Symmetry of An Integral System with Risez Potential in the Cylinder Type Domain

This paper studies the overdetermined problem for the integral system with Riesz potential in cylinder type domain. On the one hand, we show that the cylinder type domain is indeed a cylinder and the solution is also rotationally symmetric with respect to the cylinder axis provide that the standard overdetermined problem admits a positive solution. On the other hand, under some requirements on the geometry for the partial boundary, we illustrate that the partial overdetermined problem also satisfies the symmetry of the domain and the solution. This paper first shows the symmetry of integral system in the unbounded cylinder type domain, which generalized the similar results for the single equation.