Linear Elliptic Operators

Prem K. Kythe
Additional contact information
Prem K. Kythe: University of New Orleans, Department of Mathematics

Chapter 2 in Fundamental Solutions for Differential Operators and Applications, 1996, pp 37-59 from Springer

Abstract: Abstract Fundamental solutions are very useful in the theory of partial differential equations. Among many applications, they are used, e.g., in solving non-homogeneous equations like (1.4.13), and in telling us about the regularity and growth of solutions. We have proved a general existence and uniqueness theorem (Theorem 1.4.2). We will now prove that every partial differential operator L(D) with constant coefficients has a fundamental solution. This result which was a conjecture until 1954 when it was established independently by Ehrenpreis (1954) and Malgrange (1955), together with the definition of hypoellipticity, implies that a differential operator with constant coefficients is hypoelliptic iff it has a fundamental solution which belongs to the class C ∞ in a region that does not contain the origin. The existence of a tempered fundamental solution for a partial differential operator with constant coefficients was proved by Hörmander (1958). We will derive fundamental solutions for the classical elliptic differential operators, like the Laplace, Helmholtz, and Cauchy-Riemann operators, and also a method for constructing fundamental solutions for homogeneous elliptic operators, and discuss maximum principle. Specific applications in the area of boundary element methods will be discussed in Chapter 10.

Keywords: Maximum Principle; Fundamental Solution; Boundary Element Method; Constant Coefficient; Partial Differential Operator (search for similar items in EconPapers)
Date: 1996
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4612-4106-5_3

Ordering information: This item can be ordered from
http://www.springer.com/9781461241065

DOI: 10.1007/978-1-4612-4106-5_3

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().