Regularity of Solutions to an Integral Equation on a Half-Space *R*_{+}^{n}

Affiliation(s)

College of Mathematics and Information Science, Henan Normal University, Xinxiang, China.

Department of Computer Science, Henan Normal University, Xinxiang, China.

College of Mathematics and Information Science, Henan Normal University, Xinxiang, China.

Department of Computer Science, Henan Normal University, Xinxiang, China.

ABSTRACT

In this paper, we discuss the integral equation on a half space *R _{+}^{n}*

where is the reflection of the point *x* about the . We study the regularity for the positive solutions of (0.1). A regularity lifting method by contracting operators is used in proving the boundedness of solutions, and the Lipschitz continuity is derived by combinations of contracting and shrinking operators introduced by Ma-Chen-Li ([1]).

Cite this paper

L. Cao and Z. Dai, "Regularity of Solutions to an Integral Equation on a Half-Space*R*_{+}^{n}," *Advances in Pure Mathematics*, Vol. 3 No. 1, 2013, pp. 153-158. doi: 10.4236/apm.2013.31A021.

L. Cao and Z. Dai, "Regularity of Solutions to an Integral Equation on a Half-Space

References

[1] C. Ma, W. Chen and C. Li, “Regularity of Solutions for an Integral System of Wolff Type,” Advances in Mathematics, Vol. 226, No. 3, 2011, pp. 2676-2699.

[2] D. Li and R. Zhuo, “An Integral Equation on Half Space,” Proceedings of the American Mathematical Society, Vol. 138, 2010, pp. 2779-2791. doi:10.1090/S0002-9939-10-10368-2

[3] W. Chen and C. Li, “The Equivalence between Integral Systems and PDE Systems,” Preprint, 2010.

[4] W. Chen and C. Li, “Regularity of Solutions for a System of Integral Equations,” Communications on Pure and Applied Analysis, Vol. 4, No. 1, 2005, pp. 1-8.

[5] L. Ma and D. Chen, “Radial Symmetry and Monotonicity Results for an Integral Equation,” Journal of Mathematical Analysis and Applications, Vol. 342, No. 2, 2008, pp. 943-949. doi:10.1016/j.jmaa.2007.12.064

[6] L. Ma and D. Chen, “Radial Symmetry and Uniqueness of Non-Negative Solutions to an Integral System,” Mathematical and Computer Modelling, Vol. 49, No. 1-2, 2009, pp. 379-385. doi:10.1016/j.mcm.2008.06.010

[7] X. Han and G. Lu, “Regularity of Solutions to an Integral Equation Associated with Bessel Potential,” Communications on Pure and Applied Analysis, Vol. 10, No. 4, 2011, pp. 1111-1119.

[8] W. Chen and C. Li, “Methods on Nonlinear Elliptic Equations. AIMS Series on Differential Equations and Dynamical Systems,” American Institute of Mathematical Science (AIMS), Springfield, 2010.

[1] C. Ma, W. Chen and C. Li, “Regularity of Solutions for an Integral System of Wolff Type,” Advances in Mathematics, Vol. 226, No. 3, 2011, pp. 2676-2699.

[2] D. Li and R. Zhuo, “An Integral Equation on Half Space,” Proceedings of the American Mathematical Society, Vol. 138, 2010, pp. 2779-2791. doi:10.1090/S0002-9939-10-10368-2

[3] W. Chen and C. Li, “The Equivalence between Integral Systems and PDE Systems,” Preprint, 2010.

[4] W. Chen and C. Li, “Regularity of Solutions for a System of Integral Equations,” Communications on Pure and Applied Analysis, Vol. 4, No. 1, 2005, pp. 1-8.

[5] L. Ma and D. Chen, “Radial Symmetry and Monotonicity Results for an Integral Equation,” Journal of Mathematical Analysis and Applications, Vol. 342, No. 2, 2008, pp. 943-949. doi:10.1016/j.jmaa.2007.12.064

[6] L. Ma and D. Chen, “Radial Symmetry and Uniqueness of Non-Negative Solutions to an Integral System,” Mathematical and Computer Modelling, Vol. 49, No. 1-2, 2009, pp. 379-385. doi:10.1016/j.mcm.2008.06.010

[7] X. Han and G. Lu, “Regularity of Solutions to an Integral Equation Associated with Bessel Potential,” Communications on Pure and Applied Analysis, Vol. 10, No. 4, 2011, pp. 1111-1119.

[8] W. Chen and C. Li, “Methods on Nonlinear Elliptic Equations. AIMS Series on Differential Equations and Dynamical Systems,” American Institute of Mathematical Science (AIMS), Springfield, 2010.