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DOI QR Code

A CHARACTERIZATION OF SOBOLEV SPACES BY SOLUTIONS OF HEAT EQUATION AND A STABILITY PROBLEM FOR A FUNCTIONAL EQUATION

  • Chung, Yun-Sung (BK21 MATH MODELING HRD DIV. DEPARTMENT OF MATHEMATICS SUNGKYUNKWAN UNIVERSITY) ;
  • Lee, Young-Su (DEPARTMENT OF MATHEMATICSL SCIENCES KOREA ADVANCED INSTITUTE OF SCIENCE AND TECHNOLOGY) ;
  • Kwon, Deok-Yong (DEPARTMENT OF MATHEMATICS SOGANG UNIVERSITY) ;
  • Chung, Soon-Yeong (DEPARTMENT OF MATHEMATICS AND PROGRAM OF INTEGRATED BIOTECHNOLOGY SOGANG UNIVERSITY)
  • Published : 2008.07.31

Abstract

In this paper, we characterize Sobolev spaces $H^s(\mathbb{R}^n),\;s{\in}\mathbb{R}$ by the initial value of solutions of heat equation with a growth condition. By using an idea in its proof, we also discuss a stability problem for Cauchy functional equation in the Sobolev spaces.

Keywords

References

  1. J. A. Baker, Distributional methods for functional equations, Aequationes Math. 62 (2001), 136-142 https://doi.org/10.1007/PL00000134
  2. J. Chung, Stability of functional equations in the space of distributions and hyperfunctions, J. Math. Anal. Appl. 286 (2003), 177-186 https://doi.org/10.1016/S0022-247X(03)00468-2
  3. J. Chung, S.-Y. Chung, and D. Kim, The stability of Cauchy equations in the space of Schwartz distributions, J. Math. Anal. Appl. 295 (2004), 107-114 https://doi.org/10.1016/j.jmaa.2004.03.009
  4. S.-Y. Chung, A regularity theorem for the initial traces of the solutions of the heat equation, J. Korean Math. Soc. 33 (1996), no. 4, 1039-1046
  5. S.-Y. Chung, Reformulation of some functional equations in the space of Gevrey distributions and regularity of solutions, Aequationes Math. 59 (2000), 108-123 https://doi.org/10.1007/PL00000118
  6. S.-Y. Chung, J. Cho, and D. Kim, Bochner-Schwartz theorem for ultradistributions, J. Math. Anal. Appl. 228 (1998), 166-180 https://doi.org/10.1006/jmaa.1998.6124
  7. Y.-S. Chung, J.-H. Kim, and S.-Y. Chung, Stability of a quadratic functional equation in the space of distributions, Math. Inequal. Appl. 9 (2006), No. 2, 325-334
  8. L. C. Evans, Partial Differential Equations, G.S.M. Vol. 19, Americam Mathematical Society, Providence, Rhode Island, 1983
  9. L. Hormander, The Analysis of Linear Partial Differential Operators, Vol.1, Springer-Verlag, Berlin and New york, 1983
  10. Y.-S. Lee and S.-Y. Chung, Stability of a quadratic Jensen type functional equation in the spaces of generalized functions, J. Math. Anal. Appl. 324 (2006), 1395-1406 https://doi.org/10.1016/j.jmaa.2006.01.041
  11. T. Matsuzawa, A calculus approach to hyperfunctions I, Nagoya Math. J. 108 (1987), 53-66 https://doi.org/10.1017/S0027763000002646