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 main research topics are the following.
(1) Stationary isothermic surfaces and stationary critical points
To know the shapes of graphs of functions, one may begin by investigating their level surfaces and critical points. In particular, an isothermic surface (a critical point) of the solution of the heat equation is called stationary if its temperature depends only on time (if it is invariant with time). The existence of a stationary isothermic surface (or a stationary critical point) is deeply related to the symmetry of the heat conductor. The right helicoid is an interesting example of stationary isothermic surfaces in the three-dimensional Euclidean space.
(2) Interaction between diffusion and geometry of domain
We consider linear and nonlinear diffusion equations (the heat equation, the porous medium equation, etc.). In the problem where the initial value equals zero and the boundary value equals 1, the short-time behavior of solutions is deeply related to the curvatures of the boundary.
(3) Shape of solutions of linear and nonlinear elliptic equations
In general, solutions of elliptic equations describe steady states after a sufficiently long time. We consider linear and nonlinear elliptic equations. Liouville-type theorems characterize hyperplanes as graphs of entire solutions with some reasonable restriction. Overdetermined boundary value problems characterize balls, ellipsoids, or some symmetrical domains in general.
(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.