关于k-Hessian方程C2＋α局部解的存在性

C^(2+α) Local Solvability of the k-Hessian Equations

该文克服椭圆型k-Hessian算子的线性化算子不满足极大值原理的困难,利用NashMoser迭代,证明当非齐次项f∈C~α变号或非负时,k-Hessian方程C^(2+α)局部解的存在性,当然当f为C~∞时,存在C~∞局部解.其技巧是首先证明线性化方程解的唯一性,以此为基础得到线性化方程解的存在性,进而得到线性化方程解的高阶正则性和先验估计.

Overcoming the difficulty arising from the fact that the linearized operators of the elliptic k-Hessian ones do not satisfy the Maximum principle and employing Nash-Moser iteration, we prove the existence of C2＋α local solutions of k-Hessian equation when the non- homogeneous term f∈ Cα changes sign or is nonnegative. Of course there exists C∞ local solution if f∈ C∞. The technique is that, for the solution to the linearized equation, we prefer at first to prove its uniqueness from which the existence of solution, together with the higher regularity and a priori estimates of solutions, follows.