A variety of theorems named the Liouville property play fundamental roles in modern theory of mathematical analysis. Understanding each of them individually is not so difficult nowadays. What are they pointing toward in their totality, however? This monograph asserts that self-organization may be an answer, searching recent studies on linear and nonlinear partial differential equations. Critical breakthroughs have emerged in accordance with the widely spread classical ideas of J. Liouville in various forms, that is, the spectral, elliptic, surface, and transport theories, in turn. A fusion of them is seen in recently developing research fieldsmathematical oncology, non-equilibrium statistical mechanics, dynamical theory, and shape optimizationwhich taken together may be called the potential for self-organization.

Addresses most recent nonlinear analysis to Liouville's theory in a variety of areas in classical analysis Presents new insight into mathematical science based on partial differential equations Enables readers to expand their knowledge, skill, motivation, and talent to related but unknown scientific regions

Autorentext

Takashi Suzuki is a Vice-Director of the Center for Mathematical Modeling and Data Science at Osaka Univeristy.

Inhalt

1 Spectral Theory.- 2 Surface Theory.- 3 Elliptic Theory: A Classical and Modern Study.- 4 Transformation Theory.- 5 Potentials of Self-Organization: A Fusion.