## Applied Analysis

Seminar information archive ～08/10｜Next seminar｜Future seminars 08/11～

Date, time & place | Thursday 16:00 - 17:30 002Room #002 (Graduate School of Math. Sci. Bldg.) |
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**Seminar information archive**

### 2022/06/30

16:00-17:00 Online

A brief introduction to a class of new phase field models (English)

https://forms.gle/esc7Y6KGASwbFro97

**Xingzhi Bian**(Shanghai University)A brief introduction to a class of new phase field models (English)

[ Abstract ]

Existence of weak solutions for a type of new phase field models, which are the system consisting of a degenerate parabolic equation of order parameter coupled to a linear elasticity sub-system. The models are applied to describe the phase transitions in elastically deformable solids.

[ Reference URL ]Existence of weak solutions for a type of new phase field models, which are the system consisting of a degenerate parabolic equation of order parameter coupled to a linear elasticity sub-system. The models are applied to describe the phase transitions in elastically deformable solids.

https://forms.gle/esc7Y6KGASwbFro97

### 2022/04/21

16:00-17:30 Online

Effect of decay rates of initial data on the sign of solutions to Cauchy problems of some higher order parabolic equations (Japanese)

[ Reference URL ]

https://forms.gle/96bBNEAEHrsdXfH57

**( )**Effect of decay rates of initial data on the sign of solutions to Cauchy problems of some higher order parabolic equations (Japanese)

[ Reference URL ]

https://forms.gle/96bBNEAEHrsdXfH57

### 2021/12/16

16:00-17:00 Online

Existence of solutions for fractional semilinear parabolic equations in Besov-Morrey spaces (Japanese)

[ Reference URL ]

https://forms.gle/whpkgAwYvyQKQMzM8

**Zhanpeisov Erbol**( )Existence of solutions for fractional semilinear parabolic equations in Besov-Morrey spaces (Japanese)

[ Reference URL ]

https://forms.gle/whpkgAwYvyQKQMzM8

### 2021/12/02

### 2021/11/25

### 2021/10/28

16:00-17:00 Online

Quasiconformal and Sobolev mappings on metric measure

https://forms.gle/QATECqmwmWGvXoU56

**Xiaodan Zhou**(OIST)Quasiconformal and Sobolev mappings on metric measure

[ Abstract ]

The study of quasiconformal mappings has been an important and active topic since its introduction in the 1930s and the theory has been widely applied to different fields including differential geometry, harmonic analysis, PDEs, etc. In the Euclidean space, it is a fundamental result that three definitions (metric, geometric and analytic) of quasiconformality are equivalent. The theory of quasiconformal mappings has been extended to metric measure spaces by Heinonen and Koskela in the 1990s and their work laid the foundation of analysis on metric spaces. In general, the equivalence of the three characterizations will no longer hold without appropriate assumptions on the spaces and mappings. It is a question of general interest to find minimal assumptions on the metric spaces and on the mapping to guarantee the metric definition implies the analytic characterization or geometric characterization. In this talk, we will give an brief review of the above mentioned classical theory and present some recent results we achieved in obtaining the analytic property, in particular, the Sobolev regularity of a metric quasiconformal mapping with relaxed spaces and mapping conditions. Unexpectedly, we can apply this to prove results that are new even in the classical Euclidean setting. This is joint work with Panu Lahti (Chinese Academy of Sciences).

[ Reference URL ]The study of quasiconformal mappings has been an important and active topic since its introduction in the 1930s and the theory has been widely applied to different fields including differential geometry, harmonic analysis, PDEs, etc. In the Euclidean space, it is a fundamental result that three definitions (metric, geometric and analytic) of quasiconformality are equivalent. The theory of quasiconformal mappings has been extended to metric measure spaces by Heinonen and Koskela in the 1990s and their work laid the foundation of analysis on metric spaces. In general, the equivalence of the three characterizations will no longer hold without appropriate assumptions on the spaces and mappings. It is a question of general interest to find minimal assumptions on the metric spaces and on the mapping to guarantee the metric definition implies the analytic characterization or geometric characterization. In this talk, we will give an brief review of the above mentioned classical theory and present some recent results we achieved in obtaining the analytic property, in particular, the Sobolev regularity of a metric quasiconformal mapping with relaxed spaces and mapping conditions. Unexpectedly, we can apply this to prove results that are new even in the classical Euclidean setting. This is joint work with Panu Lahti (Chinese Academy of Sciences).

https://forms.gle/QATECqmwmWGvXoU56

### 2021/10/14

### 2021/07/29

16:00-17:00 Online

Lotka-Volterra competition-diffusion system: the critical case

https://forms.gle/LHj5mVUdpQ3Jxkrd6

**Dongyuan Xiao**( )Lotka-Volterra competition-diffusion system: the critical case

[ Abstract ]

We consider the reaction-diffusion competition system u_t=u_{xx}+u(1-u-v), v_t=dv_{xx}+rv(1-v-u), which is the so-called critical case. The associated ODE system then admits infinitely many equilibria, which makes the analysis quite intricate. We first prove the non-existence of monotone traveling waves by applying the phase plane analysis. Next, we study the long time behavior of the solution of the Cauchy problem with a compactly supported initial datum. We not only reveal that the ''faster'' species excludes the ''slower'' species (with an identified ''spreading speed''), but also provide a sharp description of the profile of the solution, thus shedding light on a new ''bump phenomenon''.

[ Reference URL ]We consider the reaction-diffusion competition system u_t=u_{xx}+u(1-u-v), v_t=dv_{xx}+rv(1-v-u), which is the so-called critical case. The associated ODE system then admits infinitely many equilibria, which makes the analysis quite intricate. We first prove the non-existence of monotone traveling waves by applying the phase plane analysis. Next, we study the long time behavior of the solution of the Cauchy problem with a compactly supported initial datum. We not only reveal that the ''faster'' species excludes the ''slower'' species (with an identified ''spreading speed''), but also provide a sharp description of the profile of the solution, thus shedding light on a new ''bump phenomenon''.

https://forms.gle/LHj5mVUdpQ3Jxkrd6

### 2021/06/17

### 2021/04/22

16:30-18:00 Online

Relaxation of Optimal Transport problem on finite state space via Bregman divergence (Japanese)

[ Reference URL ]

https://forms.gle/yg9XZDVdxYG6qMos8

**( )**Relaxation of Optimal Transport problem on finite state space via Bregman divergence (Japanese)

[ Reference URL ]

https://forms.gle/yg9XZDVdxYG6qMos8

### 2021/04/15

### 2020/11/05

16:00-17:30 Room #オンライン開催 (Graduate School of Math. Sci. Bldg.)

Hölder gradient estimates on L^p-viscosity solutions of fully nonlinear parabolic equations with VMO coefficients (Japanese)

https://docs.google.com/forms/d/e/1FAIpQLSf4Rmd6B0m9_t_-xdy2hT1ZC1Ziz2qEc3yLRCQNZBilAOB1Ag/viewform?usp=sf_link

**( )**Hölder gradient estimates on L^p-viscosity solutions of fully nonlinear parabolic equations with VMO coefficients (Japanese)

[ Abstract ]

We discuss fully nonlinear second-order uniformly parabolic equations, including parabolic Isaacs equations. Isaacs equations arise in the theory of stochastic differential games. In 2014, N.V. Krylov proved the existence of L^p-viscosity solutions of boundary value problems for equations with VMO (vanishing mean oscillation) “coefficients” when p>n+2. Furthermore, the solutions were in the parabolic Hölder space C^{1,α} for 0<α<1. Our purpose is to show C^{1,α} estimates on L^p-viscosity solutions of fully nonlinear parabolic equations under the same conditions as in Krylov’s result.

[ Reference URL ]We discuss fully nonlinear second-order uniformly parabolic equations, including parabolic Isaacs equations. Isaacs equations arise in the theory of stochastic differential games. In 2014, N.V. Krylov proved the existence of L^p-viscosity solutions of boundary value problems for equations with VMO (vanishing mean oscillation) “coefficients” when p>n+2. Furthermore, the solutions were in the parabolic Hölder space C^{1,α} for 0<α<1. Our purpose is to show C^{1,α} estimates on L^p-viscosity solutions of fully nonlinear parabolic equations under the same conditions as in Krylov’s result.

https://docs.google.com/forms/d/e/1FAIpQLSf4Rmd6B0m9_t_-xdy2hT1ZC1Ziz2qEc3yLRCQNZBilAOB1Ag/viewform?usp=sf_link

### 2020/10/08

16:00-17:30 Room #オンライン開催 (Graduate School of Math. Sci. Bldg.)

(Japanese)

[ Reference URL ]

https://docs.google.com/forms/d/e/1FAIpQLSd7MT077191TeM4aQzeo2hK9Bqn6HQudr3pjLRdmEqND2heqQ/viewform?usp=sf_link

**( )**(Japanese)

[ Reference URL ]

https://docs.google.com/forms/d/e/1FAIpQLSd7MT077191TeM4aQzeo2hK9Bqn6HQudr3pjLRdmEqND2heqQ/viewform?usp=sf_link

### 2019/12/19

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

### 2019/10/31

16:00-17:30 Room #128 (TBD) (Graduate School of Math. Sci. Bldg.)

Behaviour around the isolated singularity for solutions of some nonlinear elliptic inequalities and systems (English)

**Marius Ghergu**(University College Dublin)Behaviour around the isolated singularity for solutions of some nonlinear elliptic inequalities and systems (English)

[ Abstract ]

We present some results on the behaviour around the isolated singularity for solutions of nonlinear elliptic inequalities driven by the Laplace operator. We derive optimal conditions that imply either a blow-up or the existence of pointwise bounds for solutions. We obtain that whenever a pointwise bound exists, then an optimal bound is given by the fundamental solution of the Laplace operator. This situation changes in case of systems of inequalities where other types of optimal bounds may occur. The approach relies on integral representation of solutions combined with various nonlinear potential estimates. Further extensions to the parabolic case will be presented. This talk is based on joint works with S. Taliaferro (Texas A&M University) and I. Verbitsky (Missouri University).

We present some results on the behaviour around the isolated singularity for solutions of nonlinear elliptic inequalities driven by the Laplace operator. We derive optimal conditions that imply either a blow-up or the existence of pointwise bounds for solutions. We obtain that whenever a pointwise bound exists, then an optimal bound is given by the fundamental solution of the Laplace operator. This situation changes in case of systems of inequalities where other types of optimal bounds may occur. The approach relies on integral representation of solutions combined with various nonlinear potential estimates. Further extensions to the parabolic case will be presented. This talk is based on joint works with S. Taliaferro (Texas A&M University) and I. Verbitsky (Missouri University).

### 2019/10/24

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

### 2019/06/20

16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

### 2019/04/25

16:00-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

The porous medium equation on noncompact Riemannian manifolds with initial datum a measure

(English)

On sharp large deviations for the bridge of a general diffusion

(English)

**Matteo Muratori**(Polytechnic University of Milan) 16:00-17:00The porous medium equation on noncompact Riemannian manifolds with initial datum a measure

(English)

[ Abstract ]

We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds. We show existence of solutions that take a finite Radon measure as initial datum, possibly sign-changing. We then prove uniqueness in the class of nonnegative solutions, upon assuming a quadratic lower bound on the Ricci curvature. Our result is "optimal" in the sense that any weak solution necessarily solves a Cauchy problem with initial datum a finite Radon measure. Moreover, as byproducts of the techniques we employ, we obtain some new results in potential analysis on manifolds, concerning the validity of a modified version of the mean-value inequality for superharmonic functions and related properties of potentials of positive Radon measures. Finally, we briefly discuss some work in progress regarding stability of the porous medium equation with respect to the Wasserstein distance, on Riemannian manifolds with Ricci curvature bounded below.

We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds. We show existence of solutions that take a finite Radon measure as initial datum, possibly sign-changing. We then prove uniqueness in the class of nonnegative solutions, upon assuming a quadratic lower bound on the Ricci curvature. Our result is "optimal" in the sense that any weak solution necessarily solves a Cauchy problem with initial datum a finite Radon measure. Moreover, as byproducts of the techniques we employ, we obtain some new results in potential analysis on manifolds, concerning the validity of a modified version of the mean-value inequality for superharmonic functions and related properties of potentials of positive Radon measures. Finally, we briefly discuss some work in progress regarding stability of the porous medium equation with respect to the Wasserstein distance, on Riemannian manifolds with Ricci curvature bounded below.

**Maurizia Rossi**(University of Pisa) 17:00-18:00On sharp large deviations for the bridge of a general diffusion

(English)

[ Abstract ]

In this talk we provide sharp Large Deviation estimates for the probability of exit from a domain for the bridge of a d-dimensional general diffusion process X, as the conditioning time tends to 0. This kind of results is motivated by applications to numerical simulation. In particular we investigate the influence of the drift b of X. It turns out that the sharp asymptotics for the exit time probability are independent of the drift, provided b enjoyes a simple condition that is always satisfied in dimension 1. On the other hand, we show that the drift can be influential if this assumption is not satisfied. This talk is based on a joint work with P. Baldi and L. Caramellino.

In this talk we provide sharp Large Deviation estimates for the probability of exit from a domain for the bridge of a d-dimensional general diffusion process X, as the conditioning time tends to 0. This kind of results is motivated by applications to numerical simulation. In particular we investigate the influence of the drift b of X. It turns out that the sharp asymptotics for the exit time probability are independent of the drift, provided b enjoyes a simple condition that is always satisfied in dimension 1. On the other hand, we show that the drift can be influential if this assumption is not satisfied. This talk is based on a joint work with P. Baldi and L. Caramellino.

### 2018/11/15

16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Inhomogeneous Dirichlet-boundary value problem for one dimensional nonlinear Schr\"{o}dinger equations (Japanese)

**Nakao Hayashi**(Osaka University)Inhomogeneous Dirichlet-boundary value problem for one dimensional nonlinear Schr\"{o}dinger equations (Japanese)

[ Abstract ]

We consider the inhomogeneous Dirichlet-boundary value problem for the cubic nonlinear Schr\"{o}dinger equations on the half line. We present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to equations by using the classical energy method and factorization techniques.

We consider the inhomogeneous Dirichlet-boundary value problem for the cubic nonlinear Schr\"{o}dinger equations on the half line. We present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to equations by using the classical energy method and factorization techniques.

### 2018/10/11

16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

(Japanese)

**Takahito Kashiwabara**(University of Tokyo)(Japanese)

### 2018/10/04

16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

(Japanese)

**Hiroko Yamamoto**(University of Tokyo)(Japanese)

### 2018/07/19

16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Uniqueness and nondegeneracy of ground states to scalar field equation involving critical Sobolev exponent

(Japanese)

**Norihisa Ikoma**(Keio University)Uniqueness and nondegeneracy of ground states to scalar field equation involving critical Sobolev exponent

(Japanese)

[ Abstract ]

This talk is devoted to studying the uniqueness and nondegeneracy of ground states to a nonlinear scalar field equation on the whole space. The nonlinearity consists of two power functions, and their growths are subcritical and critical in the Sobolev sense respectively. Under some assumptions, it is known that the equation admits a positive radial ground state and other ground states are made from the positive radial one. We show that if the dimensions are greater than or equal to 5 and the frequency is sufficiently large, then the positive radial ground state is unique and nondegenerate. This is based on joint work with Takafumi Akahori (Shizuoka Univ.), Slim Ibrahim (Univ. of Victoria), Hiroaki Kikuchi (Tsuda Univ.) and Hayato Nawa (Meiji Univ.).

This talk is devoted to studying the uniqueness and nondegeneracy of ground states to a nonlinear scalar field equation on the whole space. The nonlinearity consists of two power functions, and their growths are subcritical and critical in the Sobolev sense respectively. Under some assumptions, it is known that the equation admits a positive radial ground state and other ground states are made from the positive radial one. We show that if the dimensions are greater than or equal to 5 and the frequency is sufficiently large, then the positive radial ground state is unique and nondegenerate. This is based on joint work with Takafumi Akahori (Shizuoka Univ.), Slim Ibrahim (Univ. of Victoria), Hiroaki Kikuchi (Tsuda Univ.) and Hayato Nawa (Meiji Univ.).

### 2018/05/24

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Sign-changing solutions for a one-dimensional semilinear parabolic problem (Japanese)

**Eiji Yanagida**(Tokyo Institute of Technology)Sign-changing solutions for a one-dimensional semilinear parabolic problem (Japanese)

[ Abstract ]

This talk is concerned with a nonlinear parabolic equation on a bounded interval with the homogeneous Dirichlet or Neumann boundary condition. Under rather general conditions on the nonlinearity, we consider the blow-up and global existence of sign-changing solutions. It is shown that there exists a nonnegative integer $k$ such that the solution blows up in finite time if the initial value changes its sign at most $k$ times, whereas there exists a stationary solution with more than $k$ zeros. The proof is based on an intersection number argument combined with a topological method.

This talk is concerned with a nonlinear parabolic equation on a bounded interval with the homogeneous Dirichlet or Neumann boundary condition. Under rather general conditions on the nonlinearity, we consider the blow-up and global existence of sign-changing solutions. It is shown that there exists a nonnegative integer $k$ such that the solution blows up in finite time if the initial value changes its sign at most $k$ times, whereas there exists a stationary solution with more than $k$ zeros. The proof is based on an intersection number argument combined with a topological method.

### 2017/12/21

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

### 2017/12/14

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain

(English)

**I-Kun, Chen**(Kyoto University)Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain

(English)

[ Abstract ]

We consider the diffuse reflection boundary problem for the linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or Maxwellian molecular gases in a $C^2$ strictly convex bounded domain. We obtain a pointwise estimate for the derivative of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. Velocity averaging effect for stationary solutions as well as observations in geometry are used in this research.

We consider the diffuse reflection boundary problem for the linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or Maxwellian molecular gases in a $C^2$ strictly convex bounded domain. We obtain a pointwise estimate for the derivative of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. Velocity averaging effect for stationary solutions as well as observations in geometry are used in this research.