| This thesis consists of three sections.The first section is the introduction, and we give the main theorem:LetΩbe a smooth bounded domain in R2 and u∈C4(Ω)∩C2(Ω) be a solution of the elliptic equation inΩ, i.e.Δu=2u2|▽u|2. Assume|▽u|≠0 inΩ, the level sets of u are strictly convex with respect to normal▽u. Then we have the following fact:the functionψ=k andφ=e-2uk attain their minimum on the boundary (?)Ω. Where k is the curvature of the level sets of u.The second section is the preliminaries. We introduce some basic concepts on the con-vexity of the level sets in classical differential geometry, formulas, maximum principle and the brief proof.The third section includes the main theorem. At first, we compute the second derivative of test functionsφ=e-2uk andψ=k, then we obtain the last formulas ofΔφandΔψby separating and combining the same species terms. Finally, we prove that the functionφandψattain their minimum on the boundary by maximum principle. |
