticelli.

### 2019年03月05日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室
Nicholas Edelen 氏 (Massachusetts Institute of Technology)
The structure of minimal surfaces near polyhedral cones (English)
[ 講演概要 ]
We prove a regularity theorem for minimal varifolds which resemble a cone $C_0$ over an equiangular geodesic net. For varifold classes admitting a `no-hole'' condition on the singular set, we additionally establish regularity near the cone $C_0 \times R^m$. Our result implies the following generalization of Taylor's structure theorem for soap bubbles: given an $n$-dimensional soap bubble $M$ in $R^{n+1}$, then away from an $(n-3)$-dimensional set, $M$ is locally $C^{1,\alpha}$ equivalent to $R^n$, a union of three half-$n$-planes meeting at $120$ degrees, or an $(n-2)$-line of tetrahedral junctions. This is joint work with Maria Colombo and Luca Spolaor.

### 2019年01月22日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室

Construction of solutions to Schrodinger equations with sub-quadratic potential via wave packet transform (Japanese)
[ 講演概要 ]
In this talk, we consider linear Schrodinger equations with sub-quadratic potentials, which can be transformed by the wave packet transform with time dependent wave packet to a PDE of first order with inhomogeneous terms including unknown function and second derivatives of the potential. If the second derivatives of the potentials are bounded, the homogenous term of the first oder equation gives a construction of solutions to Schrodinger equations with sub-quadratic potentials by the similar way as in D. Fujiwara's work for Feynman path integral. We will show numerical computations by using our construction, if we have enough time.

### 2018年12月25日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室

Modified scattering for nonlinear dispersive equations with critical non-polynomial nonlinearities (Japanese)
[ 講演概要 ]
In this talk, I will introduce resent progress on modified scattering for Schrodinger equation and Klein-Gordon equation with a non-polynomial nonlinearity. We use Fourier series expansion technique to find the resonant part of the nonlinearity which produces phase correction factor.

### 2018年11月06日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室

Global behavior of bifurcation curves and related topics (日本語)
[ 講演概要 ]
In this talk, we consider the asymptotic behavior of bifurcation curves for ODE with oscillatory nonlinear term. First, we study the global and local behavior of oscillatory bifurcation curves. We also consider the bifurcation problems with nonlinear diffusion.

### 2018年10月30日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室

Spectral structure of the Neumann-Poincaré operator in three dimensions: Willmore energy and surface geometry (日本語)
[ 講演概要 ]
The Neumann-Poincaré operator (abbreviated by NP) is a boundary integral operator naturally arising when solving classical boundary value problems using layer potentials. If the boundary of the domain, on which the NP operator is defined, is $C^{1, \alpha}$ smooth, then the NP operator is compact. Thus, the Fredholm integral equation, which appears when solving Dirichlet or Neumann problems, can be solved using the Fredholm index theory.
Regarding spectral properties of the NP operator, the spectrum consists of eigenvalues converging to $0$ for $C^{1, \alpha}$ smooth boundaries. Our main purpose here is to deduce eigenvalue asymptotics of the NP operators in three dimensions. This formula is the so-called Weyl's law for eigenvalue problems of NP operators. Then we discuss relationships among the Weyl's law, the Euler characteristic and the Willmore energy on the boundary surface.

### 2018年10月16日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室

[ 講演概要 ]

### 2018年07月31日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室

[ 講演概要 ]

### 2018年06月26日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室

[ 講演概要 ]
We consider the Cauchy problem of the drift-diffusion system in the whole space. Introducing the scaling critical case, we consider the solvability of the drift-diffusion system in the whole space and give some large time behavior of solutions. This talk is based on a collaboration with Masaki Kurokiba and Hiroshi Wakui.

### 2018年06月19日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室
Rowan Killip 氏 (UCLA)
KdV is wellposed in $H^{-1}$ (English)

### 2016年12月13日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
Hans Christianson 氏 (North Carolina State University)
Distribution of eigenfunction mass on some really simple domains (English)
[ 講演概要 ]
Eigenfunctions are fundamental objects of study in spectral geometry and quantum chaos. On a domain or manifold, they determine the behaviour of solutions to many evolution type equations using, for example, separation of variables. Eigenfunctions are very sensitive to background geometry, so it is important to understand what the eigenfunctions look like: where are they large and where are they small? There are many different ways to measure what "large" and "small" mean. One can consider local $L^2$ distribution, local and global $L^p$ distribution, as well as restrictions and boundary values. I will give an overview of what is known, and then discuss some very recent works in progress demonstrating that complicated things can happen even in very simple geometric settings.

### 2016年12月06日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
Horia Cornean 氏 (オールボー大学、デンマーク)
On the trivialization of Bloch bundles and the construction of localized Wannier functions (English)
[ 講演概要 ]
We shall present an introductory lecture on the trivialization of Bloch bundles and its physical implications. Simply stated, the main question we want to answer is the following: given a rank $N＼geq 1$ family of orthogonal projections $P(k)$ where $k＼in ＼mathbb{R}^d$, $P(＼cdot)$ is smooth and $＼mathbb{Z}^d$-periodic, is it possible to construct an orthonormal basis of its range which consists of vectors which are both smooth and periodic in $k$? We shall explain in detail the connection with solid state physics. This is joint work with I. Herbst and G. Nenciu.

### 2016年11月29日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室

Interior transmission eigenvalue problems on manifolds (Japanese)

### 2016年10月25日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室

### 2016年07月12日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
X. P. Wang 氏 (Université de Nantes, France)
Gevrey estimates of the resolvent and sub-exponential time-decay (English)
[ 講演概要 ]
For a class of non-selfadjoint Schrodinger operators satisfying some weighted coercive condition, we prove that the resolvent satisfies the Gevrey estimates at the threshold. As applications, we show that the heat and Schrodinger semigroups decay sub-exponentially in appropriately weighted spaces. We also study compactly supported perturbations of this class of operators where zero may be an embedded eigenvalue.

### 2016年06月28日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
Georgi Raikov 氏 (The Pontificia Universidad Católica de Chile)
Discrete spectrum of Schr\"odinger operators with oscillating decaying potentials (English)
[ 講演概要 ]
I will consider the Schr\"odinger operator $H_{\eta W} =-\Delta + \eta W$, self-adjoint in $L^2(\re^d)$, $d \geq 1$. Here $\eta$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. I will discuss the asymptotic behaviour of the discrete spectrum of $H_{\eta W}$ near the origin. Due to the irregular decay of $\eta W$, there exist some non semiclassical phenomena; in particular, $H_{\eta W}$ has less eigenvalues than suggested by the semiclassical intuition.

### 2016年06月21日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室

[ 講演概要 ]