Elliptic equations with diffusion coefficient vanishing at the boundary: Theoretical and computational aspects
Authors:
Chung-min Lee and Jacob Rubinstein
Journal:
Quart. Appl. Math. 64 (2006), 735-747
MSC (2000):
Primary 35J70
DOI:
https://doi.org/10.1090/S0033-569X-06-01033-1
Published electronically:
November 8, 2006
MathSciNet review:
2284468
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: A class of degenerate elliptic PDEs is considered. Specifically, it is assumed that the diffusion coefficient vanishes on the boundary of the domain. It is shown that if the diffusion coefficient vanishes fast enough, then the problem has a unique solution in the class of smooth functions even if no boundary conditions are supplied. A numerical method is derived to compute solutions for such degenerate equations. The problem is motivated by a certain approach to the recovery of the phase of a wave from intensity measurements.
grn2 T.E. Gureyev and K.A. Nugent, Phase retrieval with the transport of intensity equation II: Orthogonal series solution for nonuniform illumination, J. Opt. Soc. Amer. A 13 (1995), 1670–1682.
hjm E. Houseworth, M.S. Jolly and G.O. Mohler, Continuum limit of a discrete genetic problem, preprint.
kha R. Z. Khas’minskii, Diffusion processes and elliptic differential equations degenerating at the boundary of the domain, Theory of Probability and its Applications 3 (1958), 400–419.
- M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate-elliptic operators, Ann. Mat. Pura Appl. (4) 80 (1968), 1–122 (English, with Italian summary). MR 249828, DOI https://doi.org/10.1007/BF02413623
rod1 F. Roddier, Curvature sensing and compensation: A new concept in adaptive optics, Appl. Opt. 27 (1998), 1223–1225.
- Jacob Rubinstein and Gershon Wolansky, A variational principle in optics, J. Opt. Soc. Amer. A 21 (2004), no. 11, 2164–2172. MR 2122591, DOI https://doi.org/10.1364/JOSAA.21.002164
- Tokuzo Shiga, Continuous time multi-allelic stepping stone models in population genetics, J. Math. Kyoto Univ. 22 (1982/83), no. 1, 1–40. MR 648554, DOI https://doi.org/10.1215/kjm/1250521859
tea M.R. Teague, Deterministic phase retrieval: A Green’s function solution, J. Opt. Soc. Amer. 73 (1983), 1434–1441.
grn2 T.E. Gureyev and K.A. Nugent, Phase retrieval with the transport of intensity equation II: Orthogonal series solution for nonuniform illumination, J. Opt. Soc. Amer. A 13 (1995), 1670–1682.
hjm E. Houseworth, M.S. Jolly and G.O. Mohler, Continuum limit of a discrete genetic problem, preprint.
kha R. Z. Khas’minskii, Diffusion processes and elliptic differential equations degenerating at the boundary of the domain, Theory of Probability and its Applications 3 (1958), 400–419.
must M.K.V. Murthy and G. Stampacchia, Boundary value problems for some degenerate-elliptic operators, Annali di Matematica, LXXX (1968), 1–122.
rod1 F. Roddier, Curvature sensing and compensation: A new concept in adaptive optics, Appl. Opt. 27 (1998), 1223–1225.
ruwo J. Rubinstein and G. Wolansky, A variational principle in optics, J. Opt. Soc. Amer. A 21 (2004), 2164–2172.
shi T. Shiga, Continuous time multi-allelic stepping stone models in population genetics, J. Math. Kyoto Univ. 22 (1982), 1–40.
tea M.R. Teague, Deterministic phase retrieval: A Green’s function solution, J. Opt. Soc. Amer. 73 (1983), 1434–1441.
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2000):
35J70
Retrieve articles in all journals
with MSC (2000):
35J70
Additional Information
Chung-min Lee
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email:
chunglee@indiana.edu
Jacob Rubinstein
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405
MR Author ID:
204416
Email:
jrubinst@indiana.edu
Received by editor(s):
March 13, 2006
Published electronically:
November 8, 2006
Article copyright:
© Copyright 2006
Brown University
The copyright for this article reverts to public domain 28 years after publication.
