Gradient Estimates For The Neumann Problem Of Mean Curvature Equation

In the theory of second order elliptic equations, the study of boundary value prob-lems is one of the most important questions which are posed in the theory of elliptic equations. The boundary value problems for second order elliptic equations, at least for now, are mainly of the following types:Dirichlet problem, Neumann problem, oblique derivative problem. For the study of boundary value problems of prescribed mean curvature equations, Serrin etc. studied the Dirichlet problem and achieved the fundamental result, the details also can be seen in the book of Gilbarg-Trudinger. The oblique derivative problem of the prescribed mean curvature equations has also been studied by Ural’tseva, Simon-Spruck, Gerhardt etc., and they achieved existence results. However, so far, the Neumann problem of prescribed mean curvature equations has yet not been studied.In page 360 of the book written by Lieberman, it is posed that the boundary gradient estimate of prescribed mean curvature equations is still open. In this thesis, combining the techniques of Spruck, Lieberman, Xu-Jia Wang etc., we main-ly prove the boundary gradient estimates for the Neumann problem of prescribed mean curvature equations under the condition that the solution is bounded. Consequently, we can obtain the solvability of a class of mean curvature equations with Neumann bound-ary condition. Meanwhile, we can give a new proof of the boundary gradient estimates for the prescribed contact angle problem of prescribed mean curvature equation.Besides that, we can also show that the boundary gradient estimates for the Neu-mann problem Hessian equations in Rn.More precisely, the main results of this thesis are as follows.I. Boundary gradient estimates for the Neumann problem of prescribed mean curvature equationsLet Ω C Rn be a bounded domain in Rn, n>2,(?)Ω∈C3. Denote byConsider the following Neumann problem of prescribed mean curvature equation: where Ω(?)Rn is a bounded domain, n≥2, (?)Ω∈C3,γ is the inner unit normal vector of (?)Ω. f,ψ are given functions defined on Ω x R and (?)Ω× R respectively. We assume further there exist positive constants M0, L1, L2 such thatOur main results are the following theorems. 定理0.1. Suppose u ∈ C2(Ω) ∩ C3(Ω) is a solution of the problem (11), (12) satisfy-ing (13). If f,ψ satisfy the conditions (14), (15), (16) respectively, then there exists a small positive constant μ0 such that where M1 is a positive constant depending only on n, μ0, M0, L1, which is from the inte-rior gradient estimates; M2 is a positive constant depending only on n, Ω,μ0, M0, L1, L2. Therefore, we can obtain an existence result of the following problem. 定理 0.2. Let Ω (?) Rn be a bounded domain, n≥ 2, (?)Ω ∈ C3.γ is the inner unit normal vector of (?)Ω. If for 0<α<1,ψ(x) ∈ C3,α(Ω), then the following problem exists a unique C2(Ω) solution.At the same time, we can give a new proof of the following theorem. This the-orem was first given by Ural’ tseva and it can also be seen in Simon-Spruck, Gerhardt. The detailed proof can be seen in the book written by Lieberman. 定理0.3. Let Ω (?) Rn be a bounded domain, n≥ 2,(?)Ω∈C3.γ is the inner unit normal vector of (?)Ω.θ is a given function defined on (?)Ω. Suppose u ∈ C2(Ω) ∩ C3(Ω) is a solution of the following problem satisfying the condition (13). If f satisfies the conditions (14), (15), and θ(x) satisfies θ(x) ∈ C2(Ω),θ(x) E (0, π),(?) constant b0, L2 such that then there exists a small positive constant μ0 such that where M1 is a positive constant depending only on n,μ0, M0, L1, which is from the inte-rior gradient estimates; M2 is a positive constant depending only on n,Ω,μ0, M0, L1, L2, inf x∈Ω sin2 θ.Ⅱ. Boundary gradient estimates for the Neumann problem of Hessian equa-tionsIn the following, we shall give our another result.定理 0.4. Let Ω (?) Rn be a bounded domain, (?)Ω ∈ C3,γ is the inner unit normal vector of (?)Ω. Suppose u ∈ C2(Ω) ∩ C3(ΩQ) is an admissible solution of the following Neumann problem of Hessian equation satisfying|u|< M0.f, ψ are given functions defined on Ω×[-M0, M0] and Ω respec-tively. If f,ψ satisfy the conditions:(?)positive constants L0, L1, L2 such that then there exists a small positive constant no such thatwhere M1 is a positive constant depending only on n,μ0, M0, L0, L1, which is from the interior gradient estimates; M2 is a positive constant depending only on n,Ω,μ0, M0, L0, L1,L2.