Overdetermined Problem Of Integral Equations

This paper considers the symmetry of global overdetermined problem and partial overdetermined problem for a system of integral equations with Riesz potential in both a bounded domain and a cylinder type domain of upper half-space.In Chapter 1,we introduce the research backgrounds,summarize the main works of this paper and some related preliminary results.Chapter 2 studies the partial overdetermined problem for a system of integral equations with Riesz potential in bounded domains.Under some natural integrability conditions on the positive solution to the partial overdetermined problem,we show that the domain is a ball,and that the solution is radially symmetric and monotone decreasing with respect to the radius.In chapter 3,the overdetermined problem for a system of integral equations with Riesz potential is considered in a cylinder type domain of upper half-space.First,for the global overdetermined problem,we obtain that the cylinder type domain is indeed a cylinder and that the solution is rotationally symmetric about the cylinder axis.Furthermore,the corresponding partial overdetermined problem is also studied.When the partial boundary condition satisfies some geometric characteristics,it is proved that the partial boundary condition is valid on the whole boundary.Thus,the partial overdetermined problem is indeed a global one,and the same symmetry of both the domain and solution are also obtained.