Abstract : In the advanced sciences and technologies, singularity has been one of characteristic keywords of complex and dynamic nonlinear phenomena, such as phase transitions, crystallization processes, image denoising processes, and so on. Also, in recent years, the theoretical/numerical methods to deal with such singularity have been developed by a lot of researchers, from various viewpoints. The objective of this mini-symposium is to let wide range of experts of this field meet together, and to exchange the latest hot topics on the mathematical models of nonlinear phenomena, such as solvability, regularities, stability, optimizations, and so on.
Organizer(s) : Ken Shirakawa, Salvador Moll, Hiroshi Watanabe
Marcos Solera Diana (Universidad Autónoma de Madrid/Universitat de València)
Abstract : We obtain existence of minimizers for the $p$-capacity functional defined with a symmetric anisotropy for $1 < p<\infty$ and the associated Euler-Lagrange equation. After a change of variables and letting $p\downarrow 1$ we are led to the existence of solutions to the elliptic PDE associated with the level set formulation of the crystalline inverse mean curvature flow.
[04365] Pseudo-parabolic model of grain boundary motion coupled with solidification effect
Format : Talk at Waseda University
Author(s) :
Daiki Mizuno (Chiba University)
Ken Shirakawa (Chiba University)
Abstract : In this talk, we consider a coupled nonlinear system, which consists of an Allen-Cahn type equation and a pseudo-parabolic KWC type system. The system is based on the $\phi$-$\eta$-$\theta$ model of grain boundary motion with solidification (cf. RIMS Kokyuroku, 1210, 2001). Under suitable assumptions, the mathematical results concerned with the well-posedness, including open question of uniqueness, and fine-regularity of the solution will be discussed in the Main Theorems of this talk.
[04655] Elliptic problems involving a Hardy potential
Format : Talk at Waseda University
Author(s) :
Alexis Molino (University of Almeria)
Abstract : In this talk we consider different elliptic differential equations with a potential Hardy type singularity in a domain and Dirichlet conditions on the boundary. Specifically, the regularizing effect of lower order terms is revealed, as well as the existence of solutions beyond the well-known Hardy constant.
[04200] Solvability of a phase-field model of 3D-grain boundary motion
Format : Talk at Waseda University
Author(s) :
Salvador Moll (Universitat de Valencia)
Ken Shirakawa (Chiba University)
Hiroshi Watanabe (Oita University)
Abstract : We consider a phase-field model of 3D-grain boundary motion. The model is based on the three dimensional Kobayashi--Warren model for the dynamics of polycrystals. To formulate our 3D-model, we use a quaternion formulation for the orientation variable.
In this talk, we obtain existence of solutions to the $L^2$-gradient descent flow of the constrained energy functional via several approximating problems. Moreover, we also obtain an invariance principle for the orientation variable.
[03754] Variational models for segmentation in non-euclidian settings
Format : Talk at Waseda University
Author(s) :
Salvador Moll (Universitat de Valencia)
Abstract : I will present some new results on image segmentation in the general framework of perimeter measure spaces; including the anisotropic case and non-euclidian settings such as random walk spaces or metric graphs.
I will show the linkage between the ROF model for denoising and the two phases piecewise constant segmentation and I will show different applications of the results to nonlocal image segmentation, via discrete weighted graphs, and to multiclass classification on high dimensional spaces.
[04356] Periodic solution to KWC-type system under dynamic boundary condition
Format : Talk at Waseda University
Author(s) :
Ryota Nakayashiki (Salesian Polytechnic)
Ken Shirakawa (Chiba University)
Abstract : The aim of this study is to observe the periodic stability for KWC-type system of grain boundary motion under dynamic boundary condition. The KWC-type system is a collective term of PDE model of grain boundary motion (cf. Kobayashi et al, Physica D 140, 2000), which is governed by variable-dependent singular diffusion equation. As one of key-results of the study, we will prove the main theorem concerned with the existence of periodic solution.
[03968] Cahn-Hilliard equations with forward-backward dynamic boundary condition and non-smooth potentials
Format : Talk at Waseda University
Author(s) :
Pierluigi Colli (Università degli Studi di Pavia)
Takeshi Fukao (Ryukoku University)
Luca Scarpa (Politecnico di Milano)
Abstract : The asymptotic behavior as the coefficient of the surface diffusion acting on the boundary phase variable goes to 0 is investigated. By this analysis we obtain a forward-backward dynamic boundary condition at the limit. We can deal with a general class of potentials having a double-well structure, including the non-smooth double-obstacle potential. We illustrate that the limit problem is well-posed by also proving a continuous dependence estimate
[04411] Optimal control for shape memory alloys of the simplipied Fr'emond model in the one-dimensional case
Format : Talk at Waseda University
Author(s) :
Noriaki Yamazaki (Kanagawa University)
Ken Shirakawa (Chiba University)
Pierluigi Colli (Universita a degli Studi di Pavia)
M. Hassan Farshbaf-Shaker (Weierstrass Institute for Applied Analysis and Stochastics)
Abstract : In this talk, we consider optimal control problems for the one-dimensional Fremond model for shape memory alloys.
Then, we prove the existence of an optimal control that minimizes the cost functional for a nonlinear and nonsmooth
state problem. Moreover, we show the necessary condition of the optimal pair by using optimal control problems for
approximating systems.
[03661] Geometric convergence in regularization of inverse problems
Format : Talk at Waseda University
Author(s) :
Jose A. Iglesias (University of Twente)
Gwenael Mercier (University of Vienna)
Kristian Bredies (University of Graz)
Otmar Scherzer (University of Vienna)
Abstract : We present some results bridging classical regularization theory of ill-posed inverse problems and regularity properties of almost-minimizers of the corresponding regularization energies. In the regime of vanishing noise and regularization parameter, we obtain results of convergence in Hausdorff distance of level sets of minimizers (which can be interpreted as objects to be recovered in an imaging context) and uniform $L^\infty$ bounds. These hold both for the classical total variation, and for some fractional energies.
[04617] Numerical algorithms for optimization problems of grain boundary motions
Format : Talk at Waseda University
Author(s) :
Shodai Kubota (National Institute of Technology, Miyakonojo College)
Ken Shirakawa (Chiba University)
Makoto Okumura (Konan University)
Abstract : We consider a class of optimal control problems for state problems of one-dimensional systems. Each state problem is associated with the phase-field model of grain boundary motion, proposed by Ryo Kobayashi et al. In this regard, each optimal control problem is prescribed as a minimization problem of a cost. Under suitable assumptions, the convergence of numerical algorithms for optimization problems governed by state systems will be reported as the main theorem of this talk.
[04370] Temperature optimization problems governed by pseudo-parabolic model of grain boundary motion
Format : Talk at Waseda University
Author(s) :
Ken Shirakawa (Chiba University)
Daiki Mizuno (Chiba University)
Abstract : In this talk, we consider a class of optimal temperature control problems governed by pseudo-parabolic PDE systems. The PDE systems are based on the KWC-model of grain boundary motion (cf. Kobayashi et al, Physica D, 140, 2000). Under suitable assumptions, we will focus on the Main Theorems, concerned with: the mathematical solvability and parameter dependence of pseudo-parabolic PDE systems and optimal controls; and the first-order necessary optimality conditions for the optimal control problems.
[04841] On well-posedness of 1-harmonic map flows
Format : Online Talk on Zoom
Author(s) :
Lorenzo Giacomelli (Sapienza University of Rome)
Michal Lasica (Institute of Mathematics of the Polish Academy of Sciences)
Salvador Moll (Universitat de ValenciaUniversity of Valencia)
Abstract : We look at the formal gradient flow of the total variation of a manifold-valued unknown function. After introducing the problem and the state-of-the-art, I will discuss recent results with M. Lasica and S. Moll concerning local/global-in-time well-posedness of Lipschitz solutions and global existence of BV-solutions for one-dimensional domains. Uniqueness, gradient flow structures, and open questions will also be discussed.
