Week 1   09-13  January
01/10 ---Important inequalities & weak derivatives
01/12 --- Holder spaces & Sobolev spaces 

Week 2   16-20   January
01/17 --- Approximations and Extensions
01/19 ---  Traces & Morrey's inequalities

Week 3  23-27   January
01/24 ---Compactness --- Rellich-Kondrachor
01/26 -- Fourier Transform Methods & Poincare's inequality

Week 4  30 January -03 February
01/31 ---The space H^{-1} & Spaces involving time 
02/02---  Second order elliptic equations & Lax-Milgram  theorem 

Week 5    06-10  February 
02/07 --- Regularity,  Maximum principle, Eigenvalues & Eigen functions
02/09 --- Linear Parabolic equations--Road map for wellposedness

02/09 --- Week 6  13--17   February
02/14 ---  Energy estimates,  Regularity & Maximum principle
02/16 ---  Linear Hyerpbolic equation & Finite propagation speed

Week 7   20--24  February 
02/21 --- Symmetric Hyperbolic system--Vanishing Viscosity vs. Finite Difference 
02/23 --- Midterm

Week 8   27 February -- 03 March   (the first half semester ends on March 03)
02/28 --- The Calculus of Variations & Euler-Lagrangian equation
03/02 --- Existence of  minimizers   

Week 9  06-10  March (second half semester begins)
03/07 ---Convexity and weakly lower semi-continuity
03/09 --- Weak solutions of Euler-Lagrange equation
 

*** Spring Break 13-17  March, classes recessed ***

Week  10  20--24  March  
03/21 --- Regularity  
03/23 --- Constrained minimization    

Week 11   27-31  March
03/28 --- Harmonic map and incompressibility
03/30 --- Critical points and Mountain pass theorem 

Week 12  03-- 07 April 
04/04--- Fixed  point methods
04/06 ---Non-existence--blow up

Week 13  10--14  April
04/11 --- Hamilton-Jacobi equations 
04/13 --- Viscosity  
 

Week 14  17--21 April
04/18 --- Uniqueness
04/20 --- System of Conservation laws

Week 15  24 -- 28  April
04/25 --- Traveling solutions 
04/27 --- Entropy

Final Week  01-- 05   May TBA.