The Constant Rank Theorem Of Spacetime Convex Solutions Of Parabolic Equations

The convexity of the solutions is an important field in the study of partial differential equations. Naturally we want to discuss the spacetime convexity of the solutions of parabolic equations. The key of the microscopic convexity method in the partial differential equations is establishing constant rank theorem. However, the results about the constant rank theorem for parabolic equations, which have been obtained, are only with respect to the spatial Hessian of the solutions. In this dissertation, we concentrate on the spacetime Hessian of the solutions, and establish a constant rank theorem for the spacetime convex solutions of the heat equation in Euclidean space by using strong maximum principle, then we generalize the results to the Riemannian manifolds and the Kahler manifolds. At the same time, we give the structure conditions which take the constant rank theorem hold on for the fully nonlinear equations. Moreover, we discuss the constant rank property for the spacetime second fundamental forms of the mean curvature flow. We list the main results of this dissertation as below:Constant rank theorem for the spacetime convex solutions in Euclidean spaceTheorem 0.1. LetΩbe a domain in Rn, and u∈C3,1(Ω×[0,T)) be a spacetime convex solution of the heat equation ut=△u. If D2u attains its minimum rank at some point (x0,t0)∈Ω×(0,T), then the rank of D2u is constant onΩ×(0,t0]. Moreover, let l(t) be the minimum rank of D2u inΩfor fixed t, then l(t1)≤l(t2) for all 0<t1≤t2<T.Theorem 0.2. LetΩbe a domain in Rn, and v∈C3(Ω×[0,T)) be a spacetime convex solution of the equation svs+2△v+y·▽v-2v|▽v|2=0. If D2v attains its minimum rank at some point(y0,s0)∈Ω×(0,T),then the rank of D2v is constant onΩ×(0,s0].Moreover, Let l(s) be the minimum rank of D2v inΩfor fixed s, then l(s1)≤l(s2) for all 0<s1≤s2<T.Theorem 0.3.LetΩbe a domain in Rn, and F=F(A,p,u,x,t)∈C2,1(Sn×Rn×R×Ω×[0,T)) satisfies stucture conditions:ⅰ) F(A,p,u,x,t)is elliptic with respect to A;ⅱ) F(A-1,p,u,x,t)is locally convex in(A,u,x,t) for each p;ⅲ) for any constant C,ΓFC={(A,u,x,t):F(A,p,u,x,t)≤C} is convex. Let u∈C3,1(Ω×[0,T)) be a spacetime convex solution of the equation ut=F(▽2u,▽u,u,x,t), then for any t∈(0,T),the rank of D2u is constant inΩ.Moreover,let l(t) be the minimum rank of D2u in for fixed t∈(0,T), then l(t1)≤l(t2) for all 0<t1≤t2<T.Constant rank theorem for the spacetime convex solutions of the heat equation on Riemannian manifoldsTheorem 0.4.Let M 6e an n-dimension compact Riemannian manifolds with nonnegative sectional curvature, and satisfies Ricci parallel. Let u∈C3,1(M×[0,T))be a spacetime convex solution of the heat equation ut=△u on M.If D2u attains its minimum rank at some point(x0,t0)∈M×(0,T),then the rank of D2u is constant on M×(0,t0].Moreover, let l(t) be the minimum rank of D2u in M for fixed t, then l(t1)≤l(t2) for all 0<t1≤t2<T.Constant rank theorem for the spacetime convex solutions of the heat equation on Kahler manifolds Therem 0.5.Let M be a compact Kahler manifold with nonnegative holomorphic bisectional curvature,which has complex dimension n.Let u∈C3,1(M×[0,T)) be a spacetime convex solution of heat equation ut=△u on M.If D2u attains its minimum rank at some point(z0,t0)u∈M×(0,T),then the rank of D2u is constant on M×(0,t0].Moreover,let l(t) be the minimum rank of D2u in M for fixed t, then l(t1)≤2(t2), for all 0<t1≤t2<TConstant rank theorem for the spucetime second fundamental forms of the mean curvature flowTheorem 0.6.Let compact hypersurface Mt (?) Rn+1 be a smooth solution of the mean curvatur flow ((?)X)/((?)t)=-Hn. on [0,T),and the matrix is positive semi-definite on M×[0,T),then the rank of h is n or n+1 on M×(0,T). Moreover, if the rank of h takes n at some point (x0,t0)∈M×(0,T),then for any 0<t≤t0,the rank of h always be n on M.