Overdetermined Problems In Elliptic Equations And Integral Equations

This thesis deals with overdetermined problems in elliptic equations and integral equations. Overdetermiend elliptie problem means that the equation satisfies both0-Dirichlet condition and constant Neumann condition on the same boundary. In general, overdetermiend elliptic problem has no solutions, unless there are suitably assumption imposed about the geometry of domain. Overdetermined problem in integral equations means the solution satisfies both the integral equation and constant boundary condition. Similarly, there is no solution for overdetermined problem in integral equations if we do not impose more assumption about the geometry of domain. The purpose of this thesis is to characterize the geometry of domain when overdetermined problem admits a solution.The thesis is composed of five chapters:In Chapter1, we summarize the backgrounds of related problems and state the main results of the thesis. We also give out some preliminary results used in the whole thesis.In Chapter2. we study the overdetermined problems with p-Finsler-Laplace in con-nected bounded domain Ω C Rn. If the overdetermined problem admits a solution, with the help of maximum principle for P-function and Pohozaev type identity, we prove that Ω has the Wulff shape.In Chapter3, the overdetermined problem with general degenerate ellipticity in con-nected bounded domain Ω C Rn is discussed. Using1-homogeneous of F and proper convolution argument, we overcome the degenerate come from F and A(t). then obtain the maximum principle for P-function. Further more, by Pohozaev type identity, we prove that Ω must be of Wulff shape if the overdetermined problem admits a solution.In Chapter4. we are interesting with overdetermined problems of some integral equations in a cylinder type domain Ω=D x R+C R+n. Firstly, we get the rotational symmetry of solution for an integral equation with Bessel potential in R+n. Furthermore, for the overdetermined problem of an integral equation with Bessel potential in Ω. we prove that Ω is actually a cylinder and the solution is rotationally symmetric about the cylinder axis. Finally, we also obtain the symmetry of both domain and solution for overdetermined problem of an integral equation with Riesz potential in Ω.In Cliapter5. partially overdetermined problems in integral equations are considered. We first study the partially overdetermined problem in bounded domain Ω(?) Rn and prove that Ω is ball and the solution is radially symmetry under some assumption about the geometry of partial boundary P(?)Ω We also discuss the partially overdetermined problem of an integral equation with Riesz potential in a cylinder type domain and obtain that the domain is actually a cylinder and the solution is rotationally symmetrie about the cylinder axis.