ࡱ>  Rbjbjss2/. 8';ir %ghhhhhh$vlo8%h(((%h4h9F9F9F(4g9F(g9F9F^[aPpXECZ_gh0i_PoD`o4[aPo4o$[a-Dq9FL[ ---%h%hE:---i((((o--------- : 2014t^pef[b\f[uTTfgf[!hDe 2014t^pef[b\f[uTTfgf[!h_svf[!h g Ye'Yf[0WS_'Yf[0V]'Yf[0S'Yf[ T*Nf[!hv_s zSvsQ[cDN0SNؚ!h gSN'Yf[0WSN'Yf[0 Tg'Yf[0SN^'Yf[0fkIl'Yf[0pQ]'Yf[0 1uNfgT|ߘ['} _ gTSNؚ!h=\_\,gf[!h >mf[uOo`S~_sؚ!hT|N0f[uOo`Sbf[S Y T '`+R t^~ { 5u݋ NI{0 Tؚ!hT|N SN'Yf[gl_oRb HYPERLINK "mailto:bliu@math.pku.edu.cn" bliu@math.pku.edu.cn Ye'Yf[^ZWRoRb HYPERLINK "mailto:jgying@fudan.edu.cn" jgying@fudan.edu.cn WS_'Yf[pef[yf[f[b[NL06g30eb0R 7g1e7g19e N0 7g20eY!hf[uy!h0(Wُg _OP_Re z NO~bQb{N$N z0R+RXsQlYecS'Yf[ CSNR eQ 0VOhio]z'Yf[Yec ;NOP_Re z z ^t YecV[pgQRt^Wё_ NSN^'Yf[Yec ;NNO~bQb{N z0 OP_Re z zN~NYef['Y~. 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.In the second part we studythe Krylov-Safanov theory of Harnack inequality and Evans-Krylov estimates. If time allows we shall give a brief introduction tothe theory of fully nonlinear elliptic equations. We shall also make necessary adjustments according tothe level of students. Contents: 1. The mean value theorem and consequences. 2. The maximum principle and Hopf Lemma. 3 Harnack inequality. 4.Schauder 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: sQlYecS'Yf[ CSNR eQ 0VOhio]z'Yf[Yec zN~ 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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