Level Set Of A Class Of Elliptic Equations Are Solution Of Convex Curvature Estimation

This thesis consists of four sections.The first section is the introduction. We mainly introduce the result of convexity of thelevel sets of elliptic partial diferential equation and give the main two theorems of the thesis.The second section is the preliminaries, We firstly introduce some fundamental knowledgethe convexity of the level sets in classical diferential geometry, then we introduce some basicconcepts of curve and surface, it includes the calculating of curvature, the first and the secondfundamental forms for surface, and derive the curvature matrix of the level sets. Some lemmasand propositions will also be given which are used in the following proof.The third and the fourth section introduce the proof of our main theorem. Depending onreference of related papers, we choose suitable test function=k, which has proper constraints.then we obtain the last formulas of Δ by separating and combing the same terms.Finally, wecan obtain Gaussian curvature estimates for the convex level sets of solutions for elliptic partialdiferential equations in dimension2and3by minimum principle.