| The convexity is one of the basic properties of the objects in geometry.It can beexpressed in term of differentiation when the object is smooth.Hence,the study of convexity of an object not only is the issue in geometry,but also comes naturally into analysis.Furthermore,along with the development of the study of partial differential equation,people recognize the study of convexity becomes the need of the study of partial differentialequation itself.For example,in the study of free boundary problem(see text after).Nosurprisingly,the study of convexity has a long time in history,and tracts more and moreinteresting among the people.In this thesis,the author summarizes the history of the studyof convexity sketchily,and then give some estimation on the Gauss curvature of the levelset of some p-harmonic functions,we also give some estimation on the Gauss curvatureof the level set of 3 dimensional minimal hypersurface.On the other hand,the ConstantRank Theorem is a refined statement of convexity.It has become a powerful tool in thestudy of the geometric properties of solutions of partial differential equations.An attemptalso be take to the constant rank property of the level set of some prescribed mean curvature hypersurfaces.Besides,we find some harmonic or subharmonic functions relative tothe level sets on the 2 dimensional minimal hypersurface.Finally,we give a new proof ofthe nonexistent result of a class of Hessian inequalities by using the method of integrateby parts.The main results are as follows:定ç†0.0.1. LetΩbe a domain in R2,and u be a p-harmonic function inΩ,3/2≤p≤3,i.e.If|â–½u|≠0 and the level sets of u are all strictly convex,then,the Gauss curvature of thelevel sets of u can not attain the minimum inΩ,unless it's a constant。定ç†0.0.2. LetΩbe a domain in R3,and u be a p-harmonic function inΩ,p≥2,i.e. If|â–½u|≠0 and the level sets of u are all strictly convex,then,the Gauss curvature of thelevel sets of u can not attain the minimum inΩ,unless it's a constant。定ç†0.0.3. (For the notations see the following text)Let M2 be a minimal hypersurfacein R3 ,let the high function u of M2 corresponding to the directionξhave no criticalpoint,i.e.|â–½u|≠0,and let K be the curvature of level lines of u,G be the curvature ofsteepest descent of u,then(i)|â–½u|-1K,|â–½u|-1G are harmonic functions on M2.(ii)ln 1/|K|,ln 1/|G| are both subharmonic on M2.定ç†0.0.4.Let M3 be a minimal hypersurface in R4,let the high function u of M3corresponding to the directionξhave no critical point,and let the level sets of u be allstrictly convex.Then,the Gauss curvature of the level sets ofu can not attain the minimumin M3,unless it's a constant。推论0.0.5. Let M3 be a minimal graph over a convex ring in R3 with homogeneous Dirichlet boundary conditions,then the Gauss curvature of the level sets of M3 can not attain the minimum in M3.Let Mn be a smooth hypersurface in Rn+1 and X:M→Rn+1 be the immerse,satisfyingwhere H,N is the mean curvature and unit normal vector of Mn at X respectively,g>0 is a smooth function.Letξbe a fixed unit vector in Rn+1.Then the high function of Mncorresponding toξcan be expressed as u(X)=,here <·,·>means the usual Euclidean product in Rn+1.Now,the level set of Mn corresponding toξwith high t isWith the notations as above,the constant rank theorem on the level set of the prescribed mean curvature hypersurface is as following:定ç†0.0.6.Let Mn be as above the prescribed mean curvature hypersurface in Rn+1.If the high function u of Mn corresponding toξhave no critical point,and the level sets are all convex,i.e.their second fundamental forms are semidefinite,then the secondfundamental forms of all the level sets have constant rank,provided g-1/2 is concave.For the Hessian inequlities:whereσk(-D2u)are the k-Hessian of (-D2u) as usual.Consider the admissible solutions u∈Γk:={u∈C2(Rn)|σr(-D2u)>0,r=1,…,k},we have the followingnonexistent results:定ç†0.0.7.For 2k*:=nk/n-2k,then (?)α∈(-∞,k*],(0.0.5)have no positivesolution inΓk.
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