Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Polynomial growth solutions of uniformly elliptic operators of non-divergence form


Authors: Peter Li and Jiaping Wang
Journal: Proc. Amer. Math. Soc. 129 (2001), 3691-3699
MSC (2000): Primary 35J15
Published electronically: May 10, 2001
MathSciNet review: 1860504
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

We give an explicit description of polynomial growth solutions to a uniformly elliptic operator of non-divergence form with periodic coefficients on the Euclidean spaces. We also show that the solutions are of one-to-one correspondence to harmonic polynomials if the coefficients of the operator are continuous.


References [Enhancements On Off] (What's this?)

  • [ALn] Marco Avellaneda and Fang-Hua Lin, Un théorème de Liouville pour des équations elliptiques à coefficients périodiques, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 5, 245–250 (French, with English summary). MR 1010728
  • [KS1] N. V. Krylov and M. V. Safonov, An estimate for the probability of a diffusion process hitting a set of positive measure, Dokl. Akad. Nauk SSSR 245 (1979), no. 1, 18–20 (Russian). MR 525227
  • [KS2] N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 161–175, 239 (Russian). MR 563790
  • [KP] P. Kuchment and Y. Pinchover, Integral representations and Liouville theorems for solutions of periodic equations, preprint.
  • [L1] Peter Li, Harmonic sections of polynomial growth, Math. Res. Lett. 4 (1997), no. 1, 35–44. MR 1432808, 10.4310/MRL.1997.v4.n1.a4
  • [L2] P. Li, Curvature and function theory on Riemannian manifolds, Survey in Differential Geometry (to appear).
  • [LW] P. Li and J. Wang, Counting dimensions of L-harmonic functions, Ann. Math. 152 (2000), 645-658.
  • [Ln] Fang Hua Lin, Asymptotically conic elliptic operators and Liouville type theorems, Geometric analysis and the calculus of variations, Int. Press, Cambridge, MA, 1996, pp. 217–238. MR 1449409
  • [MS] Jürgen Moser and Michael Struwe, On a Liouville-type theorem for linear and nonlinear elliptic differential equations on a torus, Bol. Soc. Brasil. Mat. (N.S.) 23 (1992), no. 1-2, 1–20. MR 1203171, 10.1007/BF02584809
  • [Z] Zhang Liqun, On the generic eigenvalue flow of a family of metrics and its application, Comm. Anal. Geom. 7 (1999), no. 2, 259–278. MR 1685606, 10.4310/CAG.1999.v7.n2.a2

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35J15

Retrieve articles in all journals with MSC (2000): 35J15


Additional Information

Peter Li
Affiliation: Department of Mathematics, University of California, Irvine, California 92697-3875
Email: pli@math.uci.edu

Jiaping Wang
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: jiaping@math.umn.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06167-6
Received by editor(s): May 2, 2000
Published electronically: May 10, 2001
Additional Notes: The first author’s research was partially supported by NSF grant #DMS-9971418
The second author’s research was partially supported by NSF grant #DMS-9704482
Communicated by: Bennett Chow
Article copyright: © Copyright 2001 American Mathematical Society