Shigeru Sakaguchi

The main purpose is to know geometric properties of solutions of partial differential equations. Since solutions are functions, it is natural to want to know their shapes and geometric properties. The current research topics are the following.

(1) Stationary level surfaces of solutions of diffusion equations: To know the shapes of graphs of functions, one may begin by investigating their level surfaces. An isothermic surface of the solution of the heat equation is called stationary if its temperature depends only on time. The existence of a stationary isothermic surface is deeply related to the symmetry of the heat conductor. The right helicoid, the circular cylinder, the sphere and the plane are examples of stationary isothermic surfaces in Euclidean 3-space. The characterization of the circular cylinder, the sphere and the plane by using stationary isothermic surfaces in Euclidean 3-space is almost completed, and in particular, similar good characterization of the right helicoid is wanted.

(2) Problems of partial differential equations on composite materials: Recently, we considered the heat diffusion over composite media and we got a characterization of planar layers by using either stationary isothermic surfaces or surfaces with the constant flow property among multi-layered heat conductors. In particular, we are interested in problems dealing with composite materials.

(3) Interaction between diffusion and geometry of domain: The shape of the heat conductor is deeply related to the initial heat diffusion. Diffusion equations we consider are the heat equation, the porous medium type equation, and their related equations.

(4) Shapes of solutions of elliptic equations: In general, solutions of elliptic equations describe steady states after a sufficiently long time. Liouville-type theorems characterize hyperplanes as graphs of entire solutions with some restriction. Overdetermined boundary value problems characterize some symmetrical domains. Isoperimetric inequalities accompanied by boundary value problems characterize shapes of the solutions which give the equalities.

(5) The point of view of inverse problems: Partial differential equations appear in models describing natural phenomena. There are many interesting problems which characterize some geometry in some reasonable way from the point of view of inverse problems.