Convex Of The Level Set Of Solutions Of Harmonic Functions

Convex is a fundamental feature of geometric objects that can be described by differentials under smooth conditions.The study of convexity can be traced back to Shiffman's research on minimal surfaces in the 1950s.At present,many scholars have carried out in-depth and detailed analysis of partial differential equations,and sometimes the equation itself needs to be studied from the perspective of convexity.Therefore,the study of convexity not only has a long history,but also has become an increasingly interesting topic.The research on convexity can be divided into two categories: one is the research on convexity of solutions,and the other is the research on convexity of level sets of solutions.Comparatively speaking,the latter is more complex and difficult to study,so it is relatively less discussed and analyzed.On the convexity of level sets,the methods adopted by experts and scholars mainly include Gabriel's method,constant rank theorem,quasi-concave envelope method,curvature estimation and so on.This article research level set by the curvature estimation method of convexity,curvature estimation method of the solution of the basic idea is first assumed level set is convex in border,secondly,the need to level set in the area,the border of convexity to comprehensive comparative analysis of the system,and thus prove level set inside the gaussian curvature is greater than the boundaries of gaussian curvature,the solution of the level set is convex in the area.In dealing with convexity,the constant rank theorem plus continuity method is a powerful tool for proving convexity,but quantitative data such as the degree of specific convexity need to be characterized by curvature estimates.In this paper,the first chapter is mainly about a partial differential equation of convexity generalizes the research achievement,and proposes the main discussion,the second chapter gives the required basic knowledge,the third chapter using the gaussian curvature method,through the clever and calculated in detail and in three-dimensional Euclidean space,the level set of solutions of the harmonic function rise in gaussian curvature and sex,which can use the extremum principle of the quantitative study on the level set convexity.