Research on the Symmetry of Integral Equations with Dirichlet Boundary Conditions

In this paper, we consider the following integral equations with Dirichlet boundary conditions: \left\ \begingathered u_i(x) = \int\limits_R_ + ^n G_i(x,y)f_i(\boldsymbolu)\rmdy,\text i = 1,2, \cdots ,m \text \\ \boldsymbolu = (u_1,u_2, \cdots ,u_m) \\ \endgathered \right. where R_ + ^n is the n-dimensional upper half Euclidean space, G_i(x,y) is the Green’s function with Dirichlet boundary conditions in R_ + ^n , f_i(\boldsymbolu) ( i = 1,2, \cdots ,m ) is real-valued function. In this paper, assuming that the system of integral equations has a positive solution u_i that satisfies certain integrability in R_ + ^n , and using the method of moving planes in integral forms, we prove that the positive solutions u_i is rotationally symmetric about some line.