Diffusion and Geometry of Domain

描述扩散现象的偏微分方程已经被广泛地考虑。为了了解解的性质与区域几何之间的关系，我们给出了初始性质与边界曲率之间的关系，以及定常水平面的存在性与区域对称性之间的关系。特别地，我们得到了涉及静止水平面的球、超平面和圆柱的特征。这就产生了确定区域几何形状的反问题的新发展。另外，作为副产品，我们得到了Liouville型定理的粘性解的完全非线性椭圆型偏微分方程描述的一类重要的Weingarten超曲面。

Partial differential equations describing diffusion phenomena have been widely considered. To know the relationship between the behavior of solutions and the geometry of domain, we showed both the relationship between the initial behavior and the curvatures of the boundary and that between the existence of a stationary level surface with time and the symmetry of domain. In particular, we obtained characterizations of the sphere, the hyperplane, and the circular cylinder involving a stationary level surface. These yielded a new development of inverse problems determining the geometry of domain. Also, as a by-product, we obtained Liouville-type theorems for viscosity solutions of fully nonlinear elliptic partial differential equations describing an important class of Weingarten hypersurfaces.