AM410 - Elliptic Partial Differential Equations

A.Y. 2014 - 2015 (I term)

Lecturer: Alfonso Sorrentino

 

NEWS & ANNOUNCEMENTS:


No news. I have moved to another university. For all information, please write an email to sorrentino@mat.uniroma2.it.

GENERAL INFORMATION:


  1. -Course Schedule:  Tuesday and Thursday 11-13, room F.


  1. -Office Hours: By appointment.


  1. -Email: sorrentino@mat.uniroma3.it   


  1. -Office: C.203 

   

  1. -Telephone: 06 5733 8227

LECTURE DIARY

  1. 1.[23.09.2014] Introduction to Partial Differential Equations (PDEs), examples, classification (linear, semi-linear, quasi-linear, fully non-linear). Elliptic, parabolic and hyperbolic PDEs of second order. Laplace’s equation and harmonic functions. Relation between harmonic functions in dimension 2 and holomorphic complex functions.

  2. 2.[25.09.2014] Properties of Harmonic functions and orthogonal linear change of coordinates. Functions that satisfy the Mean Value Property (MVP) and their properties: maximum (minimum) principle, non-isolated zeros, regularity. Harmonic functions satisfy MVP and viceversa.

  3. 3.[30.09.2014] Local estimates: estimates on the derivatives of harmonic functions. Liouville’s theorem and generalizations to non-negative harmonic functions. Harnack’s inequality.

  4. 4.[02.10.2014] Harmonic functions are analytic. Radial solutions of Laplace equations and fundamental solutions. Green’s functions.

  5. 5.[07.10.2014] Properties of Green’s functions. Poisson’s Kernel and Green’s representation formulae.

  6. 6.[09.10.2014] Computing Green’s function for spheres and solvability of the Dirichlet problem on spheres.

  7. 7.[14.10.2014] Removable singularities of harmonic functions. Subharmonic and Superharmonic functions: definitions and properties.

  8. 8.[16.10.2014]  (Cancelled)

  9. 9.[21.10.2014] Harmonic lifting of subharmonic functions. Convergent subsequences of equibounded harmonic functions. Perron’s method: definition of subfunctions relative to the boundary datum, definition of Perron’s solution and proof that it is harmonic.

  10. 10.[23.10.2014] Behaviour of Perron’s solution on the boundary. Definition of barrier and regular points. Continuity of Perron’s solution up to the boundary iff all boundary points are regular.

  11. 11.[28.10.2014] Sufficient conditions to have regular points in dimension 2 and in dimension >2. The exterior sphere condition. C^2 boundaries satisfy the exterior sphere condition.

  12. 12.[30.10.2014 (am)] Lebesgue’s thorn (or spine): an example of domain in which a Dirichlet problem is not solvable. Introduction to Dirichlet energy methods and variational methods for the laplacian.

  13. 13.[30.10.2014 (pm)] General elliptic operators: definitions and examples. Weak maximum principles and consequences.

  14. 14.[11.11.2014]  Hopf Lemma. Strong maximum principle and consequences (comparison principle). Serrin’s generalization of comparison principle.

  15. 15.[13.11.2014] A-priori estimates. Introduction to weak derivatives: motivations, definitions, uniqueness and examples.

  16. 16.[18.11.2014] Properties of weak derivatives. Sobolev spaces and Sobolev norms. Sobolev spaces are complete.

  17. 17.[20.11.2014] Approximation by smooth functions: local and global approximation. Approximation up to the boundary (if the boundary is C^1)

  18. 18.[25.11.2014] Trace operator and trace of a function. Characterization of functions in W_0^{1,p} in terms of their trace being the zero function.

  19. 19.[27.11.2014] Elliptic operators in divergence form. Variational formulation of problem. Lax-Milgram theorem. Existence of weak solutions and Poincare’ inequality.

  20. 20.[2.12.2014] Difference quotients (definition and properties). Inner regularity of weak solutions. Interior H^2 regularity.

  21. 21.[4.12.2014] Higher interior regularity and infinity differentiability in the interior. General Sobolev inequality (without proof). Boundary regularity (no proof).

BIBLIOGRAPHY:

[E] Lawrence C. Evans, Partial Differential Equations (AMS, 2010)

[HL] Qing Han & Fanghua Lin, Elliptic Partial Differential Equations  (AMS, 1997)

[GT] David Gilbarg & Neil S. Trudinger, Elliptic PDE of second order  (Springer, 2001)