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

Title & Authors
A CHARACTERIZATION OF SOBOLEV SPACES BY SOLUTIONS OF HEAT EQUATION AND A STABILITY PROBLEM FOR A FUNCTIONAL EQUATION
Chung, Yun-Sung; Lee, Young-Su; Kwon, Deok-Yong; Chung, Soon-Yeong;

Abstract
In this paper, we characterize Sobolev spaces $\small{H^s(\mathbb{R}^n),\;s{\in}\mathbb{R}}$ 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
heat kernel;Sobolev space;Cauchy functional equation;
Language
English
Cited by
References
1.
J. A. Baker, Distributional methods for functional equations, Aequationes Math. 62 (2001), 136-142

2.
J. Chung, Stability of functional equations in the space of distributions and hyperfunctions, J. Math. Anal. Appl. 286 (2003), 177-186

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

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

6.
S.-Y. Chung, J. Cho, and D. Kim, Bochner-Schwartz theorem for ultradistributions, J. Math. Anal. Appl. 228 (1998), 166-180

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

11.
T. Matsuzawa, A calculus approach to hyperfunctions I, Nagoya Math. J. 108 (1987), 53-66