AM410 - Elliptic Partial Differential Equations
A.Y. 2014 - 2015 (I term)
Lecturer: Alfonso Sorrentino
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:
-Course Schedule: Tuesday and Thursday 11-13, room F.
-Office Hours: By appointment.
-Email: sorrentino@mat.uniroma3.it
-Office: C.203
-Telephone: 06 5733 8227
LECTURE DIARY
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.[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.[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.[02.10.2014] Harmonic functions are analytic. Radial solutions of Laplace equations and fundamental solutions. Green’s functions.
5.[07.10.2014] Properties of Green’s functions. Poisson’s Kernel and Green’s representation formulae.
6.[09.10.2014] Computing Green’s function for spheres and solvability of the Dirichlet problem on spheres.
7.[14.10.2014] Removable singularities of harmonic functions. Subharmonic and Superharmonic functions: definitions and properties.
8.[16.10.2014] (Cancelled)
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.[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.[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.[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.[30.10.2014 (pm)] General elliptic operators: definitions and examples. Weak maximum principles and consequences.
14.[11.11.2014] Hopf Lemma. Strong maximum principle and consequences (comparison principle). Serrin’s generalization of comparison principle.
15.[13.11.2014] A-priori estimates. Introduction to weak derivatives: motivations, definitions, uniqueness and examples.
16.[18.11.2014] Properties of weak derivatives. Sobolev spaces and Sobolev norms. Sobolev spaces are complete.
17.[20.11.2014] Approximation by smooth functions: local and global approximation. Approximation up to the boundary (if the boundary is C^1)
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.[27.11.2014] Elliptic operators in divergence form. Variational formulation of problem. Lax-Milgram theorem. Existence of weak solutions and Poincare’ inequality.
20.[2.12.2014] Difference quotients (definition and properties). Inner regularity of weak solutions. Interior H^2 regularity.
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)