The Role of Metrics in the Theory of Partial Differential

This volume is edited as the proceedings of the 11th Mathematical Society of Japan, Seasonal Institute (MSJ-SI) "The Role of Metrics in the Theory of Partial Differential Equations", held for the period of July 2–13, 2018 in Hokkaido University, Sapporo, Japan. This volume consists of peer-reviewed articles written by speakers who actively lead the fields of partial differential equations, numerical analysis, harmonic analysis, and geometric analysis. It should be underlined that 3 survey articles provided by course lecturers show an interesting direction of analysis under various metrics. This volume is suitable for any graduate students and young researchers to motivate the research on these fields from the viewpoint of metrics.

Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

• Course Lecturers:
  • Long Time Existence Results for Hamiltonian Non-Linear Klein-Gordon Equations on Some Compact Manifolds (Jean-Marc Delort)
  • Small Deformations of Spherical Biomembranes (Charles M Elliott, Luke Hatcher and Philip J Herbert)
  • The Thresholding Scheme for Mean Curvature Flow and De Giorgi's Ideas for Minimizing Movements (Tim Laux and Felix Otto)
• Invited Lecturers:
  • Stochastic Approaches to Lagrangian Coherent Structures (Sanjeeva Balasuriya)
  • Geometric Nonlinear Problems and GIT Stability Through Moment Maps (Akito Futaki and Hajime Ono)
  • On the Inclusion of Damping Terms in the Hyperbolic Mbo Algorithm (Elliott Ginder and Ayumu Katayama)
  • An Analytic Solution for a Gompertz-like Reaction–Diffusion Model for Tumour Growth (Bronwyn H Bradshaw-hajek and Philip Broadbridge)
  • Pinned Pulse Solutions in a Fitzhugh–Nagumo Model with a Bump-Type Heterogeneity (Peter Van Heijster and Takashi Teramoto)
  • Rigidity for Positive Ricci Curvature via Metric Measure Geometry (Shouhei Honda)
  • Blow-Up of Solutions to Semilinear Wave Equations with a Scaling Invariant Critical Damping (Masahiro Ikeda)
  • Asymptotic Behavior of the Energy Critical Nonlinear Wave Equation with a Special Scale Invariant Damping (Takahisa Inui)
  • Convergence of a Threshold-Type Algorithm for Curvature-Dependent Motions of Hypersurfaces (Katsuyuki Ishii)
  • Spectral Theory on Manifolds (Kenichi Ito and Erik Skibsted)
  • A Convergence of Generalized Lagrangian Mean Curvature Flow in Kähler Manifold with Positive Weighted Ricci Form (Toru Kajigaya and Keita Kunikawa)
  • Global Existence and the Asymptotic Behavior for Systems of Nonlinear Wave Equations Violating the Null Condition (Soichiro Katayama)
  • A Note on Traveling Waves for Area-preserving Geometric Flows (Takashi Kagaya and Yoshihito Kohsaka)
  • Uniqueness Problem for Closed Non-Smooth Hypersurfaces with Constant Anisotropic Mean Curvature (Miyuki Koiso)
  • Stability Properties of Blow Up Solutions for Critical Wave Maps (Joachim Krieger)
  • Weighted L² – L² Estimate for Wave Equation in R3 and Its Applications (Ning-an Lai)
  • On An Obstacle Problem Arising in Large Exponent Asymptotics for One Dimensional Fully Nonlinear Diffusions of Power Type (Qing Liu)
  • Singular Limit Problem for the Navier–Stokes Equations in a Curved Thin Domain (Tatsu-hiko Miura)
  • Lower Bound for the Lifespan of Solutions to the Generalized KdV Equation with Low-Degree of Nonlinearity (Hayato Miyazaki)
  • On Some Effects of Background Metrics for Several Partial Differential Equations (Makoto Nakamura)
  • Elastic Networks, Statics and Dynamics (Matteo Novaga and Alessandra Pluda)
  • Random Data Cauchy Problem for the Quadratic Nonlinear Schrödinger Equation without Gauge Invariance (Mamoru Okamoto)
  • Numerical Analysis of Constrained Total Variation Flows (Koya Sakakibara)
  • The Blow-Up Curve of Solutions for Semilinear Wave Equations with Dirichlet Boundary Conditions in One Space Dimension (Tetsuya Ishiwata and Takiko Sasaki)
  • Local Energy Decay and the Vector Field Method (Jacob Sterbenz)
  • Remark on Global Existence of Solutions to the 1D Compressible Euler Equation with Time-Dependent Damping (Yuusuke Sugiyama)
  • Short Time Blow-Up by Negative Mass Term for Semilinear Wave Equations with Small Data and Scattering Damping (Ning-An Lai, Nico Michele Schiavone and Hiroyuki Takamura)
  • Separability of Maxwell Equation in Rotating Black Hole Spacetime and Its Geometric Aspects (Tsuyoshi Houri, Norihiro Tanahashi and Yukinori Yasui)
  • Traveling Fronts in Balanced Bistable Reactiondiffusion Equations (Masaharu Taniguchi)
  • An Existence Theorem for Brakke Flow (Yoshihiro Tonegawa)
  • Equivalent Extensions of Partial Differential Equations on Surfaces (Lindsay Martin, Jay Chu and Richard Tsai)
  • Quasi-Local Mass and Isometric Embedding with Reference to a Static Spacetime (Mu-tao Wang)
  • Quasi-Periodic Solutions to Nonlinear PDEs (Wei-min Wang)
  • Simons Cone and Saddle Solutions of the Allen-Cahn Equation (Kelei Wang)
  • Harmonic Maps with Prescribed Singularities and Applications in General Relativity (Gilbert Weinstein)
  • Regularity of Minimal and CMC Stable Hypersurfaces: A Brief Overview (Neshan Wickramasekera)
  • A Novel Integral Transform Approach to Solving Hyperbolic Equations in the Curved Space-Times (Karen Yagdjian)
  • Harmonic Maps and the Einstein Equation (Sumio Yamada)
  • Global Regularity for Einstein–Klein Gordon System with U(1) x R Isometry Group (Haoyang Chen and Yi Zhou)