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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. In particular, 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. Similar good characterization of the right helicoid is wanted.
(2) Interaction between diffusion and geometry of domain: We consider diffusion equations. In the problems where the initial value equals zero and the boundary value is positive, the short-time behavior of solutions is deeply related to the curvatures of the boundary. Diffusion equations we consider are the heat equation, the porous medium type equation, the p-Laplace diffusion equation, and their related equations.
(3) 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 reasonable 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.
(4) The point of view of inverse problems: Partial differential equations appear in models describing natural phenomena. It is an interesting problem that characterizes some geometry in some reasonable way from the point of view of inverse problems. Very recently, we got an affirmative answer to an inverse problem which determines the spherical shell by using the conductivity equation in Euclidean 3-space.
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