Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MME1001	Sobolev Spaces and Partial Differential Equations	

Section	Sem.	Hours: C+S+L	Category	Type	
Applied Mathematics	1	2+1+0	speciality	compulsory	

Teaching Staff in Charge	Assoc.Prof. BUICA Adriana, Ph.D., abuica@math.ubbcluj.ro	
Aims	Basic results from the theory of distributions and Sobolev spaces; modern theiry of partial differential equations.	
Content	1.	Fundamental spaces of the theory of distributions
2.	The notion of distribution. Classification. 
3.      Operations. Derivative. Convolution.
4.	The Fourier transform
5.	Fundamental solutions. Examples
6.	Sobolev spaces
7.	The dual of a Sobolev space
8.	Sobolev embedding theorem
9.	Rellich-Kondrachov compact embedding theorem
10.	Variational theory of elliptic equations
11.	Semilinear elliptic problems. The Nemytskii operator
12.	Existence and uniqueness by the contraction principle
13.	Applications of Schauder@s fixed point theorem
14.	Application of the Leray-Schauder principle 
	
Bibliography	1. R. Precup, Lectii de ecuatii cu derivate partiale, Presa Universitara Clujeana, 2004.
2. J. Rauch, Partial Differential Equations, Springer, 1991.
3. H. Brezis, Analyse nonlineaire, Hermann, 1983.
4. M. Taylor, Partial Differential Equations, Springer, 1996.1.	L. Schwartz, Theorie des distributions, Hermann, 1959.	
5. D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations, Springer, 1983.
6. J.M. Bony, Theorie des distributions et analyse de Fourier, Ecole Polytechnique, Palaiseau, 1997.
7. R.A. Adams, Sobolev Spaces, Academic Press, 1975.

Assessment	20% activity to courses and seminaries
20% report
60% written exam