Study of path-behaviors of Biharmonic Psuedo Process with boundaries

点击翻译按钮获取中文摘要

-△ィイD12ィエD1 is called a biharamonic operator, which we often encounter in the theory of elasticity and fluid dynamics. Following pioneer works of Hochberg (1978) and Funaki (1979), many attempts are done to study the operator from the view point of probability theory.We consider 'particles' whose 'transition probability density' is the fundamental solution of ∂ィイD2tィエD2u = -△ィイD12ィエD1u, though it may have negative values. We call orbits of these 'particles' biharmonic pseudo process, or BPP in short. Our aim is to study relations between analytic boundary conditions of -△ィイD12ィエD1 and behaviors of BPP.By Nishioka (1997), we know that BPP behave as a mixture of two sorts of particles which we call 'monopoles' and 'dipoles'. So we need two boundary conditions to solve a boundary value problem related to -△ィイD12ィエD1, since each boundary condition controls behavior of each sort of particles at the boundary. We set up one of 'absorbing', 'sticking', and various 'reflecting' barriers on monopoles and dipoles one by one. Then their combinations solve various kinds of boundary value problems to -△ィイD12ィエD1.