Some Curvature Estimates Of Convex Level Sets Of Harmonic Function And Minimal Surface

In this dissertation, we discuss the smallest principal curvature estimates of convex level sets of three-dimensional harmonic function and minimal graph, Gaussian curvature estimates of convex level sets of minimal surface. Logarithmic gradient estimates of admissible solution to Hessian type equation arising in geometric optics is also considered. The dissertation consists of six chapters.In chapter 1, we briefly recall the history of convex level sets of solutions to elliptic equations. In chapter 2, we introduce the symmetric curvature matrix {aij} to describe the principal curvatures of level sets of a function. In chapter 3 and 4, we use this curvature matrix to prove the following results:Theorem 1 Let be a bounded smooth domain, 3 < n < 5 and u be a p-harmonic function inΩ, i.e.Assume in < p < 3 and the level sets of u are strictly convex with respect to gradient direction Vu, then the principal curvature of the level sets of u cannot attain its minimum inΩ,unless it is constant.Using the fact that the level sets of p-harmonic function over convex ring are strictly convex, we haveCorollary 0.1 Let u be the solution of the following boundary value problem,where and are two bounded convex sets in satisfying C If 3 < n < 5 and,then the principal curvature of the level sets of u attains its minimum on . For the case of the three-dimensional minimal graph, we haveTheorem 2 Let be a domain in and u be a minimal graph over,i.e., u satisfy the minimal surface equationAssume in . If the level sets of u are strictly convex with respect to gradient direction Vu, then the principal curvature of the level sets of u cannot attain its minimum inΩ, unless it is constant.Using the fact that the level sets of the minimal graph over convex ring are strictly convex , we get the followingCorollary 0.2 Let u satisfywhere and are bounded convex domains in R3; satisfying Then the principal curvature of the level sets of u attains its minimum value on d .In chapter 5, we discuss the Gaussian curvature estimates of convex level sets of immersed minimal surface using moving frame method and obtain the followingTheorem 3 Let be an immersed minimal surface. Let u be the height function of Mn corresponding to the direction .Ifu has no critical point, and let the level sets of u are all strictly convex with respect to direction Vu. Denote the Gaussian curvature by K and let . Then the function can not attain its minimum at an interior point of Mn, unless it is constant. In chapter 6, we give logarithmic gradient estimate of admissible solution to Hessian type equation and obtain the followingTheorem 4 be a smooth admissible solution to the following equationwhereLet f be a positive function. Then we have...