Quantitative Estimation Of Solutions For A Class Of Elliptic Partial Differential Equations

The paper consists of four sections.The first section is the introduction. It mainly introduces geometric properties as-sociated to the study of level sets of general elliptic partial differential equations. At the same time, introduce two main theorems.The second section is preliminaries. we give the brief definition of the convex level sets of a function, derive the curvature matrix of the level sets of a function. At last, we obtain about maximum principle and other conclusions.The third section and the fourth section is the estimate of the convex level sets of elliptic partial differential equations in R2and R3,by depending on a auxiliary function, maximum principle prove the main theorems, we will derive the following results:Theorom1.1. Let Ω, be a bounded smooth domain in R2, u∈C4(Ω)∩C2(Ω) is the solution of elliptic partial differential equation in Ω,, i.e. Δu＝f(u,▽u)＝f(u).|▽u|5Ω Assume|▽u|≠0in Ω, the level sets of u and the level sets of u are strictly convex with respect to normal▽u, let k be the curvature of the level sets, then the function k attains its minimum on the boundary (?)ΩTheorom1.2. Let Ω be a bounded smooth domain in R3, u∈C4(Ω)∩C2(Ω) is the solution of elliptic partial differential equation in Ω, i.e. Δu=f(u,▽u)=f(u)·|▽u|2in Ω the level sets of u and the level sets of u are strictly convex, let k be the Gauss curva-ture of the level sets of▽u. then the function k attains its minimum on the boundary (?)Ω...