Partial differential equations

2022/2023
Programme:
Mathematics, Second Cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
core mandatory
Group:
M1
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
2
Seminar
1
Tutorial
2
Lab
0
Content (Syllabus outline)

Spaces of differentiable functions. Hölder spaces. Schwarz class of rapidly decreasing functions.Test functions. Distributions. Sobolev spaces. Fundamental solution. Characteristics of linear partial differential operator. Cauchy problem. Cauchy-Kowalevski theorem. Lewy's example.
Laplace equation. Newton potential as fundamental solution of the Laplace equation. Dirichlet problem. Subharmonic functions. Perron method. Weak solutions of the Dirichlet problem. Eigenfunctions and eigenvalues of the Laplace operator. Regularity of solutions.
The heat equation in higher dimensions. Gauss kernel. Fundamental solution of the heat equation. Inhomogeneous heat equation. Weierstrass theorem. The heat equation on bounded domains. Maximum principle. Fourier method of separation of variables.
The wave equation in higher dimensions. Spherical means. The wave equation in the space and in the plane. Fundamental solution of the wave equation. Inhomogeneous wave equation. The wave equation on bounded domains. Fourier method of separation of variables.

Readings

L. C. Evans: Partial Differential Equations, 2nd edition, AMS, Providence, 2010.
G. B. Folland: Introduction to Partial Differential Equations, 2nd edition, Princeton Univ. Press, Princeton, 1995.
L. Hörmander: The Analysis of Linear Partial Differential Operators I : Distribution Theory and Fourier Analysis, 2nd edition, Springer, Berlin, 2003.
F. John: Partial Differential Equations, 4th edition, Springer, New York, 1991.
F. Križanič: Parcialne diferencialne enačbe, DMFA-založništvo, Ljubljana, 2004.
E. H. Lieb, M. Loss: Analysis, 2nd edition, AMS, Providence, 2001.
Y. Pinchover, J. Rubinstein: An Introduction to Partial Differential Equations, CUP, Cambridge, 2005
A. Suhadolc: Integralske transformacije/Integralske enačbe, DMFA-založništvo, Ljubljana, 1994.
M. E. Taylor: Partial differential equations I: Basic theory, 2nd edition, Springer, New York, 2011

Objectives and competences

Student becomes familiar with partial differential equations in arbitrary dimensions. Introduced are distributions as generalized solutions of linear partial differential equations.
Proved are existence and basic regularity theorems for solutions of the Laplace equation, the heat equation and the wave equation.

Intended learning outcomes

Knowledge and understanding: Understanding the notion of a generalized solution of a partial differential equation. Skills to analytically find solutions of certain types of partial differential equation in higher dimensions. Understanding the properties of solutions of different types of second order partial differential equations.
Application: Formulation of certain mathematical and non-mathematical problems in the form of partial differential equations. Solving these partial differential equations.
Reflection: Understanding of the theory from the applications.
Transferable skills: The ability to identify, formulate, analyze and solve mathematical and non-mathematical problems with the help of partial differential equations. Skills in using the domestic and foreign literature.

Learning and teaching methods

Lectures, exercises, homeworks, seminar, consultations

Assessment

Exercise based exam / Theoretical knowledge exam
written exam / oral exam

Grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Miran Černe:
ČERNE, Miran, ZAJEC, Matej. Boundary differential relations for holomorphic functions on the disc. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2011, vol. 139, no. 2, str. 473-484. [COBISS-SI-ID 15710553]
ČERNE, Miran, FLORES, Manuel. Generalized Ahlfors functions. Transactions of the American Mathematical Society, ISSN 0002-9947, 2007, vol. 359, no. 2, str. 671-686. [COBISS-SI-ID 14227801]
ČERNE, Miran, FLORES, Manuel. Quasilinear [overline{partial}]-equation on bordered Riemann surfaces. Mathematische Annalen, ISSN 0025-5831, 2006, vol. 335, no. 2, str. 379-403. [COBISS-SI-ID 13970777]
Franc Forstnerič:
FORSTNERIČ, Franc. Runge approximation on convex sets implies the Oka property. Annals of mathematics, ISSN 0003-486X, 2006, vol. 163, no. 2, str. 689-707. [COBISS-SI-ID 13908825]
FORSTNERIČ, Franc. Noncritical holomorphic functions on Stein manifolds. Acta mathematica, ISSN 0001-5962, 2003, vol. 191, no. 2, str. 143-189. [COBISS-SI-ID 13138009]
FORSTNERIČ, Franc, ROSAY, Jean-Pierre. Approximation of biholomorphic mappings by automorphisms of C [sup] n. Inventiones Mathematicae, ISSN 0020-9910, 1993, let. 112, št. 2, str. 323-349. [COBISS-SI-ID 9452121]
Pavle Saksida:
SAKSIDA, Pavle. Lattices of Neumann oscillators and Maxwell-Bloch equations. Nonlinearity, ISSN 0951-7715, 2006, vol. 19, no. 3, str. 747-768. [COBISS-SI-ID 13932377]
SAKSIDA, Pavle. Maxwell-Bloch equations, C Neumann system and Kaluza-Klein theory. Journal of physics. A, Mathematical and general, ISSN 0305-4470, 2005, vol. 38, no. 48, str. 10321-10344. [COBISS-SI-ID 13802073]
SAKSIDA, Pavle. Nahm's equations and generalizations Neumann system. Proceedings of the London Mathematical Society, ISSN 0024-6115, 1999, let. 78, št. 3, str. 701-720. [COBISS-SI-ID 8853849]