Some Studies On Two Classical Problems Related To Geometric Partial Differential Equations

This thesis consists of two parts concerning two classical problems in the geometric partial differential equations.In the first part,we will study the capillary boundary problem,namely chapter 2 and chapter 3.Firstly we consider the classical capillary problem,under the condition that the domain is strictly convex and the contact angle is close to ?/2,we show the existence of a classical solution to the capillary problem.Secondly,we consider its parabolic counterpart,that is,studying the long-time existence and asymptotic behavior of mean curvature type flows with capillary boundary conditions.There are two cases depending on the supporting hypersurface:a cylinder(non-parametric mean curvature flow)and a unit ball(volume-preserving mean curvature type flow)in the Euclidean space.In the second part,we will study the relative isoperimetric problem,namely chapter 4 and chapter 5.Firstly,based on integrating some previous ideas,we reprove the rel-ative isoperimetric inequality for domains in Euclidean space.Secondly,we prove the relative isoperimetric inequality for domains in the general submanifold,which partially solve the open problem raised by Choe in 2005.Thirdly,based on the ABP method,we prove the weighted isoperimetric inequality for domains in general submanifold.Besides,we reprove the weighted Heintze-Karcher inequality and the weighted Reilly inequality using the ABP method.