In this class, as a sequel of "Functional Analysis", I will explain how to deal with differential equations from the viewpoint
of functional analysis. For a differential equation which may have no solutions, we introduce the concept of weak solutions
by regarding the derivatives in the equation as those in a weak sense. Applying theorems of functional analysis to a functional
defined on an appropriate space of functions (Sobolev space), we can often assure the existence of weak solutions of the equation.
If it is possible to derivate the weak solution in the usual sense, then we can conclude that the weak solution is really
a solution, hence the equation has the classical solution. Such an approach to differential equations is called the variational
method, which is a standard strategy in modern analysis.
Modeling a phenomenon in terms of differential equations, we usually can't solve the equation because of its nonlinearlity.
We are not sure if the equation is authentic as a model of the phenomenon, unless the existence of solutions is assured. Therefore,
for a given differential equation, we need a mathematical theory to study the existence and property of the solution without
solving the equation. The purpose of the class is to understand the variational method, a theory based on functional analysis,
and the basic study of differential equations. The theory of Sobolev spaces and vatiational methods is standard in modern
analysis and important as a mathematical theory of the finite element method (FEM) in numerical analysis as well as a pure-mathematical
theory.
- To understand the concept of weak solutions.
- To understand the character of Soboleb spaces.
- To recognize that functional analysis is effective in studying differential equations.
| Class schedule | HW assignments (Including preparation and review of the class.) | Amount of Time Required | |
|---|---|---|---|
| 1. | Weak solutions of differential equations | Check out "inhomogeneous linear differential equations" and "integration by parts" in advance. | 190minutes |
| 2. | Sobolev space W^{1,p} | Check out "Lebesgue integral" and "L^p spaces" and "weak derivative" in advance. | 190minutes |
| 3. | Convolution | Check out "convolution" in advance. | 190minutes |
| 4. | Regularization | Check out "uniform continuous" in advance. | 190minutes |
| 5. | Extension operators | Check out "bounded linear operators" in advance. | 190minutes |
| 6. | Density | Check out "dense" in advance. | 190minutes |
| 7. | Sobolev embedding | Review "W^{1,p}" in advance. | 190minutes |
| 8. | Properties of weak differentiation | Review the precious session in advance. | 190minutes |
| 9. | Sobolev space W^{m,p}, W^{1,p}_0 | Review the precious session in advance. | 190minutes |
| 10. | Poincare's inequality | Review the precious session in advance. | 190minutes |
| 11. | Stampacchia's theorem | Review the precious session in advance. | 190minutes |
| 12. | Lax-Milgram's theorem | Review the precious session in advance. | 190minutes |
| 13. | Application to boundary value problem | Review the precious session in advance. | 190minutes |
| 14. | Final exam | Review all the sessions in advance. | 190minutes |
| Total. | - | - | 2660minutes |
| Final exam | Total. | |
|---|---|---|
| 1. | 40% | 40% |
| 2. | 30% | 30% |
| 3. | 30% | 30% |
| Total. | 100% | - |
No textbook is necessary, but I will give lectures along Chapter XIII in: Haim Brezis, "KANSUU KAISEKI", Sangyo Tosho (translated
by Fujita and Konishi). Also, as a reference, I recommend the textbook of "Functional Analysis", i.e. Higuchi, Serizawa and
Jimbo, "KANSUU KAISEKI NO KISO/KIHON", Makino Shoten.
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Last modified : Wed Oct 17 07:33:08 JST 2018