Electron. J. Differential Equations, Vol. 2025 (2025), No. 14, pp. 1-62.

Properties of the Dirichlet Green's function for linear diffusions on a half line

Joseph G. Conlon, Michael Dabkowski

Abstract:
This article concerns the study of Green's functions for one dimensional diffusions with constant diffusion coefficient and linear time inhomogeneous drift. It is well know that the whole line Green's function is given by a Gaussian. Formulas for the Dirichlet Green's function on the half line are only known in special cases. The main object of study in the paper is the ratio of the Dirichlet to whole line Green's functions. Bounds, asymptotic behavior in the limit as the diffusion coefficient vanishes, and a log concavity result are obtained for this ratio. These results have been used in the proof of asymptotic behavior for a simple model of Ostwald ripening.

Submitted December 13, 2023. Published February 19, 2025.
Math Subject Classifications: 35F21, 35K20, 49N10.
Key Words: Nonlinear PDE; coarsening.
DOI: 10.58997/ejde.2025.14

Show me the PDF file (600 KB), TEX file for this article.

Joseph G. Conlon
University of Michigan
Department of Mathematics
Ann Arbor, MI 48109-1109, USA
email: conlon@umich.edu
Michael Dabkowski
University of Michigan-Dearborn
Department of Mathematics and Statistics
Dearborn, MI 48128, USA
email: mgdabkow@umich.edu

Return to the EJDE web page