Two Problems Related To Mean Curvature Equations

Mean curvature equation is very important in differential geometry and partial differential equation.In recent years,many scholars studied this kind of equations,and the most popular ones are the mean curvature flow equation and the minimal graph equation(You can look up articles in the literature list).Regarding the mean curvature flow equation,the long-term existence and convergence behavior of the solution have attracted much attention,such as [6],[12],[27],[32],[34],etc.As for the minimal graph equation,the convexity of its level set is favored by mathematicians,such as [24],[25],[30],[31],etc.It can be seen that the study of the mean curvature equation is very meaningful.Next,this paper divides into two chapters to study the two geometric properties of the mean curvature equation by constructing suitable auxiliary functions.Firstly,the first chapter explores the global gradient estimation of the solution of the mean curvature flow equation(1.1)under the condition of oblique derivative boundary value,and gives the long-time existence of solutions.Particularly,when the oblique derivatives have a constant value of zero,we generalize the results of Huisken’s prescribed perpendicular contact angle and prove the asymptotic behavior of the mean curvature flow equation.After collating the above results,as follows:Theorem 1 Let Ω ? R~n be a bounded domain with (?),n≥ 2.Assume (?)and (?)satisfies fτ≥-k for some nonnegative constant k≥ 0,then for some α∈(0,1),there exists a unique (?) solution towhere β is a C~3 inward unit strictly oblique vector field along (?).Theorem 2 Let Ω? Rn be a smooth bounded domain,n≥ 2.β is a smooth inward unit strictly oblique vector field along (?),then the solution of the following mean curvature flow will converge to a constant function as the time goes to infinitywhere γ is the downward unit normal fields on the graph of u and u0(x) is a given smooth function satisfying the compatible condition:u(0,β)=0 on (?).In the second chapter,we discuss minimal surface with strictly convex level sets in R2.By constructing an auxiliary function to explore the curvature of the level sets,we can get the convexity of the curvature function with respect to the height of the minimal surface.We also indicate that our result is optimal.The main results of this chapter are as follows:Theorem 3 Let (?),t0≤u(x)≤1 be a minimal graph defined on Ω,i.e.where Ω be a bounded smooth domain in R2.Suppose (?),letΓt={x∈Ω|u(x)=t} for t0<t<t1and K be the curvature of the level sets.Setandf(t)=min{κ(x)|x∈Γt},g(t)=max{κ(x)|x∈Γt}.If the level sets of u are strictly convex with respect to normal ▽u,we have:D2f(t)≤0 D2g(t)≥0,in(t0,t1)Corollary 1 Let u satisfywhere Ω0 and Ω1 are bounded smooth convex domains in R2,(?).For every point x∈Γt,0<t<1,we have the following estimates:and...