The Equation Of Mean Curvature Under Neumann Boundary Conditions

In differential geometry,the mean curvature of a surface is an important concept.At the same time,the mean curvature equation also plays an important role in partial differenti-ation.The mean curvature equation with Neumann boundary value condition has important geometric meaning and it also has been studied by many mathematicians.This paper mainly studies from the following two parts:The first part mainly studies the existence of solutions of parabolic equations with Neumann boundary conditions on[0,1],and concludes that the curvilinear flow corresponding to the solution of the equation will converge to a section of are.The second part studies the existence theorem of solutions for elliptic equations in strictly convex bounded domain under Neumann boundary conditions.The main results of this paper are as follows:Theorem 1 Suppose f(x,?)is a function defined on[0,1]×R and satisfies that(?)with ??0.Let u(x,t)be the solution to the following equation Then for t ?[0,T)we have the estimate|ux(·,t)|?C,where C=C(T,?|f|C0([0,1]×R)i|fx|C0([0,1)).Corollary 1 Under the same conditions as described in Theorem 1,the parabolic equation has a smooth solution u=u(x,t).Theorem 2 Let u(x,t)be the solution to the parabolic equation Then u(x,t)will converge to a translating solution as ?t+??,where ? is a suitable solution to equation as followsTheorem 3 For the following equation Let ? be a strictly convex bounded domain in Rn with(?),n? 2.Suppose f satisfies(1)(?);(2)(?).For(?),there exists a unique ? ? R and ??C2,?((?)),such that(?,?)satisfies the above equation.Moreover,the solution w is unique up to a constant.