The Constant Rank Theorem Of K-convex Solutions Of Semilinear Elliptic Equation

The geometric property of solutions is a fundamental problem in the theory ofnonlinear elliptic partial di?erential equations. The study on convexity of solutionsto elliptic equation, not only is the issue in geometry, but also comes naturally inanalysis. Whether the solutions are convex; if not, we will ask whether the solutionshave partial convexity. Such question is an issue of interest for a long time. The k-convexity is one of the partial convexity, in this dissertation we want to show undercertain conditions the sum of any k eigenvalues of the Hessian matrix of solution isnonnegative. The k-convexity is an interesting issue in partial di?erential equationand di?erential geometry. In this dissertation, using the strong maximum principle,we obtain a constant rank theorem for the k-convex solutions of semilinear ellipticpartial di?erential equations in Rn. Applying the constant rank theorem andthe method of continuity, we obtain an existence theorem of k-convex starshapedhypersurface with prescribed mean curvature.There are four chapters.In chapter 1, we will review some background by giving some classical ex-amples in the process of the study on convexity, and introduce some relevantterminologies which will be involved later.In chapter 2, we obtain a constant rank theorem for the k-convex solutionsof Poisson equation in Rn. In section 1, we get the constant rank theorem in thecase of k = 2; in section 2, we get the general case for k≥3.In chapter 3, we obtain a constant rank theorem for the k-convex solutions ofsemilinear elliptic equation on Sn.In the last chapter, we give an existence theorem of k-convex starshapedhypersurface with prescribed mean curvature in Rn+1.