Regularity Problems for Some Semi-linear Problems

Serguei Dachkovski ()
Additional contact information
Serguei Dachkovski: Universität Bremen, Zentrum für Technomathematik

A chapter in Function Spaces, Differential Operators and Nonlinear Analysis, 2003, pp 255-266 from Springer

Abstract: Abstract In this article we consider the regularity problems for some partial differential and integral equations (also and specially related to fractals) containing the semilinearity of the following type 1 $$T^ + f(x) = f_ + (x) = \max (0,f(x)).$$ T + is defined on real functions and it is usually calledtruncation operatorIts properties were studied by many authors, see for example [0sw92], [Tri0113], [Zie89]. We will consider this operator in (real) Besov $$B_{pq}^s (\mathbb{R}^n )$$ and Hardy-Sobolev $$H_p^s (\mathbb{R}^n )$$ spaces and we look in these spaces for the best regularity of the solutions of semi-linear equations below. The final statement on the boundedness and Lipschitz continuity of the operatorT + in these spaces can be found in [Tri00] or [Tri0113], §25. We will see that the operatorT + has rather bad continuity properties in the spaces involved. To overcome the difficulties caused by this circumstance one can try to apply bootstrapping arguments using the lifting property of the corresponding (hypo)elliptic operators as we do it in Section 2. But it may happen (as we see in Sections 4 and 5) that there is no space to start from. An efficient tool to obtain the optimal smoothness also in such case is the so-called Q-method recently invented by H. Triebel in [TriOla] (see also [TriOlb], §27). We describe it briefly in Section 3.

Keywords: Elliptic Operator; Besov Space; Radon Measure; Regularity Problem; Lebesgue Measure Zero (search for similar items in EconPapers)
Date: 2003
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:978-3-0348-8035-0_15

Ordering information: This item can be ordered from
http://www.springer.com/9783034880350

DOI: 10.1007/978-3-0348-8035-0_15

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 ().