邢唷> 欹 鳵殭bjbjs鱯2/. 8';ir %ghhhhhh$vlo8%h(((%h4緃9F9F9F(4g9F(g9F9F骬[aP鼢pXECZ縚雊詇0i賍Po烡`╫4[aPo4刼$[a-Dq9FL[ ---%h%hE:---i((((╫--------- : 2014t^pef[詁\f[uTT慺gf[!hD崣e 2014t^pef[詁\f[uTT慺gf[!h_韘剉f[!h g Y鎒'Yf[0WS_'Yf[0踁輂'Yf[0钑'Yf[ T*Nf[!h剉_韘 z蔛鴙sQ塠抍翂D桍N0耂N貧!h gS琋'Yf[0WS琋'Yf[0 T梘'Yf[0S琋^'Yf[0fkIl'Yf[0pQ轢'Yf[0 1u嶯慺gT鹼邩縖儚'} _ gT耂N貧!h=\隷\,gf[!h >mf[u酧o`裇愘~_韘貧!hT鹼篘0f[u酧o`S靊f[鱏 覻 T '`+R t^ 異眥 5u輯 珟齆翄I{0 T貧!hT鹼篘 S琋'Yf[骻l_oRb HYPERLINK "mailto:bliu@math.pku.edu.cn" bliu@math.pku.edu.cn Y鎒'Yf[擽ZWRoRb HYPERLINK "mailto:jgying@fudan.edu.cn" jgying@fudan.edu.cn WS_'Yf[pef[褃f[f[b枊[NL06g30錯0R 7g1錯髞7g19錯 N0 7g20錯Y!hf[u粂!h0(W購g魰 _緥OP甠R筫 z NO魚觔Qb{薔$N钑 z0R+RX鲖sQ鈒Ye坈钑'Yf[ CS篘R eQ 0齎Ohio轢藌'Yf[Ye坈 ;N矉OP甠R筫 z z 睈^t 昚e坈齎禰pg鶴R梩^鶺褢穬梍 NSN^'Yf[Ye坈 ;N矉NO魚觔Qb{薔 z0 OP甠R筫 z z薔蛜NYef['Y瞺. In this course we study uniformlyelliptic secondorder PDEs. It roughly consists of two parts. The first part is anintroduction to thebasic theory, includingthemaximum principle, Hopf Lemma, and Schauder estimates for Poisson equation.營n the second part we study爐he Krylov-Safanov theory of Harnack inequality and Evans-Krylov estimates. If time allows we shall give a brief introduction to爐he theory of fully nonlinear elliptic equations. We shall also make necessary adjustments according to爐he level of students.Contents:1. The mean value theorem and consequences. 2. The maximum principle and Hopf Lemma. 3 Harnack inequality. 4.燬chauder estimates 5. Alexandrov maximum principle. 6. Krylov-Safanov theory. 7. Evans-Krylov theorem. Reference: L. C. 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This course is closely related with Professor Guan Bo s course. There are three basic topics: 1. Maximum Principle. 2. Schauder Estimates. 3. Harnack Inequality. We will discuss these three topics for elliptic and parabolic equations. z2: Geometric PDEs for curves and surfaces ;N矉: sQ鈒Ye坈钑'Yf[ CS篘R eQ 0齎Ohio轢藌'Yf[Ye坈 z薔蛜 1. Geometric variational problems: a. Geodesics, minimal surfaces. Curvatures. b. Isoperimetric inequality c. Surfaces of constant curvatures 2. Hyperbolic plane a. The metric, length of curves b. The geodesics, completeness c. Triangles, the curvature d. Different models. 3. The unit sphere a. The induced metric b. geodesics, triangles, and curvature 4. Simple applications of ODE and PDE to geometry a. Alexandrov moving plane method: (i) for curves b. The Laplace equation and Isoperimetric inequality c. The maximum principle and Hopf lemma d. Alexandrov moving plane method: (ii) for surfaces 5. Curvature flow of curves a. Length shortening b. Convergence to a point c. Application: proof of the isoperimetric inequality 6. Fully nonlinear equations in classical differential geometry a. Convex surfaces, Lieberman theorem b. Weyl problem c. 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