Equivalence between a Boundary Value Problem with Obstacles on the Boundary and a Variational Problem

Contact problems widely exist in diverse fields. Many contact problems are induced to boundary values and variational problems. Boundary variational inequality methods have been playing crucial roles in contact problems fields, which induce all the boundary conditions and contact conditions to one variational inequality and make the theoretical analysis very easy. Variational problem is a bridge to solve boundary values by variational inequality. In the paper, a function is derived on the base of the functional of potential energy, and their equivalence between a boundary value problem with obstacles on the boundary and equal variational problem is proved. So the partial differential equation´s boundary value with obstacles can be solved by the corresponding variational problem.