## 解析学火曜セミナー

開催情報 火曜日　16:00～17:30　数理科学研究科棟(駒場) 156号室 石毛 和弘, 坂井 秀隆, 伊藤 健一 https://www.ms.u-tokyo.ac.jp/seminar/analysis/

### 2023年03月14日(火)

16:00-17:30   数理科学研究科棟(駒場) 126号室

Piermarco Cannarsa 氏 (University of Rome "Tor Vergata")
Parameter reconstruction for degenerate parabolic equations (English)
[ 講演概要 ]
First, we study degenerate parabolic equations arising in climate dynamics, providing uniqueness and stability estimates for the determination of the insolation function. Then, we address several aspects of the reconstruction of the degenerate diffusion coefficient. Finally, we discuss systems of two equations including a vertical component into the model.
[ 参考URL ]
https://forms.gle/nejpQS824vFKRbMQ6

### 2022年12月20日(火)

16:00-17:30   数理科学研究科棟(駒場) 126号室

[ 講演概要 ]
Jan Boman's (Stockholm Univ.) recent two papers:
, Regularity of a distribution and of the boundary of its support, The Journal of Geometric Analysis vol.32, Article number: 300 (2022).
, A hypersurface containing the support of a Radon transform must be an ellipsoid. II: The general case; J. Inverse Ill-Posed Probl. 2021; 29(3): 351–367.
In  he proved "Let $f(x_1,…,x_n,y)$ be a non-zero distribution with support in a $C^1$ surface $N=\{y=F(x)\}$. If $f(x,y)$ is depending real analytically on x-variables, then $F(x)$ is analytic". As an application, he reinforced the main result of . These results are obtained essentially by means of matrix algebra and a number theoretic method.
[ 参考URL ]
https://forms.gle/BpciRTzKh9FPUV8D7

### 2022年12月13日(火)

16:00-17:30   数理科学研究科棟(駒場) 126号室

Continuum limit problem of discrete Schrödinger operators on square lattices (Japanese)
[ 講演概要 ]
We consider discrete Schrödinger operators on the square lattice with its mesh size very small. The aim of this talk is to introduce the rigorous setting of continuum limit problems in the view point of operator theory and then to give its proof for the above operators, the one of which is defined on the vertices and the other of which is defined on the edges. This talk is based on joint works with Shu Nakamura (Gakushuin University) and Pavel Exner (Czech Academy of Science, Czech Technical University).
[ 参考URL ]
https://forms.gle/CRha8hydEuXzh71S7

### 2022年11月29日(火)

16:00-17:30   数理科学研究科棟(駒場) 126号室

Bernstein type theorem for the parabolic 2-Hessian equation under weaker assumptions (Japanese)
[ 講演概要 ]
In the early twentieth century, Bernstein proved that a minimal surface which can be expressed as the graph of a function defined in $\mathbb{R}^2$ must be a plane. For Monge-Ampère equation, it is known that a convex solution to $\det D^2 u=1$ in $\mathbb{R}^n$ must be a quadratic polynomial. Such kind of theorems, which we call Bernstein type theorems in this talk, have been extensively studied for various PDEs. For the parabolic $k$-Hessian equation, Bernstein type theorem has been proved by Nakamori and Takimoto (2015, 2016) under the convexity and some growth assumptions on the solution. In this talk, we shall obtain Bernstein type theorem for the parabolic 2-Hessian equation under weaker assumptions.
[ 参考URL ]
https://forms.gle/93YQ9C6DGYt5Vjuf7

### 2022年10月04日(火)

16:00-17:30   数理科学研究科棟(駒場) 126号室

[ 講演概要 ]
4階の偏微分方程式であるCahn-Hilliard方程式は相分離現象を記述する方程式としてよく知られている. J. W. Cahn, "Science during Paradigm Creation", (2011)によると時間後方問題となる難点が4階の項によって解決される点は現象解明の副産物であったようである. 近年, 前方後方問題への接近としてこのCahn-Hilliard方程式における粘性消滅法の考察がBui-Smarrazzo-Tesei, J. Math. Anal. Appl, (2014)やKagawa-Otani, J. Math. Anal. Appl, (2022)などで行われている. 本講演ではこれらの粘性消滅法の考えを時間微分を境界条件に含む, いわゆる動的境界条件で考察する. 講演の前半では研究動機と動的境界条件下でのCahn-Hilliard方程式についての先行研究を紹介しつつ, 抽象発展方程式の枠組みで適切性を論じる流れを解説する. 後半では動的境界条件下でのCahn-Hilliard方程式の1つとしてよく知られるGMSモデルを元に証明の大枠, すなわち一様評価と極限操作を解説し, 最後にLWモデルの場合との違いについて述べる. なお, 本講演はPavia大学のP. Colli氏とMilano工科大学のL. Scarpa氏との共同研究に基づく.
[ 参考URL ]
https://forms.gle/nPfEgKUX2tfUrg5LA

### 2022年08月23日(火)

16:00-17:30   数理科学研究科棟(駒場) 126号室

Stefan Neukamm 氏 (Dresden University/RIMS)
Quantitative homogenization for monotone, uniformly elliptic systems with random coefficients (English)
[ 講演概要 ]
Motivated by homogenization of nonlinearly elastic composite materials, we study homogenization rates for elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. Under the assumption of a fast decay of correlations on scales larger than the microscale $\varepsilon$, we establish estimates of optimal order for the approximation of the homogenized operator by the method of representative volumes. Moreover, we discuss applications to nonlinear elasticity random laminates.
[ 参考URL ]
https://forms.gle/V1wxbYhT4mkPF4gY9

### 2022年07月26日(火)

16:00-17:30   数理科学研究科棟(駒場) 126号室

Periodic homogenization of non-symmetric jump-type processes with drifts (Japanese)
[ 講演概要 ]
Homogenization problem is one of the classical problems in analysis and probability which is very actively studied recently. In this talk, we consider homogenization problem for non-symmetric Lévy-type processes with drifts in periodic media. Under a proper scaling, we show the scaled processes converge weakly to Lévy processes on ${\mathds R}^d$. In particular, we completely characterize the limiting processes when the coefficient function of the drift part is bounded continuous, and the decay rate of the jumping measure is comparable to $r^{-1-\alpha}$ for $r>1$ in the spherical coordinate with $\alpha \in (0,\infty)$. Different scaling limits appear depending on the values of $\alpha$.
This talk is based on joint work with Xin Chen, Zhen-Qing Chen and Jian Wang (Ann. Probab. 2021).
[ 参考URL ]
https://forms.gle/ewZEy1jAXrAhWx1Q8

### 2022年06月28日(火)

16:00-17:30   数理科学研究科棟(駒場) 126号室

Mourre inequality for non-local Schödinger operators (Japanese)
[ 講演概要 ]
We consider the Mourre inequality for the following self-adjoint operator $H=\Psi(-\Delta/2)+V$ acting on $L^2(\mathbb{R}^d)$, where $\Psi: [0,\infty)\rightarrow\mathbb{R}$ is an increasing function, $\Delta$ is Laplacian and $V: \mathbb{R}^d\rightarrow\mathbb{R}$ is an interaction potential. Mourre inequality immediately yields the discreteness and finite multiplicity of the eigenvalues. Moreover, Mourre inequality has the application to the absence of the singular continuous spectrum by combining the limiting absorption principle and, in addition, Mourre inequality is also used for proof of the minimal velocity estimate that plays an important role in the scattering theory. In this talk, we report that Mourre inequality holds under the general $\Psi$ and $V$ by choosing the conjugate operator $A=(p\cdot x+x\cdot p)/2$ with $p=-\sqrt{-1}\nabla$, and that the discreteness and finite multiplicity of the eigenvalues hold. This talk is a joint work with J. Lőrinczi (Hungarian Academy of Sciences) and I. Sasaki (Shinshu University).
[ 参考URL ]
https://forms.gle/sBSeNH9AYFNypNBk9

### 2022年05月31日(火)

16:00-17:30   数理科学研究科棟(駒場) 126号室

Convergence of Sobolev gradient trajectories to elastica (Japanese)
[ 講演概要 ]
In this talk we consider a higher order Sobolev gradient flow for the modified elastic energy defined on closed space curves. The $L^2$-gradient flow for the modified elastic energy has been well studied, and standard results are solvability of the flow for smooth initial curve and subconvergence of solutions to elastica. Moreover, stronger convergence results, so called full limit convergence, are generally up to reparametrisation and sometimes translation. In this talk, we consider $H^2$-gradient flow for the modified elastic energy and prove (i) the solvability of the flow for initial curve in the energy class, (ii) full limit convergence to elastica by way of a Lojasiewicz—Simon gradient inequality. This talk is based on a joint work with Philip Schrader (Murdoch University).
[ 参考URL ]
https://forms.gle/wkCbqdmNuz9zr3vA8

### 2022年05月24日(火)

16:00-17:30   オンライン開催
Michael Goesswein 氏 (東京大学/University of Regensburg)
Stability analysis for the surface diffusion flow on double bubbles using the Lojasiewicz-Simon (English)
[ 講演概要 ]
Many strategies for stability analysis use precise knowledge of the set of equilibria. For example, Escher, Mayer, and Simonett used center manifold analysis to study the surface diffusion flow on closed manifolds. Especially in higher dimensional situations with boundaries, this can cause problems as the set of equilibria will have a lot of degrees of freedom. In such situations approaches with a Lojasiewicz-Simon inequality gives an elegant way to avoid this problem. In this talk, we will both explain the general tools and ideas for this strategy and use them to prove the stability of standard double bubbles with respect to the surface diffusion flow. The talk is based on joint work with H. Garcke.
[ 参考URL ]
https://forms.gle/Cam3mpSSEKKVppZr9

### 2022年04月26日(火)

16:00-17:30   数理科学研究科棟(駒場) 126号室

Existence of a bounded forward self-similar solution to a minimal Keller-Segel model (Japanese)
[ 講演概要 ]
In this talk, we consider existence of a bounded forward self-similar solution to the initial value problem of a minimal Keller-Segel model. It is well known that the mass conservation law plays an important role to classify its large time behavior of solutions to Keller-Segel models. On the other hand, we could not expect existence of self-similar solutions to our problem with the mass conservation law except for the two dimensional case due to the scaling invariance of our problem. We will show existence of a forward self-similar solution to our problem. The key idea to guarantee boundedness of its self-similar solution is to choose a concrete upper barrier function using the hypergeometric function.
[ 参考URL ]
https://forms.gle/mrXnjsgctSJJ1WSF6

### 2022年04月12日(火)

16:00-17:30   オンライン開催
Amru Hussein 氏 (Technische Universität Kaiserslautern)
Maximal $L^p$-regularity and $H^{\infty}$-calculus for block operator matrices and applications (English)
[ 講演概要 ]
Many coupled evolution equations can be described via $2\times2$-block operator matrices of the form $\mathcal{A}=\begin{bmatrix}A & B \\ C & D \end{bmatrix}$ in a product space $X=X_1\times X_2$ with possibly unbounded entries. Here, the case of diagonally dominant block operator matrices is considered, that is, the case where the full operator $\mathcal{A}$ can be seen as a relatively bounded perturbation of its diagonal part though with possibly large relative bound. For such operators, the properties of sectoriality, $\mathcal{R}$-sectoriality and the boundedness of the $H^\infty$-calculus are studied, and for these properties perturbation results for possibly large but structured perturbations are derived. Thereby, the time-dependent parabolic problem associated with $\mathcal{A}$ can be analyzed in maximal $L^p_t$-regularity spaces, and this is applied to a wide range of problems such as different theories for liquid crystals, an artificial Stokes system, strongly damped wave and plate equations, and a Keller-Segel model.
This talk is based on a joint work with Antonio Agresti, see https://arxiv.org/abs/2108.01962
[ 参考URL ]
https://forms.gle/QbQKex12dbQrt2Lw6

### 2021年11月16日(火)

16:00-17:30   オンライン開催

[ 講演概要 ]

[ 参考URL ]
https://forms.gle/6ZCp8hQxKA3vq3DB9

### 2021年10月19日(火)

16:00-17:30   オンライン開催

Global structure of steady-states for a cross-diffusion limit in the Shigesada-Kawasaki-Teramoto model (Japanese)
[ 講演概要 ]
In 1979, Shigesada, Kawasaki and Teramoto proposed a Lotka-Volterra competition model with cross-diffusion terms in order to realize the segregation phenomena of two competing species. This talk concerns the asymptotic behavior of steady-states to the Shigesada-Kawasaki-Teramoto model in the full cross-diffusion limit where both coefficients of cross-diffusion terms tend to infinity at the same rate. In the former half of this talk, we derive a uniform estimate of all steady-states independent of the cross-diffusion terms. In the latter half, we show the global structure of steady-states of a shadow system in the full cross-diffusion limit.
[ 参考URL ]
https://forms.gle/hkfCd3fSW5A77mwv5

### 2021年07月13日(火)

16:00-17:30   オンライン開催

Li-Yau type inequality for curves and applications (Japanese)
[ 講演概要 ]
A classical result of Li and Yau asserts an optimal relation between the bending energy and multiplicity of a closed surface in Euclidean space. Here we establish an analogue for curves in a completely general form, and observe new phenomena due to low dimensionality. We also discuss its applications to elastic flows, networks, and knots.
[ 参考URL ]
https://forms.gle/gR4gfn8v59LEoqp38

### 2021年06月08日(火)

16:00-17:30   オンライン開催

Local well-posedness for the Landau-Lifshitz equation with helicity term (Japanese)
[ 講演概要 ]
We consider the initial value problem for the Landau-Lifshitz equation with helicity term (chiral interaction term), which arises from the Dzyaloshinskii-Moriya interaction. We show that it is locally well-posed in Sobolev spaces $H^s$ when $s>2$. The key idea is to reduce the problem to a system of semi-linear Schr\"odinger equations, called modified Schr\"odinger map equation. The problem here is that the helicity term appears as quadratic derivative nonlinearities, which is known to be difficult to treat as perturbation of the free evolution. To overcome that, we consider them as magnetic terms, then apply the energy method by introducing the differential operator associated with magnetic potentials.
[ 参考URL ]
https://forms.gle/nc85Mw9Jd6NgJzT98

### 2021年05月25日(火)

16:00-17:30   オンライン開催

Asymptotic limit of fast rotation for the incompressible Navier-Stokes equations in a 3D layer (Japanese)
[ 講演概要 ]
In this talk, we consider the initial value problem for the Navier-Stokes equation with the Coriolis force in a three-dimensional infinite layer. We prove the unique existence of global solutions for initial data in the scaling invariant space when the speed of rotation is sufficiently high. Furthermore, we consider the asymptotic limit of the fast rotation, and show that the global solution converges to that of 2D incompressible Navier-Stokes equations in some global in time space-time norms. This talk is based on the joint work with Hiroki Ohyama (Kyushu University).
[ 参考URL ]
https://forms.gle/wHpi7BSpppsiiguD6

### 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)