Functional Analytic Methods for Partial Differential Equations

Hiroki Tanabe

• Publisher: CRC Press
• Year: 2017
• ISBN:
  9781351446860
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Combining both classical and current methods of analysis, this text present discussions on the application of functional analytic methods in partial differential equations. It furnishes a simplified, self-contained proof of Agmon-Douglis-Niremberg's Lp-estimates for boundary value problems, using the theory of singular integrals and the Hilbert transform.

Chapter 1

Preliminaries

1.1 Preliminaries on Measure and Integration

In this chapter we collect some preliminary results from the theory of measure and integration.

Let E be a Lebesgue measurable subset of the ν-dimensional Euclidean space R^u. The Lebesgue measure of E is denoted by |E|. The supremum of |E|/|Q| over all cubes Q containing E is called the parameter of regularity of E and is denoted by r(E).

A sequence {E_n} of measurable sets is called a regular sequence if there exists a positive number a such that r(E_n) ≥ α for all n = 1,2,….

For a nonempty open subset Ω of R^ν the notation Lloc1(Ω) stands for the set of all functions which are Lebesgue integrable on each compact subset of Ω.

Theorem 1. 1

For a function f∈Lloc1(Rν) there exists a null set N such that for any x ∉ N and for any regular sequence {e_n} tending to x

limn→∞1|en|∫nf(y)dy=f(x).

Especially for any x ∉ N

limρ→01Vνρν∫|x-y|<ρf(y)dy=f(x),

where V_ν is the volume of the unit ball of R^ν.

More strongly the following theorem holds.

Theorem 1. ...