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Communications on Pure and Applied Analysis (CPAA)
 

Regularity of solutions to an integral equation associated with Bessel potential

Pages: 1111 - 1119, Volume 10, Issue 4, July 2011      doi:10.3934/cpaa.2011.10.1111

 
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Xiaolong Han - Department of Mathematics, Wayne State University, Detroit, MI 48202, United States (email)
Guozhen Lu - Department of Mathematics, Wayne State University, Detroit, MI 48202, United States (email)

Abstract: In this paper, we study the regularity of the positive solutions to an integral equation associated with the Bessel potential. The kernel estimates for the Bessel potential plays an essential role in deriving such regularity results. First, we apply the regularity lifting by contracting operators to get the $L^\infty$ estimate. Then, we use the regularity lifting by combinations of contracting and shrinking operators, which was recently developed in [4] and [5], to prove the Lipschitz continuity estimate. Our regularity results here have been recently extended to positive solutions to an integral system associated with Bessel potential [9].

Keywords:  Integral equation, Bessel potential, regularity lifting, $L^\infty$ estimate, Lipschitz continuity estimate.
Mathematics Subject Classification:  Primary: 45E10; Secondary: 35J60.

Received: July 2010;      Revised: December 2010;      Published: April 2011.

 References